![Page 1: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/1.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Every love story is a GHOsT story:Goerss–Hopkins obstruction theory for
∞-categories
Aaron Mazel-Gee
University of California, Berkeley
November 21, 2013
Aaron Mazel-Gee GHOsT for∞-categories
![Page 2: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/2.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
1. Introduction
Aaron Mazel-Gee GHOsT for∞-categories
![Page 3: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/3.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
History
Goerss–Hopkins obstruction theory is a tool for obtaining E∞-ringspectra from algebraic data.
For example, the p-adic K -theory of an E∞-ring spectrumnaturally carries the structure of a θ-algebra.
Conversely, given a θ-algebra A, Goerss–Hopkins obstruction theoryallows us to search for an E∞-ring spectrum R with (K∧p )∗R ∼= A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 4: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/4.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
History
Goerss–Hopkins obstruction theory is a tool for obtaining E∞-ringspectra from algebraic data.
For example, the p-adic K -theory of an E∞-ring spectrumnaturally carries the structure of a θ-algebra.
Conversely, given a θ-algebra A, Goerss–Hopkins obstruction theoryallows us to search for an E∞-ring spectrum R with (K∧p )∗R ∼= A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 5: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/5.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
History
Goerss–Hopkins obstruction theory is a tool for obtaining E∞-ringspectra from algebraic data.
For example, the p-adic K -theory of an E∞-ring spectrumnaturally carries the structure of a θ-algebra.
Conversely, given a θ-algebra A, Goerss–Hopkins obstruction theoryallows us to search for an E∞-ring spectrum R with (K∧p )∗R ∼= A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 6: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/6.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
History
Goerss–Hopkins obstruction theory is a tool for obtaining E∞-ringspectra from algebraic data.
For example, the p-adic K -theory of an E∞-ring spectrumnaturally carries the structure of a θ-algebra.
Conversely, given a θ-algebra A, Goerss–Hopkins obstruction theoryallows us to search for an E∞-ring spectrum R with (K∧p )∗R ∼= A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 7: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/7.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstructions to realizing a θ-algebra A as an E∞-ringspectrum lie in Andre–Quillen cohomology groups:
to existence in Hn+2(A,ΩnA);
to uniqueness in Hn+1(A,ΩnA).
These are both questions about π0 of the moduli space of(K (1)-local) E∞-ring spectra R with (KU∧p )∗R ∼= A.
In fact, Goerss–Hopkins obstruction theory allows us compute allits homotopy groups!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 8: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/8.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstructions to realizing a θ-algebra A as an E∞-ringspectrum lie in Andre–Quillen cohomology groups:
to existence in Hn+2(A,ΩnA);
to uniqueness in Hn+1(A,ΩnA).
These are both questions about π0 of the moduli space of(K (1)-local) E∞-ring spectra R with (KU∧p )∗R ∼= A.
In fact, Goerss–Hopkins obstruction theory allows us compute allits homotopy groups!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 9: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/9.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstructions to realizing a θ-algebra A as an E∞-ringspectrum lie in Andre–Quillen cohomology groups:
to existence in Hn+2(A,ΩnA);
to uniqueness in Hn+1(A,ΩnA).
These are both questions about π0 of the moduli space of(K (1)-local) E∞-ring spectra R with (KU∧p )∗R ∼= A.
In fact, Goerss–Hopkins obstruction theory allows us compute allits homotopy groups!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 10: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/10.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstructions to realizing a θ-algebra A as an E∞-ringspectrum lie in Andre–Quillen cohomology groups:
to existence in Hn+2(A,ΩnA);
to uniqueness in Hn+1(A,ΩnA).
These are both questions about π0 of the moduli space of(K (1)-local) E∞-ring spectra R with (KU∧p )∗R ∼= A.
In fact, Goerss–Hopkins obstruction theory allows us compute allits homotopy groups!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 11: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/11.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstructions to realizing a θ-algebra A as an E∞-ringspectrum lie in Andre–Quillen cohomology groups:
to existence in Hn+2(A,ΩnA);
to uniqueness in Hn+1(A,ΩnA).
These are both questions about π0 of the moduli space of(K (1)-local) E∞-ring spectra R with (KU∧p )∗R ∼= A.
In fact, Goerss–Hopkins obstruction theory allows us compute allits homotopy groups!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 12: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/12.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms,
which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra. We will see that the latter is easier because theA∞-operad is Σ-free: for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces. Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 13: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/13.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms, which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra. We will see that the latter is easier because theA∞-operad is Σ-free: for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces. Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 14: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/14.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms, which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra.
We will see that the latter is easier because theA∞-operad is Σ-free: for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces. Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 15: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/15.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms, which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra. We will see that the latter is easier because theA∞-operad is Σ-free:
for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces. Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 16: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/16.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms, which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra. We will see that the latter is easier because theA∞-operad is Σ-free: for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces. Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 17: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/17.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms, which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra. We will see that the latter is easier because theA∞-operad is Σ-free: for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces. Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 18: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/18.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms, which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra. We will see that the latter is easier because theA∞-operad is Σ-free: for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces.
Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 19: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/19.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
This was originally conceived to construct the E∞-ring spectrum oftopological modular forms, which is by definition the globalsections of a sheaf of E∞-ring spectra on the moduli stack ofelliptic curves.
It builds on the Hopkins–Miller obstruction theory for A∞-ringspectra. We will see that the latter is easier because theA∞-operad is Σ-free: for all n, Σn acts freely on π0(A∞(n)).
In turn, these are both based on the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces from Π-algebras.
These all ultimately rely on the E 2-model structure ofDwyer–Kan–Stover for simplicial based spaces. Also called theresolution model structure, this is meant to give “projectiveresolutions” in a nonabelian setting.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 20: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/20.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Overview
Given a homotopy monoid object, one can define an obstructiontheory for strictifying it to an A∞-algebra by attempting to extendvarious maps over the interiors of higher and higher associahedra.
Though the obstructions magically live in the same groups, this isdifferent!
Our obstruction theory will take place in the category of simplicialalgebras.
We’ll start with nothing more than the desired algebraic image A,and we’ll find a moduli space M∞(A) of simplicial objects which isequivalent to the moduli space M (A) of realizations of A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 21: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/21.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Overview
Given a homotopy monoid object, one can define an obstructiontheory for strictifying it to an A∞-algebra by attempting to extendvarious maps over the interiors of higher and higher associahedra.
Though the obstructions magically live in the same groups, this isdifferent!
Our obstruction theory will take place in the category of simplicialalgebras.
We’ll start with nothing more than the desired algebraic image A,and we’ll find a moduli space M∞(A) of simplicial objects which isequivalent to the moduli space M (A) of realizations of A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 22: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/22.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Overview
Given a homotopy monoid object, one can define an obstructiontheory for strictifying it to an A∞-algebra by attempting to extendvarious maps over the interiors of higher and higher associahedra.
Though the obstructions magically live in the same groups, this isdifferent!
Our obstruction theory will take place in the category of simplicialalgebras.
We’ll start with nothing more than the desired algebraic image A,and we’ll find a moduli space M∞(A) of simplicial objects which isequivalent to the moduli space M (A) of realizations of A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 23: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/23.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Overview
Given a homotopy monoid object, one can define an obstructiontheory for strictifying it to an A∞-algebra by attempting to extendvarious maps over the interiors of higher and higher associahedra.
Though the obstructions magically live in the same groups, this isdifferent!
Our obstruction theory will take place in the category of simplicialalgebras.
We’ll start with nothing more than the desired algebraic image A,and we’ll find a moduli space M∞(A) of simplicial objects which isequivalent to the moduli space M (A) of realizations of A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 24: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/24.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Overview
Given a homotopy monoid object, one can define an obstructiontheory for strictifying it to an A∞-algebra by attempting to extendvarious maps over the interiors of higher and higher associahedra.
Though the obstructions magically live in the same groups, this isdifferent!
Our obstruction theory will take place in the category of simplicialalgebras.
We’ll start with nothing more than the desired algebraic image A,
and we’ll find a moduli space M∞(A) of simplicial objects which isequivalent to the moduli space M (A) of realizations of A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 25: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/25.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Overview
Given a homotopy monoid object, one can define an obstructiontheory for strictifying it to an A∞-algebra by attempting to extendvarious maps over the interiors of higher and higher associahedra.
Though the obstructions magically live in the same groups, this isdifferent!
Our obstruction theory will take place in the category of simplicialalgebras.
We’ll start with nothing more than the desired algebraic image A,and we’ll find a moduli space M∞(A) of simplicial objects
which isequivalent to the moduli space M (A) of realizations of A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 26: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/26.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Overview
Given a homotopy monoid object, one can define an obstructiontheory for strictifying it to an A∞-algebra by attempting to extendvarious maps over the interiors of higher and higher associahedra.
Though the obstructions magically live in the same groups, this isdifferent!
Our obstruction theory will take place in the category of simplicialalgebras.
We’ll start with nothing more than the desired algebraic image A,and we’ll find a moduli space M∞(A) of simplicial objects which isequivalent to the moduli space M (A) of realizations of A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 27: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/27.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups, flip it onits side, and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side, and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 28: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/28.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups, flip it onits side, and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side, and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 29: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/29.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups,
flip it onits side, and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side, and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 30: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/30.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups, flip it onits side,
and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side, and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 31: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/31.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups, flip it onits side, and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side, and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 32: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/32.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups, flip it onits side, and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side, and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 33: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/33.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups, flip it onits side, and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side,
and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 34: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/34.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
The obstruction theory relies on the theory of Postnikov towers.
However, note that Postnikov towers don’t work so well fornonconnective spectra: there’s nowhere to start.
So instead, we take their Z’s worth of homotopy groups, flip it onits side, and then resolve upwards in a new, simplicial direction.
This gives us a new theory of Postnikov towers with respect to thesimplicial direction.
In fact, we’ll have such theories of Postnikov towers on both thetopology side and the algebra side, and the obstruction theory willbe based on very a tight relationship between them.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 35: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/35.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
These theories of Postnikov towers will both be stronglyreminiscent of the classical theory of Postnikov towers, so we takea moment to review that now.
Let X be a based connected space, and write G = π1X . Then thePostnikov tower of X is a tower
Xlim−→ · · · → P2X → P1X .
If we write M = πnX , then the map PnX → Pn−1X sits in apullback square
PnX BG
Pn−1X BG(M, n + 1).
The space BG (M, n + 1) represents G -twisted cohomology forspaces over BG ' P1X .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 36: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/36.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
These theories of Postnikov towers will both be stronglyreminiscent of the classical theory of Postnikov towers, so we takea moment to review that now.
Let X be a based connected space, and write G = π1X .
Then thePostnikov tower of X is a tower
Xlim−→ · · · → P2X → P1X .
If we write M = πnX , then the map PnX → Pn−1X sits in apullback square
PnX BG
Pn−1X BG(M, n + 1).
The space BG (M, n + 1) represents G -twisted cohomology forspaces over BG ' P1X .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 37: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/37.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
These theories of Postnikov towers will both be stronglyreminiscent of the classical theory of Postnikov towers, so we takea moment to review that now.
Let X be a based connected space, and write G = π1X . Then thePostnikov tower of X is a tower
Xlim−→ · · · → P2X → P1X .
If we write M = πnX , then the map PnX → Pn−1X sits in apullback square
PnX BG
Pn−1X BG(M, n + 1).
The space BG (M, n + 1) represents G -twisted cohomology forspaces over BG ' P1X .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 38: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/38.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
These theories of Postnikov towers will both be stronglyreminiscent of the classical theory of Postnikov towers, so we takea moment to review that now.
Let X be a based connected space, and write G = π1X . Then thePostnikov tower of X is a tower
Xlim−→ · · · → P2X → P1X .
If we write M = πnX , then the map PnX → Pn−1X sits in apullback square
PnX BG
Pn−1X BG(M, n + 1).
The space BG (M, n + 1) represents G -twisted cohomology forspaces over BG ' P1X .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 39: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/39.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
These theories of Postnikov towers will both be stronglyreminiscent of the classical theory of Postnikov towers, so we takea moment to review that now.
Let X be a based connected space, and write G = π1X . Then thePostnikov tower of X is a tower
Xlim−→ · · · → P2X → P1X .
If we write M = πnX , then the map PnX → Pn−1X sits in apullback square
PnX BG
Pn−1X BG(M, n + 1).
The space BG (M, n + 1) represents G -twisted cohomology forspaces over BG ' P1X .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 40: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/40.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
These theories of Postnikov towers will both be stronglyreminiscent of the classical theory of Postnikov towers, so we takea moment to review that now.
Let X be a based connected space, and write G = π1X . Then thePostnikov tower of X is a tower
Xlim−→ · · · → P2X → P1X .
If we write M = πnX , then the map PnX → Pn−1X sits in apullback square
PnX BG
Pn−1X BG(M, n + 1).
The space BG (M, n + 1) represents G -twisted cohomology forspaces over BG ' P1X .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 41: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/41.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Ultimately, if X is an (n − 1)-type,
then the moduli spaceM (X ⊕ (M, n)) of n-types Y with Pn−1Y ' X and withπnY ∼= M has
π0M (X ⊕ (M, n)) ∼=
∐[X ,BG ]#Top∗
Hn+1G (X ;M)
/Autho(Top∗)(X )×Aut(G ,M).
Here, # denotes those maps that are isomorphisms on π1.
The group actions keep track of two different ways of gettingequivalences of induced n-types.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 42: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/42.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Ultimately, if X is an (n − 1)-type, then the moduli spaceM (X ⊕ (M, n)) of n-types Y with Pn−1Y ' X and withπnY ∼= M has
π0M (X ⊕ (M, n)) ∼=
∐[X ,BG ]#Top∗
Hn+1G (X ;M)
/Autho(Top∗)(X )×Aut(G ,M).
Here, # denotes those maps that are isomorphisms on π1.
The group actions keep track of two different ways of gettingequivalences of induced n-types.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 43: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/43.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Ultimately, if X is an (n − 1)-type, then the moduli spaceM (X ⊕ (M, n)) of n-types Y with Pn−1Y ' X and withπnY ∼= M has
π0M (X ⊕ (M, n)) ∼=
∐[X ,BG ]#Top∗
Hn+1G (X ;M)
/Autho(Top∗)(X )×Aut(G ,M).
Here, # denotes those maps that are isomorphisms on π1.
The group actions keep track of two different ways of gettingequivalences of induced n-types.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 44: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/44.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Ultimately, if X is an (n − 1)-type, then the moduli spaceM (X ⊕ (M, n)) of n-types Y with Pn−1Y ' X and withπnY ∼= M has
π0M (X ⊕ (M, n)) ∼=
∐[X ,BG ]#Top∗
Hn+1G (X ;M)
/Autho(Top∗)(X )×Aut(G ,M).
Here, # denotes those maps that are isomorphisms on π1.
The group actions keep track of two different ways of gettingequivalences of induced n-types.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 45: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/45.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Ultimately, if X is an (n − 1)-type, then the moduli spaceM (X ⊕ (M, n)) of n-types Y with Pn−1Y ' X and withπnY ∼= M has
π0M (X ⊕ (M, n)) ∼=
∐[X ,BG ]#Top∗
Hn+1G (X ;M)
/Autho(Top∗)(X )×Aut(G ,M).
Here, # denotes those maps that are isomorphisms on π1.
The group actions keep track of two different ways of gettingequivalences of induced n-types.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 46: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/46.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Ultimately, if X is an (n − 1)-type, then the moduli spaceM (X ⊕ (M, n)) of n-types Y with Pn−1Y ' X and withπnY ∼= M has
π0M (X ⊕ (M, n)) ∼=
∐[X ,BG ]#Top∗
Hn+1G (X ;M)
/Autho(Top∗)(X )×Aut(G ,M).
Here, # denotes those maps that are isomorphisms on π1.
The group actions keep track of two different ways of gettingequivalences of induced n-types.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 47: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/47.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Motivation
The GHOsT story is extremely complicated, and relies on manytechnical assumptions on the model category of spectra thatshouldn’t really be necessary.
The obstructions are entirely algebraic, and if there’s any justice inthe world, the obstruction theory itself will be model-independent:it should factor through the passage to the underlying ∞-category.
Of course, once you’re there, you may as well do it for all(sufficiently nice) ∞-categories.
(This actually began as a joint project with Markus Spitzweck toconstruct a motivic GHOsT, but it eventually became clear that weweren’t doing anything inherently motivic anyways.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 48: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/48.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Motivation
The GHOsT story is extremely complicated, and relies on manytechnical assumptions on the model category of spectra thatshouldn’t really be necessary.
The obstructions are entirely algebraic, and if there’s any justice inthe world, the obstruction theory itself will be model-independent:it should factor through the passage to the underlying ∞-category.
Of course, once you’re there, you may as well do it for all(sufficiently nice) ∞-categories.
(This actually began as a joint project with Markus Spitzweck toconstruct a motivic GHOsT, but it eventually became clear that weweren’t doing anything inherently motivic anyways.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 49: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/49.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Motivation
The GHOsT story is extremely complicated, and relies on manytechnical assumptions on the model category of spectra thatshouldn’t really be necessary.
The obstructions are entirely algebraic, and if there’s any justice inthe world, the obstruction theory itself will be model-independent:
it should factor through the passage to the underlying ∞-category.
Of course, once you’re there, you may as well do it for all(sufficiently nice) ∞-categories.
(This actually began as a joint project with Markus Spitzweck toconstruct a motivic GHOsT, but it eventually became clear that weweren’t doing anything inherently motivic anyways.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 50: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/50.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Motivation
The GHOsT story is extremely complicated, and relies on manytechnical assumptions on the model category of spectra thatshouldn’t really be necessary.
The obstructions are entirely algebraic, and if there’s any justice inthe world, the obstruction theory itself will be model-independent:it should factor through the passage to the underlying ∞-category.
Of course, once you’re there, you may as well do it for all(sufficiently nice) ∞-categories.
(This actually began as a joint project with Markus Spitzweck toconstruct a motivic GHOsT, but it eventually became clear that weweren’t doing anything inherently motivic anyways.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 51: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/51.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Motivation
The GHOsT story is extremely complicated, and relies on manytechnical assumptions on the model category of spectra thatshouldn’t really be necessary.
The obstructions are entirely algebraic, and if there’s any justice inthe world, the obstruction theory itself will be model-independent:it should factor through the passage to the underlying ∞-category.
Of course, once you’re there, you may as well do it for all(sufficiently nice) ∞-categories.
(This actually began as a joint project with Markus Spitzweck toconstruct a motivic GHOsT, but it eventually became clear that weweren’t doing anything inherently motivic anyways.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 52: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/52.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Motivation
The GHOsT story is extremely complicated, and relies on manytechnical assumptions on the model category of spectra thatshouldn’t really be necessary.
The obstructions are entirely algebraic, and if there’s any justice inthe world, the obstruction theory itself will be model-independent:it should factor through the passage to the underlying ∞-category.
Of course, once you’re there, you may as well do it for all(sufficiently nice) ∞-categories.
(This actually began as a joint project with Markus Spitzweck toconstruct a motivic GHOsT,
but it eventually became clear that weweren’t doing anything inherently motivic anyways.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 53: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/53.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
Motivation
The GHOsT story is extremely complicated, and relies on manytechnical assumptions on the model category of spectra thatshouldn’t really be necessary.
The obstructions are entirely algebraic, and if there’s any justice inthe world, the obstruction theory itself will be model-independent:it should factor through the passage to the underlying ∞-category.
Of course, once you’re there, you may as well do it for all(sufficiently nice) ∞-categories.
(This actually began as a joint project with Markus Spitzweck toconstruct a motivic GHOsT, but it eventually became clear that weweren’t doing anything inherently motivic anyways.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 54: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/54.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 55: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/55.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”:
a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 56: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/56.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 57: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/57.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites.
(This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 58: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/58.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable:
Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 59: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/59.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth,
but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 60: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/60.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth.
Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 61: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/61.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 62: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/62.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
In fact, GHOsT undergirds the derived algebro-geometricconstruction of tmf , too.
Moreover, GHOsT gives a basis for doing derived algebraicgeometry “naively”: a derived scheme is just an ordinary scheme(X ,OX ) equipped with a sheaf Otop
X of E∞-rings such thatπ0O
topX∼= OX .
This is a reasonable definition because for R a connective E∞-ring,SpecR and Specπ0R have equivalent Zariski sites. (This shouldbe believable: Andre–Quillen cohomology groups measure thefailure of a map to be smooth, but Zariski opens are smooth. Infact, this same statement holds for their etale sites, too.)
So, this could give a basis for doing “naive” derived algebraicgeometry in other contexts, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 63: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/63.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
But more concretely, lots of people care about other ∞-categoriesthat aren’t the ∞-category of spectra,
and these people deserve anobstruction theory too!
Later in this talk, we’ll also briefly mention work in progresstowards a generalization that applies to stronger notions ofcommutativity that (ought to) exist e.g. in equivariant and motivichomotopy theory.
And if nothing else: a guy’s gotta write a thesis, right??!?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 64: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/64.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
But more concretely, lots of people care about other ∞-categoriesthat aren’t the ∞-category of spectra, and these people deserve anobstruction theory too!
Later in this talk, we’ll also briefly mention work in progresstowards a generalization that applies to stronger notions ofcommutativity that (ought to) exist e.g. in equivariant and motivichomotopy theory.
And if nothing else: a guy’s gotta write a thesis, right??!?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 65: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/65.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
But more concretely, lots of people care about other ∞-categoriesthat aren’t the ∞-category of spectra, and these people deserve anobstruction theory too!
Later in this talk, we’ll also briefly mention work in progresstowards a generalization that applies to stronger notions ofcommutativity
that (ought to) exist e.g. in equivariant and motivichomotopy theory.
And if nothing else: a guy’s gotta write a thesis, right??!?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 66: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/66.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
But more concretely, lots of people care about other ∞-categoriesthat aren’t the ∞-category of spectra, and these people deserve anobstruction theory too!
Later in this talk, we’ll also briefly mention work in progresstowards a generalization that applies to stronger notions ofcommutativity that (ought to) exist e.g. in equivariant and motivichomotopy theory.
And if nothing else: a guy’s gotta write a thesis, right??!?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 67: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/67.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
But more concretely, lots of people care about other ∞-categoriesthat aren’t the ∞-category of spectra, and these people deserve anobstruction theory too!
Later in this talk, we’ll also briefly mention work in progresstowards a generalization that applies to stronger notions ofcommutativity that (ought to) exist e.g. in equivariant and motivichomotopy theory.
And if nothing else:
a guy’s gotta write a thesis, right??!?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 68: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/68.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
HistoryOverviewMotivation
But more concretely, lots of people care about other ∞-categoriesthat aren’t the ∞-category of spectra, and these people deserve anobstruction theory too!
Later in this talk, we’ll also briefly mention work in progresstowards a generalization that applies to stronger notions ofcommutativity that (ought to) exist e.g. in equivariant and motivichomotopy theory.
And if nothing else: a guy’s gotta write a thesis, right??!?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 69: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/69.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
2. Blanc–Dwyer–Goerss obstructiontheory
Aaron Mazel-Gee GHOsT for∞-categories
![Page 70: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/70.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
As a warm-up, let’s start with the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces for Π-algebras.
This is easier for two reasons:
The functor π∗ already determines equivalences, so ourdesired moduli space is simply a subgroupoid of our category(instead of a localization thereof).
There’s no operad in the game.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 71: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/71.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
As a warm-up, let’s start with the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces for Π-algebras.
This is easier for two reasons:
The functor π∗ already determines equivalences, so ourdesired moduli space is simply a subgroupoid of our category(instead of a localization thereof).
There’s no operad in the game.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 72: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/72.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
As a warm-up, let’s start with the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces for Π-algebras.
This is easier for two reasons:
The functor π∗ already determines equivalences, so ourdesired moduli space is simply a subgroupoid of our category(instead of a localization thereof).
There’s no operad in the game.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 73: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/73.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
As a warm-up, let’s start with the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces for Π-algebras.
This is easier for two reasons:
The functor π∗ already determines equivalences, so ourdesired moduli space is simply a subgroupoid of our category(instead of a localization thereof).
There’s no operad in the game.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 74: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/74.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
As a warm-up, let’s start with the Blanc–Dwyer–Goerssobstruction theory for obtaining based spaces for Π-algebras.
This is easier for two reasons:
The functor π∗ already determines equivalences, so ourdesired moduli space is simply a subgroupoid of our category(instead of a localization thereof).
There’s no operad in the game.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 75: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/75.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalities
So, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 76: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/76.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 77: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/77.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.
(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 78: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/78.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 79: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/79.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts
and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 80: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/80.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 81: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/81.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G
(or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 82: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/82.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 83: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/83.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Categorical generalitiesSo, let C be a model category, and let G ⊂ C be a set of generators.
We assume our generators are cobased, though of courseeverything gets easier if they are even h-cogroup objects.(Throughout, feel free to assume that C is pointed.)
We also assume (without loss of generality) that G is closed underfinite coproducts and that for any Sβ ∈ G, also Sn+β = ΣnSβ ∈ G.
We write A = PδΣ(G) for the category of product-preservingpresheaves of sets on G (or really on ho(G)).
We then have the functor π∗ : C→ A given by
(π∗X )(Sβ) = [Sβ ,X ]C.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 84: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/84.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, the canonical example of an object of A is given by π∗Xfor X ∈ C.
But more generally, if we have a functor ϕ : G→M to a modelcategory M which preserves coproducts up to weak equivalence,then for any X ∈M we can also obtain πϕ∗X ∈ A by setting
(πϕ∗ X )(Sβ) = [ϕ(Sβ),X ]M.
This induces a functor πϕ∗ : ho(M)→ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 85: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/85.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, the canonical example of an object of A is given by π∗Xfor X ∈ C.
But more generally, if we have a functor ϕ : G→M to a modelcategory M which preserves coproducts up to weak equivalence,
then for any X ∈M we can also obtain πϕ∗X ∈ A by setting
(πϕ∗ X )(Sβ) = [ϕ(Sβ),X ]M.
This induces a functor πϕ∗ : ho(M)→ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 86: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/86.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, the canonical example of an object of A is given by π∗Xfor X ∈ C.
But more generally, if we have a functor ϕ : G→M to a modelcategory M which preserves coproducts up to weak equivalence,then for any X ∈M we can also obtain πϕ∗X ∈ A by setting
(πϕ∗ X )(Sβ) = [ϕ(Sβ),X ]M.
This induces a functor πϕ∗ : ho(M)→ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 87: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/87.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, the canonical example of an object of A is given by π∗Xfor X ∈ C.
But more generally, if we have a functor ϕ : G→M to a modelcategory M which preserves coproducts up to weak equivalence,then for any X ∈M we can also obtain πϕ∗X ∈ A by setting
(πϕ∗ X )(Sβ) = [ϕ(Sβ),X ]M.
This induces a functor πϕ∗ : ho(M)→ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 88: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/88.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg.
But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 89: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/89.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg. But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 90: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/90.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg. But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 91: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/91.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg. But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 92: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/92.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg. But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 93: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/93.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg. But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 94: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/94.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg. But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 95: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/95.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, historically, the original example was C = Top≥1∗ , G = Π,
and A = Π-alg. But there’s no reason to restrict to this case.
Note that one can really just keep track of the values on a subsetof G which generates it under coproducts, along with some extraalgebraic structure.
For instance, a Π-algebra is a product-preserving functorΠop → Set, but it’s determined by its values on just the spheres,along with:
their (abelian) group structures;
their compositions induced by elements of πiSj ;
their Whitehead brackets.
(This reidentification is due to the Hilton–Milnor theorem forcomputing the homotopy type of Ω(ΣX1 ∨ ΣX2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 96: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/96.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is a model structure on sA where Y → Z is an equivalenceif for all Sβ ∈ G, Y (Sβ)→ Z (Sβ) is an equivalence in sSet (withthe standard Quillen model structure).
This is compatible with the external simplicial structure on sA: forK ∈ sSet and Y ∈ sA we define K Y ∈ sA by
(K Y )n =∐Kn
A
Yn.
Explicitly, this means that it is compatible with the Quillen modelstructure on sSet.
Even more explicitly, this means that the tensoringsSet× sA→ sA descends to a tensoring H × ho(sA)→ ho(sA),where H ' sSet[W−1
Quillen] denotes the homotopy category oftopological spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 97: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/97.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is a model structure on sA where Y → Z is an equivalenceif for all Sβ ∈ G, Y (Sβ)→ Z (Sβ) is an equivalence in sSet (withthe standard Quillen model structure).
This is compatible with the external simplicial structure on sA:
forK ∈ sSet and Y ∈ sA we define K Y ∈ sA by
(K Y )n =∐Kn
A
Yn.
Explicitly, this means that it is compatible with the Quillen modelstructure on sSet.
Even more explicitly, this means that the tensoringsSet× sA→ sA descends to a tensoring H × ho(sA)→ ho(sA),where H ' sSet[W−1
Quillen] denotes the homotopy category oftopological spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 98: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/98.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is a model structure on sA where Y → Z is an equivalenceif for all Sβ ∈ G, Y (Sβ)→ Z (Sβ) is an equivalence in sSet (withthe standard Quillen model structure).
This is compatible with the external simplicial structure on sA: forK ∈ sSet and Y ∈ sA we define K Y ∈ sA by
(K Y )n =∐Kn
A
Yn.
Explicitly, this means that it is compatible with the Quillen modelstructure on sSet.
Even more explicitly, this means that the tensoringsSet× sA→ sA descends to a tensoring H × ho(sA)→ ho(sA),where H ' sSet[W−1
Quillen] denotes the homotopy category oftopological spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 99: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/99.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is a model structure on sA where Y → Z is an equivalenceif for all Sβ ∈ G, Y (Sβ)→ Z (Sβ) is an equivalence in sSet (withthe standard Quillen model structure).
This is compatible with the external simplicial structure on sA: forK ∈ sSet and Y ∈ sA we define K Y ∈ sA by
(K Y )n =∐Kn
A
Yn.
Explicitly, this means that it is compatible with the Quillen modelstructure on sSet.
Even more explicitly, this means that the tensoringsSet× sA→ sA descends to a tensoring H × ho(sA)→ ho(sA),where H ' sSet[W−1
Quillen] denotes the homotopy category oftopological spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 100: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/100.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is a model structure on sA where Y → Z is an equivalenceif for all Sβ ∈ G, Y (Sβ)→ Z (Sβ) is an equivalence in sSet (withthe standard Quillen model structure).
This is compatible with the external simplicial structure on sA: forK ∈ sSet and Y ∈ sA we define K Y ∈ sA by
(K Y )n =∐Kn
A
Yn.
Explicitly, this means that it is compatible with the Quillen modelstructure on sSet.
Even more explicitly, this means that the tensoringsSet× sA→ sA
descends to a tensoring H × ho(sA)→ ho(sA),where H ' sSet[W−1
Quillen] denotes the homotopy category oftopological spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 101: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/101.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is a model structure on sA where Y → Z is an equivalenceif for all Sβ ∈ G, Y (Sβ)→ Z (Sβ) is an equivalence in sSet (withthe standard Quillen model structure).
This is compatible with the external simplicial structure on sA: forK ∈ sSet and Y ∈ sA we define K Y ∈ sA by
(K Y )n =∐Kn
A
Yn.
Explicitly, this means that it is compatible with the Quillen modelstructure on sSet.
Even more explicitly, this means that the tensoringsSet× sA→ sA descends to a tensoring H × ho(sA)→ ho(sA),
where H ' sSet[W−1Quillen] denotes the homotopy category of
topological spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 102: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/102.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is a model structure on sA where Y → Z is an equivalenceif for all Sβ ∈ G, Y (Sβ)→ Z (Sβ) is an equivalence in sSet (withthe standard Quillen model structure).
This is compatible with the external simplicial structure on sA: forK ∈ sSet and Y ∈ sA we define K Y ∈ sA by
(K Y )n =∐Kn
A
Yn.
Explicitly, this means that it is compatible with the Quillen modelstructure on sSet.
Even more explicitly, this means that the tensoringsSet× sA→ sA descends to a tensoring H × ho(sA)→ ho(sA),where H ' sSet[W−1
Quillen] denotes the homotopy category oftopological spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 103: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/103.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
For Y ∈ A, we simply write K Y for K const(Y ).
For K ∈ sSet∗, we also define
K ∧ Y = colim (pt 0← pt Y → K Y ) ,
as one would expect.
Now, let’s define ϕn : G→ sA by
ϕn(Sβ) = Sn∆ ∧ π∗Sβ ,
where Sn∆ = ∆n/∂∆n ∈ sSet∗.
Then, for any Y ∈ sA, we can define πnY = πϕn∗ Y ∈ A:
(πnY )(Sβ) = [ϕn(Sβ),Y ]sA ∼= πn(Y (Sβ)).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 104: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/104.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
For Y ∈ A, we simply write K Y for K const(Y ).
For K ∈ sSet∗, we also define
K ∧ Y = colim (pt 0← pt Y → K Y ) ,
as one would expect.
Now, let’s define ϕn : G→ sA by
ϕn(Sβ) = Sn∆ ∧ π∗Sβ ,
where Sn∆ = ∆n/∂∆n ∈ sSet∗.
Then, for any Y ∈ sA, we can define πnY = πϕn∗ Y ∈ A:
(πnY )(Sβ) = [ϕn(Sβ),Y ]sA ∼= πn(Y (Sβ)).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 105: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/105.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
For Y ∈ A, we simply write K Y for K const(Y ).
For K ∈ sSet∗, we also define
K ∧ Y = colim (pt 0← pt Y → K Y ) ,
as one would expect.
Now, let’s define ϕn : G→ sA by
ϕn(Sβ) = Sn∆ ∧ π∗Sβ ,
where Sn∆ = ∆n/∂∆n ∈ sSet∗.
Then, for any Y ∈ sA, we can define πnY = πϕn∗ Y ∈ A:
(πnY )(Sβ) = [ϕn(Sβ),Y ]sA ∼= πn(Y (Sβ)).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 106: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/106.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
For Y ∈ A, we simply write K Y for K const(Y ).
For K ∈ sSet∗, we also define
K ∧ Y = colim (pt 0← pt Y → K Y ) ,
as one would expect.
Now, let’s define ϕn : G→ sA by
ϕn(Sβ) = Sn∆ ∧ π∗Sβ ,
where Sn∆ = ∆n/∂∆n ∈ sSet∗.
Then, for any Y ∈ sA, we can define πnY = πϕn∗ Y ∈ A:
(πnY )(Sβ) = [ϕn(Sβ),Y ]sA ∼= πn(Y (Sβ)).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 107: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/107.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
For Y ∈ A, we simply write K Y for K const(Y ).
For K ∈ sSet∗, we also define
K ∧ Y = colim (pt 0← pt Y → K Y ) ,
as one would expect.
Now, let’s define ϕn : G→ sA by
ϕn(Sβ) = Sn∆ ∧ π∗Sβ ,
where Sn∆ = ∆n/∂∆n ∈ sSet∗.
Then, for any Y ∈ sA, we can define πnY = πϕn∗ Y ∈ A:
(πnY )(Sβ) = [ϕn(Sβ),Y ]sA ∼= πn(Y (Sβ)).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 108: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/108.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
For Y ∈ A, we simply write K Y for K const(Y ).
For K ∈ sSet∗, we also define
K ∧ Y = colim (pt 0← pt Y → K Y ) ,
as one would expect.
Now, let’s define ϕn : G→ sA by
ϕn(Sβ) = Sn∆ ∧ π∗Sβ ,
where Sn∆ = ∆n/∂∆n ∈ sSet∗.
Then, for any Y ∈ sA, we can define πnY = πϕn∗ Y ∈ A:
(πnY )(Sβ) = [ϕn(Sβ),Y ]sA ∼= πn(Y (Sβ)).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 109: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/109.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
For Y ∈ A, we simply write K Y for K const(Y ).
For K ∈ sSet∗, we also define
K ∧ Y = colim (pt 0← pt Y → K Y ) ,
as one would expect.
Now, let’s define ϕn : G→ sA by
ϕn(Sβ) = Sn∆ ∧ π∗Sβ ,
where Sn∆ = ∆n/∂∆n ∈ sSet∗.
Then, for any Y ∈ sA, we can define πnY = πϕn∗ Y ∈ A:
(πnY )(Sβ) = [ϕn(Sβ),Y ]sA ∼= πn(Y (Sβ)).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 110: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/110.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In fact, for any n ≥ 1 we have a split cofiber sequence
Sn∆ (Sn
∆)+ S0∆.
For any Sβ ∈ G, this induces a natural split short exact sequence
0[Sn
∆ ∧ π∗Sβ ,Y]sA
[(Sn
∆)+ ∧ π∗Sβ ,Y]sA
[S0
∆ ∧ π∗Sβ ,Y]sA
0.
(πnY )(Sβ) (π0Y )(Sβ)
Note that (Sn∆)+ ∈ (sSet∗)/S0
∆is an h-cogroup object in the
Quillen model structure.
Thus, collecting over all Sβ, this displays πnY as the kernel of thestructure map of an abelian group object in A/π0Y .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 111: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/111.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In fact, for any n ≥ 1 we have a split cofiber sequence
Sn∆ (Sn
∆)+ S0∆.
For any Sβ ∈ G, this induces a natural split short exact sequence
0[Sn
∆ ∧ π∗Sβ ,Y]sA
[(Sn
∆)+ ∧ π∗Sβ ,Y]sA
[S0
∆ ∧ π∗Sβ ,Y]sA
0.
(πnY )(Sβ) (π0Y )(Sβ)
Note that (Sn∆)+ ∈ (sSet∗)/S0
∆is an h-cogroup object in the
Quillen model structure.
Thus, collecting over all Sβ, this displays πnY as the kernel of thestructure map of an abelian group object in A/π0Y .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 112: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/112.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In fact, for any n ≥ 1 we have a split cofiber sequence
Sn∆ (Sn
∆)+ S0∆.
For any Sβ ∈ G, this induces a natural split short exact sequence
0[Sn
∆ ∧ π∗Sβ ,Y]sA
[(Sn
∆)+ ∧ π∗Sβ ,Y]sA
[S0
∆ ∧ π∗Sβ ,Y]sA
0.
(πnY )(Sβ) (π0Y )(Sβ)
Note that (Sn∆)+ ∈ (sSet∗)/S0
∆is an h-cogroup object in the
Quillen model structure.
Thus, collecting over all Sβ, this displays πnY as the kernel of thestructure map of an abelian group object in A/π0Y .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 113: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/113.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In fact, for any n ≥ 1 we have a split cofiber sequence
Sn∆ (Sn
∆)+ S0∆.
For any Sβ ∈ G, this induces a natural split short exact sequence
0[Sn
∆ ∧ π∗Sβ ,Y]sA
[(Sn
∆)+ ∧ π∗Sβ ,Y]sA
[S0
∆ ∧ π∗Sβ ,Y]sA
0.
(πnY )(Sβ) (π0Y )(Sβ)
Note that (Sn∆)+ ∈ (sSet∗)/S0
∆is an h-cogroup object in the
Quillen model structure.
Thus, collecting over all Sβ, this displays πnY as the kernel of thestructure map of an abelian group object in A/π0Y .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 114: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/114.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In fact, for any n ≥ 1 we have a split cofiber sequence
Sn∆ (Sn
∆)+ S0∆.
For any Sβ ∈ G, this induces a natural split short exact sequence
0[Sn
∆ ∧ π∗Sβ ,Y]sA
[(Sn
∆)+ ∧ π∗Sβ ,Y]sA
[S0
∆ ∧ π∗Sβ ,Y]sA
0.
(πnY )(Sβ) (π0Y )(Sβ)
Note that (Sn∆)+ ∈ (sSet∗)/S0
∆is an h-cogroup object in the
Quillen model structure.
Thus, collecting over all Sβ, this displays πnY as the kernel of thestructure map of an abelian group object in A/π0Y .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 115: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/115.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Another way of saying all this is that we’ve endowed πnY with thestructure of a module for π0Y .
We denote the category of π0Y -modules by Modπ0Y (A); the A
emphasizes the underlying category in which the modules live.
Modules appear in our story since the obstruction groups areAndre–Quillen cohomology groups, and these are defined withrespect to a module of coefficients.
Now that we know what modules are, we’re ready to talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 116: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/116.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Another way of saying all this is that we’ve endowed πnY with thestructure of a module for π0Y .
We denote the category of π0Y -modules by Modπ0Y (A); the A
emphasizes the underlying category in which the modules live.
Modules appear in our story since the obstruction groups areAndre–Quillen cohomology groups, and these are defined withrespect to a module of coefficients.
Now that we know what modules are, we’re ready to talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 117: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/117.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Another way of saying all this is that we’ve endowed πnY with thestructure of a module for π0Y .
We denote the category of π0Y -modules by Modπ0Y (A); the A
emphasizes the underlying category in which the modules live.
Modules appear in our story since the obstruction groups areAndre–Quillen cohomology groups, and these are defined withrespect to a module of coefficients.
Now that we know what modules are, we’re ready to talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 118: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/118.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Another way of saying all this is that we’ve endowed πnY with thestructure of a module for π0Y .
We denote the category of π0Y -modules by Modπ0Y (A); the A
emphasizes the underlying category in which the modules live.
Modules appear in our story since the obstruction groups areAndre–Quillen cohomology groups, and these are defined withrespect to a module of coefficients.
Now that we know what modules are, we’re ready to talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 119: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/119.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big picture
Suppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment), we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A). In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 120: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/120.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big pictureSuppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment), we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A). In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 121: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/121.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big pictureSuppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment), we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A). In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 122: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/122.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big pictureSuppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment),
we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A). In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 123: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/123.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big pictureSuppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment), we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A). In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 124: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/124.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big pictureSuppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment), we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A). In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 125: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/125.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big pictureSuppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment), we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A).
In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 126: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/126.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The big pictureSuppose we have some A ∈ A, and we’re interested in the modulispace M (A) ⊂ C of objects X ∈ C with π∗X ∼= A.
For any Y ∈ sC, we have π∗Y ∈ sA, and there is a spectralsequence running E 2
i ,β = πiπβY ⇒ πi+β|Y |.
Using the E 2-model structure on sC (which we’ll define in amoment), we define M∞(A) ⊂ sCE2 to be the moduli space ofthose Y ∈ sC with π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Such Y ∈ sC are called ∞-stages for A.
Note that for Y ∈M∞(A), the above spectral sequence collapses,so |Y | ∈M (A). In fact, geometric realization defines anequivalence
M∞(A)∼−→M (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 127: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/127.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We will define the moduli spaces Mn(A) ⊂ sCE2 of n-stages suchthat
M∞(A)lim−→ · · · →M1(A)→M0(A).
Moreover, we will have that M0(A) ' BAut(A), and for all n ≥ 1we will have a pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 128: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/128.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We will define the moduli spaces Mn(A) ⊂ sCE2 of n-stages suchthat
M∞(A)lim−→ · · · →M1(A)→M0(A).
Moreover, we will have that M0(A) ' BAut(A),
and for all n ≥ 1we will have a pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 129: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/129.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We will define the moduli spaces Mn(A) ⊂ sCE2 of n-stages suchthat
M∞(A)lim−→ · · · →M1(A)→M0(A).
Moreover, we will have that M0(A) ' BAut(A), and for all n ≥ 1we will have a pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 130: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/130.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M) picks out the
component 0 ∈ H i = π0H i and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 131: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/131.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M) picks out the
component 0 ∈ H i = π0H i and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 132: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/132.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M) picks out the
component 0 ∈ H i = π0H i and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 133: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/133.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M) picks out the
component 0 ∈ H i = π0H i and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 134: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/134.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M)
picks out thecomponent 0 ∈ H i = π0H i and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 135: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/135.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M) picks out the
component 0 ∈ H i = π0H i
and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 136: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/136.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M) picks out the
component 0 ∈ H i = π0H i and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 137: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/137.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Let us explain the map BAut(A,ΩnA)→ H n+2(A,ΩnA).
For any i ≥ 1, H iA(X ,M) denotes the Andre–Quillen cohomology
of X ∈ A/A with coefficients in M ∈ ModA(A).
This is given by π0 of a certain mapping space, which we denoteby H i
A(X ,M).
These are based spaces, and in fact ΩH i 'H i−1.
Moreover, the basepoint map pt→H iA(X ,M) picks out the
component 0 ∈ H i = π0H i and is Aut(A,M)-equivariant.
Taking the homotopy orbits of this basepoint map yields the mapin question (for X = A, M = ΩnA, and i = n + 2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 138: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/138.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Thus, this map BAut(A,M)→ H n+2(A,ΩnA) has fiber
ΩH n+2(A,ΩnA) 'H n+1(A,ΩnA)
over the path component of 0 ∈ Hn+2 and has empty fiberotherwise.
Of course, this tells us about the fibers of Mn(A)→Mn−1(A):
If Hn+2(A,ΩnA) = 0 then this map is surjective on π0, so wecan always lift an (n − 1)-stage to an n-stage.
If Hn+1(A,ΩnA) = 0 then the fibers of this map areconnected, so that these lifts are unique up to equivalence(although the equivalence need not be essentially unique).
As we make our way up the tower for M∞(A) 'M (A), thisrecovers the obstructions to existence and uniqueness assertedearlier!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 139: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/139.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Thus, this map BAut(A,M)→ H n+2(A,ΩnA) has fiber
ΩH n+2(A,ΩnA) 'H n+1(A,ΩnA)
over the path component of 0 ∈ Hn+2 and has empty fiberotherwise.
Of course, this tells us about the fibers of Mn(A)→Mn−1(A):
If Hn+2(A,ΩnA) = 0 then this map is surjective on π0, so wecan always lift an (n − 1)-stage to an n-stage.
If Hn+1(A,ΩnA) = 0 then the fibers of this map areconnected, so that these lifts are unique up to equivalence(although the equivalence need not be essentially unique).
As we make our way up the tower for M∞(A) 'M (A), thisrecovers the obstructions to existence and uniqueness assertedearlier!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 140: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/140.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Thus, this map BAut(A,M)→ H n+2(A,ΩnA) has fiber
ΩH n+2(A,ΩnA) 'H n+1(A,ΩnA)
over the path component of 0 ∈ Hn+2 and has empty fiberotherwise.
Of course, this tells us about the fibers of Mn(A)→Mn−1(A):
If Hn+2(A,ΩnA) = 0 then this map is surjective on π0, so wecan always lift an (n − 1)-stage to an n-stage.
If Hn+1(A,ΩnA) = 0 then the fibers of this map areconnected, so that these lifts are unique up to equivalence(although the equivalence need not be essentially unique).
As we make our way up the tower for M∞(A) 'M (A), thisrecovers the obstructions to existence and uniqueness assertedearlier!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 141: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/141.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Thus, this map BAut(A,M)→ H n+2(A,ΩnA) has fiber
ΩH n+2(A,ΩnA) 'H n+1(A,ΩnA)
over the path component of 0 ∈ Hn+2 and has empty fiberotherwise.
Of course, this tells us about the fibers of Mn(A)→Mn−1(A):
If Hn+2(A,ΩnA) = 0 then this map is surjective on π0, so wecan always lift an (n − 1)-stage to an n-stage.
If Hn+1(A,ΩnA) = 0 then the fibers of this map areconnected, so that these lifts are unique up to equivalence
(although the equivalence need not be essentially unique).
As we make our way up the tower for M∞(A) 'M (A), thisrecovers the obstructions to existence and uniqueness assertedearlier!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 142: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/142.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Thus, this map BAut(A,M)→ H n+2(A,ΩnA) has fiber
ΩH n+2(A,ΩnA) 'H n+1(A,ΩnA)
over the path component of 0 ∈ Hn+2 and has empty fiberotherwise.
Of course, this tells us about the fibers of Mn(A)→Mn−1(A):
If Hn+2(A,ΩnA) = 0 then this map is surjective on π0, so wecan always lift an (n − 1)-stage to an n-stage.
If Hn+1(A,ΩnA) = 0 then the fibers of this map areconnected, so that these lifts are unique up to equivalence(although the equivalence need not be essentially unique).
As we make our way up the tower for M∞(A) 'M (A), thisrecovers the obstructions to existence and uniqueness assertedearlier!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 143: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/143.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Thus, this map BAut(A,M)→ H n+2(A,ΩnA) has fiber
ΩH n+2(A,ΩnA) 'H n+1(A,ΩnA)
over the path component of 0 ∈ Hn+2 and has empty fiberotherwise.
Of course, this tells us about the fibers of Mn(A)→Mn−1(A):
If Hn+2(A,ΩnA) = 0 then this map is surjective on π0, so wecan always lift an (n − 1)-stage to an n-stage.
If Hn+1(A,ΩnA) = 0 then the fibers of this map areconnected, so that these lifts are unique up to equivalence(although the equivalence need not be essentially unique).
As we make our way up the tower for M∞(A) 'M (A),
thisrecovers the obstructions to existence and uniqueness assertedearlier!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 144: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/144.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Thus, this map BAut(A,M)→ H n+2(A,ΩnA) has fiber
ΩH n+2(A,ΩnA) 'H n+1(A,ΩnA)
over the path component of 0 ∈ Hn+2 and has empty fiberotherwise.
Of course, this tells us about the fibers of Mn(A)→Mn−1(A):
If Hn+2(A,ΩnA) = 0 then this map is surjective on π0, so wecan always lift an (n − 1)-stage to an n-stage.
If Hn+1(A,ΩnA) = 0 then the fibers of this map areconnected, so that these lifts are unique up to equivalence(although the equivalence need not be essentially unique).
As we make our way up the tower for M∞(A) 'M (A), thisrecovers the obstructions to existence and uniqueness assertedearlier!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 145: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/145.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, what are these “n-stages”, you ask?
An n-stage Y will have π∗π∗Y (the E 2-page of its spectralsequence) concentrated at π0π∗Y ∼= A and πn+2π∗Y ∼= Ωn+1A(which we haven’t defined yet).
Of course, a spectral sequence always has an exact couple lurkingin the background, and we want to have the correct E 2-exactcouple, not just the correct E 2-page.
But even just to describe the exact couple, we’ll need to talkabout...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 146: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/146.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, what are these “n-stages”, you ask?
An n-stage Y will have π∗π∗Y (the E 2-page of its spectralsequence) concentrated at π0π∗Y ∼= A and πn+2π∗Y ∼= Ωn+1A(which we haven’t defined yet).
Of course, a spectral sequence always has an exact couple lurkingin the background, and we want to have the correct E 2-exactcouple, not just the correct E 2-page.
But even just to describe the exact couple, we’ll need to talkabout...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 147: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/147.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, what are these “n-stages”, you ask?
An n-stage Y will have π∗π∗Y (the E 2-page of its spectralsequence) concentrated at π0π∗Y ∼= A and πn+2π∗Y ∼= Ωn+1A(which we haven’t defined yet).
Of course, a spectral sequence always has an exact couple lurkingin the background, and we want to have the correct E 2-exactcouple, not just the correct E 2-page.
But even just to describe the exact couple, we’ll need to talkabout...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 148: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/148.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, what are these “n-stages”, you ask?
An n-stage Y will have π∗π∗Y (the E 2-page of its spectralsequence) concentrated at π0π∗Y ∼= A and πn+2π∗Y ∼= Ωn+1A(which we haven’t defined yet).
Of course, a spectral sequence always has an exact couple lurkingin the background, and we want to have the correct E 2-exactcouple, not just the correct E 2-page.
But even just to describe the exact couple, we’ll need to talkabout...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 149: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/149.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The E 2-model structure
Recall that the Reedy model structure on sC has the levelwiseequivalences.
The E 2-model structure is weaker: a map Y → Z in sC is anE 2-equivalence if the induced map π∗Y → π∗Z is a weakequivalence in sA.
(In this language, a Reedy equivalence is an E 1-equivalence: itinduces an isomorphism in sA.)
The E 2-equivalences are exactly those maps that induce anisomorphism of E 2-pages of the spectral sequence; in fact, theyautomatically induce isomorphisms of E 2-exact couples.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 150: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/150.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The E 2-model structure
Recall that the Reedy model structure on sC has the levelwiseequivalences.
The E 2-model structure is weaker: a map Y → Z in sC is anE 2-equivalence if the induced map π∗Y → π∗Z is a weakequivalence in sA.
(In this language, a Reedy equivalence is an E 1-equivalence: itinduces an isomorphism in sA.)
The E 2-equivalences are exactly those maps that induce anisomorphism of E 2-pages of the spectral sequence; in fact, theyautomatically induce isomorphisms of E 2-exact couples.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 151: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/151.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The E 2-model structure
Recall that the Reedy model structure on sC has the levelwiseequivalences.
The E 2-model structure is weaker: a map Y → Z in sC is anE 2-equivalence if the induced map π∗Y → π∗Z is a weakequivalence in sA.
(In this language, a Reedy equivalence is an E 1-equivalence: itinduces an isomorphism in sA.)
The E 2-equivalences are exactly those maps that induce anisomorphism of E 2-pages of the spectral sequence; in fact, theyautomatically induce isomorphisms of E 2-exact couples.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 152: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/152.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The E 2-model structure
Recall that the Reedy model structure on sC has the levelwiseequivalences.
The E 2-model structure is weaker: a map Y → Z in sC is anE 2-equivalence if the induced map π∗Y → π∗Z is a weakequivalence in sA.
(In this language, a Reedy equivalence is an E 1-equivalence: itinduces an isomorphism in sA.)
The E 2-equivalences are exactly those maps that induce anisomorphism of E 2-pages of the spectral sequence; in fact, theyautomatically induce isomorphisms of E 2-exact couples.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 153: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/153.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The E 2-model structure
Recall that the Reedy model structure on sC has the levelwiseequivalences.
The E 2-model structure is weaker: a map Y → Z in sC is anE 2-equivalence if the induced map π∗Y → π∗Z is a weakequivalence in sA.
(In this language, a Reedy equivalence is an E 1-equivalence: itinduces an isomorphism in sA.)
The E 2-equivalences are exactly those maps that induce anisomorphism of E 2-pages of the spectral sequence;
in fact, theyautomatically induce isomorphisms of E 2-exact couples.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 154: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/154.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The E 2-model structure
Recall that the Reedy model structure on sC has the levelwiseequivalences.
The E 2-model structure is weaker: a map Y → Z in sC is anE 2-equivalence if the induced map π∗Y → π∗Z is a weakequivalence in sA.
(In this language, a Reedy equivalence is an E 1-equivalence: itinduces an isomorphism in sA.)
The E 2-equivalences are exactly those maps that induce anisomorphism of E 2-pages of the spectral sequence; in fact, theyautomatically induce isomorphisms of E 2-exact couples.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 155: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/155.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In the E 2-model structure on sC, every object admits a cofibrantreplacement
by a simplicial object which is inductively given bylatching on an arbitrary coproduct of elements of G. Such anobject is called G-free.
As mentioned before, this is also called the resolution modelstructure; one should think of the elements of G as “projectivegenerators”.
In fact, we’re secretly using the E 2-model structure on the algebraside, too: our model structure on sA is just the E 2-modelstructure with respect to the full subcategory
GA = π∗G ⊂ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 156: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/156.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In the E 2-model structure on sC, every object admits a cofibrantreplacement by a simplicial object which is inductively given bylatching on an arbitrary coproduct of elements of G.
Such anobject is called G-free.
As mentioned before, this is also called the resolution modelstructure; one should think of the elements of G as “projectivegenerators”.
In fact, we’re secretly using the E 2-model structure on the algebraside, too: our model structure on sA is just the E 2-modelstructure with respect to the full subcategory
GA = π∗G ⊂ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 157: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/157.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In the E 2-model structure on sC, every object admits a cofibrantreplacement by a simplicial object which is inductively given bylatching on an arbitrary coproduct of elements of G. Such anobject is called G-free.
As mentioned before, this is also called the resolution modelstructure; one should think of the elements of G as “projectivegenerators”.
In fact, we’re secretly using the E 2-model structure on the algebraside, too: our model structure on sA is just the E 2-modelstructure with respect to the full subcategory
GA = π∗G ⊂ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 158: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/158.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In the E 2-model structure on sC, every object admits a cofibrantreplacement by a simplicial object which is inductively given bylatching on an arbitrary coproduct of elements of G. Such anobject is called G-free.
As mentioned before, this is also called the resolution modelstructure; one should think of the elements of G as “projectivegenerators”.
In fact, we’re secretly using the E 2-model structure on the algebraside, too: our model structure on sA is just the E 2-modelstructure with respect to the full subcategory
GA = π∗G ⊂ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 159: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/159.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In the E 2-model structure on sC, every object admits a cofibrantreplacement by a simplicial object which is inductively given bylatching on an arbitrary coproduct of elements of G. Such anobject is called G-free.
As mentioned before, this is also called the resolution modelstructure; one should think of the elements of G as “projectivegenerators”.
In fact, we’re secretly using the E 2-model structure on the algebraside, too:
our model structure on sA is just the E 2-modelstructure with respect to the full subcategory
GA = π∗G ⊂ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 160: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/160.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In the E 2-model structure on sC, every object admits a cofibrantreplacement by a simplicial object which is inductively given bylatching on an arbitrary coproduct of elements of G. Such anobject is called G-free.
As mentioned before, this is also called the resolution modelstructure; one should think of the elements of G as “projectivegenerators”.
In fact, we’re secretly using the E 2-model structure on the algebraside, too: our model structure on sA is just the E 2-modelstructure with respect to the full subcategory
GA = π∗G ⊂ A.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 161: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/161.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Recall that we’ve defined the functors πn : sA→ A for all n ≥ 0,and in fact we’ve seen that πn ∈ Modπ0(A) for n ≥ 1.
Note that we also have π∗ : sC→ sA, defined levelwise.
For Y ∈ sC, we call the πnπβY the classical homotopy groups;though we need the actual simplicial object to define them, theyare invariants of its E 2-equivalence class.
Thus πnπ∗Y ∈ A, and in fact πnπ∗Y ∈ Modπ0π∗Y (A) for n ≥ 1.
By definition, the E 2-equivalences are created by the classicalhomotopy groups.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 162: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/162.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Recall that we’ve defined the functors πn : sA→ A for all n ≥ 0,and in fact we’ve seen that πn ∈ Modπ0(A) for n ≥ 1.
Note that we also have π∗ : sC→ sA, defined levelwise.
For Y ∈ sC, we call the πnπβY the classical homotopy groups;though we need the actual simplicial object to define them, theyare invariants of its E 2-equivalence class.
Thus πnπ∗Y ∈ A, and in fact πnπ∗Y ∈ Modπ0π∗Y (A) for n ≥ 1.
By definition, the E 2-equivalences are created by the classicalhomotopy groups.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 163: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/163.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Recall that we’ve defined the functors πn : sA→ A for all n ≥ 0,and in fact we’ve seen that πn ∈ Modπ0(A) for n ≥ 1.
Note that we also have π∗ : sC→ sA, defined levelwise.
For Y ∈ sC, we call the πnπβY the classical homotopy groups;
though we need the actual simplicial object to define them, theyare invariants of its E 2-equivalence class.
Thus πnπ∗Y ∈ A, and in fact πnπ∗Y ∈ Modπ0π∗Y (A) for n ≥ 1.
By definition, the E 2-equivalences are created by the classicalhomotopy groups.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 164: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/164.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Recall that we’ve defined the functors πn : sA→ A for all n ≥ 0,and in fact we’ve seen that πn ∈ Modπ0(A) for n ≥ 1.
Note that we also have π∗ : sC→ sA, defined levelwise.
For Y ∈ sC, we call the πnπβY the classical homotopy groups;though we need the actual simplicial object to define them, theyare invariants of its E 2-equivalence class.
Thus πnπ∗Y ∈ A, and in fact πnπ∗Y ∈ Modπ0π∗Y (A) for n ≥ 1.
By definition, the E 2-equivalences are created by the classicalhomotopy groups.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 165: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/165.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Recall that we’ve defined the functors πn : sA→ A for all n ≥ 0,and in fact we’ve seen that πn ∈ Modπ0(A) for n ≥ 1.
Note that we also have π∗ : sC→ sA, defined levelwise.
For Y ∈ sC, we call the πnπβY the classical homotopy groups;though we need the actual simplicial object to define them, theyare invariants of its E 2-equivalence class.
Thus πnπ∗Y ∈ A, and in fact πnπ∗Y ∈ Modπ0π∗Y (A) for n ≥ 1.
By definition, the E 2-equivalences are created by the classicalhomotopy groups.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 166: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/166.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Recall that we’ve defined the functors πn : sA→ A for all n ≥ 0,and in fact we’ve seen that πn ∈ Modπ0(A) for n ≥ 1.
Note that we also have π∗ : sC→ sA, defined levelwise.
For Y ∈ sC, we call the πnπβY the classical homotopy groups;though we need the actual simplicial object to define them, theyare invariants of its E 2-equivalence class.
Thus πnπ∗Y ∈ A, and in fact πnπ∗Y ∈ Modπ0π∗Y (A) for n ≥ 1.
By definition, the E 2-equivalences are created by the classicalhomotopy groups.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 167: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/167.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is another type of homotopy group which will constitute therest of the E 2-exact couple.
First, recall that the E 2-model structure is compatible in the withthe external simplicial structure.
On the other hand, the Reedy model structure is only compatiblewith the trivial model structure on sSet (and this is tautological).We’ll need this later.
We use the same notations K Y and K ∧ Y as before.
These actually take part in the two-variable adjunctions
−− : sSet× C→ sC
map(−,−)0 : sSetop × sC→ C
homlwC (−,−) : C× sC→ sSet
− ∧− : sSet∗ × C∅ → sC
map∗(−,−)0 : sSetop∗ × sC→ C∅
homlwC (−,−) : C∅ × sC→ sSet∗.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 168: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/168.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is another type of homotopy group which will constitute therest of the E 2-exact couple.
First, recall that the E 2-model structure is compatible in the withthe external simplicial structure.
On the other hand, the Reedy model structure is only compatiblewith the trivial model structure on sSet (and this is tautological).We’ll need this later.
We use the same notations K Y and K ∧ Y as before.
These actually take part in the two-variable adjunctions
−− : sSet× C→ sC
map(−,−)0 : sSetop × sC→ C
homlwC (−,−) : C× sC→ sSet
− ∧− : sSet∗ × C∅ → sC
map∗(−,−)0 : sSetop∗ × sC→ C∅
homlwC (−,−) : C∅ × sC→ sSet∗.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 169: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/169.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is another type of homotopy group which will constitute therest of the E 2-exact couple.
First, recall that the E 2-model structure is compatible in the withthe external simplicial structure.
On the other hand, the Reedy model structure is only compatiblewith the trivial model structure on sSet (and this is tautological).
We’ll need this later.
We use the same notations K Y and K ∧ Y as before.
These actually take part in the two-variable adjunctions
−− : sSet× C→ sC
map(−,−)0 : sSetop × sC→ C
homlwC (−,−) : C× sC→ sSet
− ∧− : sSet∗ × C∅ → sC
map∗(−,−)0 : sSetop∗ × sC→ C∅
homlwC (−,−) : C∅ × sC→ sSet∗.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 170: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/170.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is another type of homotopy group which will constitute therest of the E 2-exact couple.
First, recall that the E 2-model structure is compatible in the withthe external simplicial structure.
On the other hand, the Reedy model structure is only compatiblewith the trivial model structure on sSet (and this is tautological).We’ll need this later.
We use the same notations K Y and K ∧ Y as before.
These actually take part in the two-variable adjunctions
−− : sSet× C→ sC
map(−,−)0 : sSetop × sC→ C
homlwC (−,−) : C× sC→ sSet
− ∧− : sSet∗ × C∅ → sC
map∗(−,−)0 : sSetop∗ × sC→ C∅
homlwC (−,−) : C∅ × sC→ sSet∗.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 171: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/171.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is another type of homotopy group which will constitute therest of the E 2-exact couple.
First, recall that the E 2-model structure is compatible in the withthe external simplicial structure.
On the other hand, the Reedy model structure is only compatiblewith the trivial model structure on sSet (and this is tautological).We’ll need this later.
We use the same notations K Y and K ∧ Y as before.
These actually take part in the two-variable adjunctions
−− : sSet× C→ sC
map(−,−)0 : sSetop × sC→ C
homlwC (−,−) : C× sC→ sSet
− ∧− : sSet∗ × C∅ → sC
map∗(−,−)0 : sSetop∗ × sC→ C∅
homlwC (−,−) : C∅ × sC→ sSet∗.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 172: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/172.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There is another type of homotopy group which will constitute therest of the E 2-exact couple.
First, recall that the E 2-model structure is compatible in the withthe external simplicial structure.
On the other hand, the Reedy model structure is only compatiblewith the trivial model structure on sSet (and this is tautological).We’ll need this later.
We use the same notations K Y and K ∧ Y as before.
These actually take part in the two-variable adjunctions
−− : sSet× C→ sC
map(−,−)0 : sSetop × sC→ C
homlwC (−,−) : C× sC→ sSet
− ∧− : sSet∗ × C∅ → sC
map∗(−,−)0 : sSetop∗ × sC→ C∅
homlwC (−,−) : C∅ × sC→ sSet∗.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 173: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/173.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y ) ∼= [Sn
∆∧Sβ ,Y ]sCE2∼= [Sβ ,map∗(S
n∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A, and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 174: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/174.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y )
∼= [Sn∆∧Sβ ,Y ]sC
E2∼= [Sβ ,map∗(S
n∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A, and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 175: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/175.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y ) ∼= [Sn
∆∧Sβ ,Y ]sCE2
∼= [Sβ ,map∗(Sn∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A, and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 176: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/176.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y ) ∼= [Sn
∆∧Sβ ,Y ]sCE2∼= [Sβ ,map∗(S
n∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A, and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 177: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/177.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y ) ∼= [Sn
∆∧Sβ ,Y ]sCE2∼= [Sβ ,map∗(S
n∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A,
and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 178: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/178.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y ) ∼= [Sn
∆∧Sβ ,Y ]sCE2∼= [Sβ ,map∗(S
n∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A, and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 179: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/179.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y ) ∼= [Sn
∆∧Sβ ,Y ]sCE2∼= [Sβ ,map∗(S
n∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A, and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 180: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/180.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In this language, the classical homotopy groups are given by
πnπβY = πn[Sβ ,Y ]lwC = πn(πlw
0 homlwC (Sβ ,Y )).
Using the E 2-model structure, we define the natural homotopygroups to be
πn,βY = πn homsCE2 (const(Sβ),Y ) ∼= [Sn
∆∧Sβ ,Y ]sCE2∼= [Sβ ,map∗(S
n∆,Y )0]C∅ .
The same technique as before shows that πn,∗Y ∈ A, and eventhat πn,∗Y ∈ Modπ0,∗Y (A) for n ≥ 1.
So finally we can attempt to understand the E 2-exact couple.
Of course, the best way to understand an exact couple is to unrollit into a long exact sequence, and when we do this we get...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 181: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/181.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequence
The spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 182: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/182.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 183: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/183.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple,
whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 184: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/184.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC
(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 185: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/185.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 186: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/186.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi ,
whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 187: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/187.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 188: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/188.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical!
Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 189: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/189.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups,
which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 190: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/190.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The spiral exact sequenceThe spiral exact sequence comes from unrolling the E 2-exactcouple of the spectral sequence E 2
i ,β = πiπβY ⇒ πi+β|Y |.
This is of course the derived couple of the E 1-exact couple, whichis of course not an invariant of the E 2-equivalence class of Y ∈ sC(but only its Reedy equivalence class).
Note that the natural homotopy groups πi ,βY have to do withmaps Sβ → Yi , whereas the classical homotopy groups πiπβY arenot as directly related to geometry.
But by a stroke of divine providence, the spiral exact sequence is2/3 natural and 1/3 classical! Thus, we can control the naturalhomotopy groups in order to control the classical homotopygroups, which are what show up on the E 2-page.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 191: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/191.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There are two adjoint methods for obtaining the E 1-exact couple.
(1) Note that in (sSet∗)triv, we have a (homotopy!) cofibersequence
Sn−1∆ → Dn
∆ → Sn∆,
where Dn∆ = ∆n/Λn
0 ∈ sSet∗.
For any Sβ ∈ G, this yields a (homotopy!) cofiber sequence
Sn−1∆ ∧ Sβ → Dn
∆ ∧ Sβ → Sn∆ ∧ Sβ
in sCReedy.
Applying [−,Y ]sCReedyyields a long exact sequence.
When we collect over all Sβ ∈ G and range over all n ≥ 1,these splice together into the E 1-exact couple.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 192: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/192.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There are two adjoint methods for obtaining the E 1-exact couple.
(1) Note that in (sSet∗)triv, we have a (homotopy!) cofibersequence
Sn−1∆ → Dn
∆ → Sn∆,
where Dn∆ = ∆n/Λn
0 ∈ sSet∗.
For any Sβ ∈ G, this yields a (homotopy!) cofiber sequence
Sn−1∆ ∧ Sβ → Dn
∆ ∧ Sβ → Sn∆ ∧ Sβ
in sCReedy.
Applying [−,Y ]sCReedyyields a long exact sequence.
When we collect over all Sβ ∈ G and range over all n ≥ 1,these splice together into the E 1-exact couple.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 193: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/193.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There are two adjoint methods for obtaining the E 1-exact couple.
(1) Note that in (sSet∗)triv, we have a (homotopy!) cofibersequence
Sn−1∆ → Dn
∆ → Sn∆,
where Dn∆ = ∆n/Λn
0 ∈ sSet∗.
For any Sβ ∈ G, this yields a (homotopy!) cofiber sequence
Sn−1∆ ∧ Sβ → Dn
∆ ∧ Sβ → Sn∆ ∧ Sβ
in sCReedy.
Applying [−,Y ]sCReedyyields a long exact sequence.
When we collect over all Sβ ∈ G and range over all n ≥ 1,these splice together into the E 1-exact couple.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 194: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/194.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There are two adjoint methods for obtaining the E 1-exact couple.
(1) Note that in (sSet∗)triv, we have a (homotopy!) cofibersequence
Sn−1∆ → Dn
∆ → Sn∆,
where Dn∆ = ∆n/Λn
0 ∈ sSet∗.
For any Sβ ∈ G, this yields a (homotopy!) cofiber sequence
Sn−1∆ ∧ Sβ → Dn
∆ ∧ Sβ → Sn∆ ∧ Sβ
in sCReedy.
Applying [−,Y ]sCReedyyields a long exact sequence.
When we collect over all Sβ ∈ G and range over all n ≥ 1,these splice together into the E 1-exact couple.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 195: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/195.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
There are two adjoint methods for obtaining the E 1-exact couple.
(1) Note that in (sSet∗)triv, we have a (homotopy!) cofibersequence
Sn−1∆ → Dn
∆ → Sn∆,
where Dn∆ = ∆n/Λn
0 ∈ sSet∗.
For any Sβ ∈ G, this yields a (homotopy!) cofiber sequence
Sn−1∆ ∧ Sβ → Dn
∆ ∧ Sβ → Sn∆ ∧ Sβ
in sCReedy.
Applying [−,Y ]sCReedyyields a long exact sequence.
When we collect over all Sβ ∈ G and range over all n ≥ 1,these splice together into the E 1-exact couple.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 196: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/196.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
(2) Adjointly (and more homotopico-algebraically),
given Y ∈ sCwe can form its nonabelian cobased normalized n-chainsobject
NnY = map∗(Dn∆,Y )0
and its nonabelian cobased n-cycles object
ZnY = map∗(Sn∆,Y )0.
These fit into fiber sequences
ZnY → NnY → Zn−1Y
in C∅.
Applying [Sβ,−]C∅ yields the same long exact sequences asbefore.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 197: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/197.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
(2) Adjointly (and more homotopico-algebraically), given Y ∈ sCwe can form its nonabelian cobased normalized n-chainsobject
NnY = map∗(Dn∆,Y )0
and its nonabelian cobased n-cycles object
ZnY = map∗(Sn∆,Y )0.
These fit into fiber sequences
ZnY → NnY → Zn−1Y
in C∅.
Applying [Sβ,−]C∅ yields the same long exact sequences asbefore.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 198: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/198.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
(2) Adjointly (and more homotopico-algebraically), given Y ∈ sCwe can form its nonabelian cobased normalized n-chainsobject
NnY = map∗(Dn∆,Y )0
and its nonabelian cobased n-cycles object
ZnY = map∗(Sn∆,Y )0.
These fit into fiber sequences
ZnY → NnY → Zn−1Y
in C∅.
Applying [Sβ,−]C∅ yields the same long exact sequences asbefore.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 199: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/199.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
(2) Adjointly (and more homotopico-algebraically), given Y ∈ sCwe can form its nonabelian cobased normalized n-chainsobject
NnY = map∗(Dn∆,Y )0
and its nonabelian cobased n-cycles object
ZnY = map∗(Sn∆,Y )0.
These fit into fiber sequences
ZnY → NnY → Zn−1Y
in C∅.
Applying [Sβ,−]C∅ yields the same long exact sequences asbefore.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 200: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/200.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
(2) Adjointly (and more homotopico-algebraically), given Y ∈ sCwe can form its nonabelian cobased normalized n-chainsobject
NnY = map∗(Dn∆,Y )0
and its nonabelian cobased n-cycles object
ZnY = map∗(Sn∆,Y )0.
These fit into fiber sequences
ZnY → NnY → Zn−1Y
in C∅.
Applying [Sβ,−]C∅ yields the same long exact sequences asbefore.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 201: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/201.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In either case, the key fact that identifies the derived exact coupleis that we have an exact sequence
[Dn+1∆ ∧ Sβ ,Y ]sCReedy → [Sn
∆ ∧ Sβ ,Y ]sCReedy → [Sn∆ ∧ Sβ ,Y ]sC
E2 → 0.
The key point is that the map Sn∆ ∧ Sβ → Dn+1
∆ ∧ Sβ is an“E 2-cone”.
That is, it’s an E 2-cofibration into an E 2-contractible object.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 202: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/202.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In either case, the key fact that identifies the derived exact coupleis that we have an exact sequence
[Dn+1∆ ∧ Sβ ,Y ]sCReedy → [Sn
∆ ∧ Sβ ,Y ]sCReedy → [Sn∆ ∧ Sβ ,Y ]sC
E2 → 0.
The key point is that the map Sn∆ ∧ Sβ → Dn+1
∆ ∧ Sβ is an“E 2-cone”.
That is, it’s an E 2-cofibration into an E 2-contractible object.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 203: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/203.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In either case, the key fact that identifies the derived exact coupleis that we have an exact sequence
[Dn+1∆ ∧ Sβ ,Y ]sCReedy → [Sn
∆ ∧ Sβ ,Y ]sCReedy → [Sn∆ ∧ Sβ ,Y ]sC
E2 → 0.
The key point is that the map Sn∆ ∧ Sβ → Dn+1
∆ ∧ Sβ is an“E 2-cone”.
That is, it’s an E 2-cofibration into an E 2-contractible object.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 204: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/204.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Finally, the spiral exact sequence runs
· · · Ωπn−1,∗Y πn,∗Y πnπ∗Y · · ·
· · · Ωπ0,∗Y π1,∗Y π1π∗Y pt.
Here, for any A ∈ A, we define ΩA ∈ A by (ΩA)(Sβ) = A(ΣSβ).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 205: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/205.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Finally, the spiral exact sequence runs
· · · Ωπn−1,∗Y πn,∗Y πnπ∗Y · · ·
· · · Ωπ0,∗Y π1,∗Y π1π∗Y pt.
Here, for any A ∈ A, we define ΩA ∈ A by (ΩA)(Sβ) = A(ΣSβ).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 206: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/206.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
This is not just a long exact sequence in A.
On the one hand, all the πnπ∗Y for n ≥ 1 live in Modπ0π∗Y (A).
On the other hand, all the πn,∗Y and Ωπn,∗Y for n ≥ 1 (andΩπ0,∗Y too, it turns out) live in Modπ0,∗Y (A).
In fact, it turns out that π0,∗Y ∼= π0π∗Y ! Really the spiral exactsequence ends with
pt = Ωπ−1,∗Y π0,∗Y π0π∗Y pt.∼
If we denote their common value by π0Y ∈ A, then the spiralexact sequence (excluding this last little bit) actually takes place inModπ0Y (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 207: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/207.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
This is not just a long exact sequence in A.
On the one hand, all the πnπ∗Y for n ≥ 1 live in Modπ0π∗Y (A).
On the other hand, all the πn,∗Y and Ωπn,∗Y for n ≥ 1 (andΩπ0,∗Y too, it turns out) live in Modπ0,∗Y (A).
In fact, it turns out that π0,∗Y ∼= π0π∗Y ! Really the spiral exactsequence ends with
pt = Ωπ−1,∗Y π0,∗Y π0π∗Y pt.∼
If we denote their common value by π0Y ∈ A, then the spiralexact sequence (excluding this last little bit) actually takes place inModπ0Y (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 208: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/208.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
This is not just a long exact sequence in A.
On the one hand, all the πnπ∗Y for n ≥ 1 live in Modπ0π∗Y (A).
On the other hand, all the πn,∗Y and Ωπn,∗Y for n ≥ 1 (andΩπ0,∗Y too, it turns out) live in Modπ0,∗Y (A).
In fact, it turns out that π0,∗Y ∼= π0π∗Y ! Really the spiral exactsequence ends with
pt = Ωπ−1,∗Y π0,∗Y π0π∗Y pt.∼
If we denote their common value by π0Y ∈ A, then the spiralexact sequence (excluding this last little bit) actually takes place inModπ0Y (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 209: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/209.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
This is not just a long exact sequence in A.
On the one hand, all the πnπ∗Y for n ≥ 1 live in Modπ0π∗Y (A).
On the other hand, all the πn,∗Y and Ωπn,∗Y for n ≥ 1 (andΩπ0,∗Y too, it turns out) live in Modπ0,∗Y (A).
In fact, it turns out that π0,∗Y ∼= π0π∗Y !
Really the spiral exactsequence ends with
pt = Ωπ−1,∗Y π0,∗Y π0π∗Y pt.∼
If we denote their common value by π0Y ∈ A, then the spiralexact sequence (excluding this last little bit) actually takes place inModπ0Y (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 210: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/210.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
This is not just a long exact sequence in A.
On the one hand, all the πnπ∗Y for n ≥ 1 live in Modπ0π∗Y (A).
On the other hand, all the πn,∗Y and Ωπn,∗Y for n ≥ 1 (andΩπ0,∗Y too, it turns out) live in Modπ0,∗Y (A).
In fact, it turns out that π0,∗Y ∼= π0π∗Y ! Really the spiral exactsequence ends with
pt = Ωπ−1,∗Y π0,∗Y π0π∗Y pt.∼
If we denote their common value by π0Y ∈ A, then the spiralexact sequence (excluding this last little bit) actually takes place inModπ0Y (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 211: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/211.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
This is not just a long exact sequence in A.
On the one hand, all the πnπ∗Y for n ≥ 1 live in Modπ0π∗Y (A).
On the other hand, all the πn,∗Y and Ωπn,∗Y for n ≥ 1 (andΩπ0,∗Y too, it turns out) live in Modπ0,∗Y (A).
In fact, it turns out that π0,∗Y ∼= π0π∗Y ! Really the spiral exactsequence ends with
pt = Ωπ−1,∗Y π0,∗Y π0π∗Y pt.∼
If we denote their common value by π0Y ∈ A, then the spiralexact sequence (excluding this last little bit) actually takes place inModπ0Y (A).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 212: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/212.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In particular, these really are abelian groups, and so it’s notdangerous to say that the sequence (which a priori just lives inSet∗) is “exact”.
(Of course, we know what an exact sequence inSet∗ is, too. And who doesn’t like to live dangerously?)
To prove that the spiral exact sequence supports this modulestructure requires lots and lots of fiddling around with the interplaybetween the Reedy and E 2-model structures on sC.
Anyways, now that we have the spiral exact sequence in hand, wecan finally talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 213: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/213.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In particular, these really are abelian groups, and so it’s notdangerous to say that the sequence (which a priori just lives inSet∗) is “exact”. (Of course, we know what an exact sequence inSet∗ is, too. And who doesn’t like to live dangerously?)
To prove that the spiral exact sequence supports this modulestructure requires lots and lots of fiddling around with the interplaybetween the Reedy and E 2-model structures on sC.
Anyways, now that we have the spiral exact sequence in hand, wecan finally talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 214: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/214.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In particular, these really are abelian groups, and so it’s notdangerous to say that the sequence (which a priori just lives inSet∗) is “exact”. (Of course, we know what an exact sequence inSet∗ is, too. And who doesn’t like to live dangerously?)
To prove that the spiral exact sequence supports this modulestructure requires lots and lots of fiddling around with the interplaybetween the Reedy and E 2-model structures on sC.
Anyways, now that we have the spiral exact sequence in hand, wecan finally talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 215: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/215.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
In particular, these really are abelian groups, and so it’s notdangerous to say that the sequence (which a priori just lives inSet∗) is “exact”. (Of course, we know what an exact sequence inSet∗ is, too. And who doesn’t like to live dangerously?)
To prove that the spiral exact sequence supports this modulestructure requires lots and lots of fiddling around with the interplaybetween the Reedy and E 2-model structures on sC.
Anyways, now that we have the spiral exact sequence in hand, wecan finally talk about...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 216: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/216.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Obstruction theory
Recall that for an object A ∈ A, we will define the moduli spacesMn(A) ⊂ sCE2 of n-stages such that
M (A)∼←−M∞(A)
lim−→ · · · →M1(A)→M0(A).
We’ve already defined the ∞-stages: these are the objectsY ∈ sC such that π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Recall also that we said that an n-stage Y would have π0π∗Y ∼= Aand πn+2π∗Y ∼= Ωn+1A, but that that wasn’t enough to pin downits E 2-exact couple. Finally, we can define Y ∈ sC to be ann-stage if it has
i 0 1 2 · · · n − 1 n n + 1 n + 2 n + 3 · · ·πiπ∗Y A 0 0 · · · 0 0 0 Ωn+1A 0 · · ·πi,∗Y A ΩA Ω2A · · · Ωn−1A ΩnA 0 0 0 · · ·
Aaron Mazel-Gee GHOsT for∞-categories
![Page 217: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/217.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Obstruction theoryRecall that for an object A ∈ A, we will define the moduli spacesMn(A) ⊂ sCE2 of n-stages
such that
M (A)∼←−M∞(A)
lim−→ · · · →M1(A)→M0(A).
We’ve already defined the ∞-stages: these are the objectsY ∈ sC such that π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Recall also that we said that an n-stage Y would have π0π∗Y ∼= Aand πn+2π∗Y ∼= Ωn+1A, but that that wasn’t enough to pin downits E 2-exact couple. Finally, we can define Y ∈ sC to be ann-stage if it has
i 0 1 2 · · · n − 1 n n + 1 n + 2 n + 3 · · ·πiπ∗Y A 0 0 · · · 0 0 0 Ωn+1A 0 · · ·πi,∗Y A ΩA Ω2A · · · Ωn−1A ΩnA 0 0 0 · · ·
Aaron Mazel-Gee GHOsT for∞-categories
![Page 218: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/218.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Obstruction theoryRecall that for an object A ∈ A, we will define the moduli spacesMn(A) ⊂ sCE2 of n-stages such that
M (A)∼←−M∞(A)
lim−→ · · · →M1(A)→M0(A).
We’ve already defined the ∞-stages: these are the objectsY ∈ sC such that π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Recall also that we said that an n-stage Y would have π0π∗Y ∼= Aand πn+2π∗Y ∼= Ωn+1A, but that that wasn’t enough to pin downits E 2-exact couple. Finally, we can define Y ∈ sC to be ann-stage if it has
i 0 1 2 · · · n − 1 n n + 1 n + 2 n + 3 · · ·πiπ∗Y A 0 0 · · · 0 0 0 Ωn+1A 0 · · ·πi,∗Y A ΩA Ω2A · · · Ωn−1A ΩnA 0 0 0 · · ·
Aaron Mazel-Gee GHOsT for∞-categories
![Page 219: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/219.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Obstruction theoryRecall that for an object A ∈ A, we will define the moduli spacesMn(A) ⊂ sCE2 of n-stages such that
M (A)∼←−M∞(A)
lim−→ · · · →M1(A)→M0(A).
We’ve already defined the ∞-stages: these are the objectsY ∈ sC such that π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Recall also that we said that an n-stage Y would have π0π∗Y ∼= Aand πn+2π∗Y ∼= Ωn+1A, but that that wasn’t enough to pin downits E 2-exact couple. Finally, we can define Y ∈ sC to be ann-stage if it has
i 0 1 2 · · · n − 1 n n + 1 n + 2 n + 3 · · ·πiπ∗Y A 0 0 · · · 0 0 0 Ωn+1A 0 · · ·πi,∗Y A ΩA Ω2A · · · Ωn−1A ΩnA 0 0 0 · · ·
Aaron Mazel-Gee GHOsT for∞-categories
![Page 220: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/220.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Obstruction theoryRecall that for an object A ∈ A, we will define the moduli spacesMn(A) ⊂ sCE2 of n-stages such that
M (A)∼←−M∞(A)
lim−→ · · · →M1(A)→M0(A).
We’ve already defined the ∞-stages: these are the objectsY ∈ sC such that π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Recall also that we said that an n-stage Y would have π0π∗Y ∼= Aand πn+2π∗Y ∼= Ωn+1A,
but that that wasn’t enough to pin downits E 2-exact couple. Finally, we can define Y ∈ sC to be ann-stage if it has
i 0 1 2 · · · n − 1 n n + 1 n + 2 n + 3 · · ·πiπ∗Y A 0 0 · · · 0 0 0 Ωn+1A 0 · · ·πi,∗Y A ΩA Ω2A · · · Ωn−1A ΩnA 0 0 0 · · ·
Aaron Mazel-Gee GHOsT for∞-categories
![Page 221: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/221.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Obstruction theoryRecall that for an object A ∈ A, we will define the moduli spacesMn(A) ⊂ sCE2 of n-stages such that
M (A)∼←−M∞(A)
lim−→ · · · →M1(A)→M0(A).
We’ve already defined the ∞-stages: these are the objectsY ∈ sC such that π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Recall also that we said that an n-stage Y would have π0π∗Y ∼= Aand πn+2π∗Y ∼= Ωn+1A, but that that wasn’t enough to pin downits E 2-exact couple.
Finally, we can define Y ∈ sC to be ann-stage if it has
i 0 1 2 · · · n − 1 n n + 1 n + 2 n + 3 · · ·πiπ∗Y A 0 0 · · · 0 0 0 Ωn+1A 0 · · ·πi,∗Y A ΩA Ω2A · · · Ωn−1A ΩnA 0 0 0 · · ·
Aaron Mazel-Gee GHOsT for∞-categories
![Page 222: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/222.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Obstruction theoryRecall that for an object A ∈ A, we will define the moduli spacesMn(A) ⊂ sCE2 of n-stages such that
M (A)∼←−M∞(A)
lim−→ · · · →M1(A)→M0(A).
We’ve already defined the ∞-stages: these are the objectsY ∈ sC such that π0π∗Y ∼= A and πiπ∗Y = 0 for i > 0.
Recall also that we said that an n-stage Y would have π0π∗Y ∼= Aand πn+2π∗Y ∼= Ωn+1A, but that that wasn’t enough to pin downits E 2-exact couple. Finally, we can define Y ∈ sC to be ann-stage if it has
i 0 1 2 · · · n − 1 n n + 1 n + 2 n + 3 · · ·πiπ∗Y A 0 0 · · · 0 0 0 Ωn+1A 0 · · ·πi,∗Y A ΩA Ω2A · · · Ωn−1A ΩnA 0 0 0 · · ·
Aaron Mazel-Gee GHOsT for∞-categories
![Page 223: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/223.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 .
Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 224: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/224.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 225: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/225.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case.
We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 226: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/226.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 227: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/227.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 228: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/228.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if
π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 229: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/229.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A
andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 230: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/230.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 231: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/231.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if
π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 232: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/232.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,
πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 233: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/233.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M,
and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 234: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/234.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
The obstruction theory exploits the interplay between Postnikovtheory in sCE2 and Postnikov theory in sA = sAE2 . Thus, we willneed to understand Eilenberg–MacLane objects in both categories.
The stories begin identically, so we use D to denote eithercategory, and we use πi(∗) : D→ A to simultaneously denote πi ,∗in the former case and πi in the latter case. We’ll later use thesuperscripts top and alg to distinguish things when necessary.
Then, for any A ∈ A, M ∈ ModA(A), and n ≥ 1, we say:
that an object Y ∈ D has type KA if π0(∗)Y ∼= A andπi(∗)Y = 0 for i 6= 0;
that an object Z ∈ D has type KA(M, n) if π0(∗)Z ∼= A,πn(∗)Z ∼= M, and πi(∗)Z = 0 for i 6= 0, n.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 235: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/235.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, these Eilenberg–MacLane objects always exist, and areunique up to equivalence.
We use them to define Andre–Quillen cohomology groups: for anyn ≥ 1 and X ∈ D/KA
, we define
HnA(X ,M) = [X ,KA(M, n)]D/KA
.
In fact, by the long exact sequence in homotopy we have thatΩKA(M, n) ' KA(M, n − 1) in D/KA
.
So, setting KA(M, 0) = ΩKA(M, 1) (which of course has typeKMoA), we have that
KAM = KA(M, n)n≥0
forms an Ω-spectrum in D/KA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 236: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/236.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, these Eilenberg–MacLane objects always exist, and areunique up to equivalence.
We use them to define Andre–Quillen cohomology groups:
for anyn ≥ 1 and X ∈ D/KA
, we define
HnA(X ,M) = [X ,KA(M, n)]D/KA
.
In fact, by the long exact sequence in homotopy we have thatΩKA(M, n) ' KA(M, n − 1) in D/KA
.
So, setting KA(M, 0) = ΩKA(M, 1) (which of course has typeKMoA), we have that
KAM = KA(M, n)n≥0
forms an Ω-spectrum in D/KA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 237: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/237.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, these Eilenberg–MacLane objects always exist, and areunique up to equivalence.
We use them to define Andre–Quillen cohomology groups: for anyn ≥ 1 and X ∈ D/KA
, we define
HnA(X ,M) = [X ,KA(M, n)]D/KA
.
In fact, by the long exact sequence in homotopy we have thatΩKA(M, n) ' KA(M, n − 1) in D/KA
.
So, setting KA(M, 0) = ΩKA(M, 1) (which of course has typeKMoA), we have that
KAM = KA(M, n)n≥0
forms an Ω-spectrum in D/KA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 238: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/238.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, these Eilenberg–MacLane objects always exist, and areunique up to equivalence.
We use them to define Andre–Quillen cohomology groups: for anyn ≥ 1 and X ∈ D/KA
, we define
HnA(X ,M) = [X ,KA(M, n)]D/KA
.
In fact, by the long exact sequence in homotopy we have thatΩKA(M, n) ' KA(M, n − 1) in D/KA
.
So, setting KA(M, 0) = ΩKA(M, 1) (which of course has typeKMoA), we have that
KAM = KA(M, n)n≥0
forms an Ω-spectrum in D/KA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 239: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/239.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, these Eilenberg–MacLane objects always exist, and areunique up to equivalence.
We use them to define Andre–Quillen cohomology groups: for anyn ≥ 1 and X ∈ D/KA
, we define
HnA(X ,M) = [X ,KA(M, n)]D/KA
.
In fact, by the long exact sequence in homotopy we have thatΩKA(M, n) ' KA(M, n − 1) in D/KA
.
So, setting KA(M, 0) = ΩKA(M, 1) (which of course has typeKMoA),
we have that
KAM = KA(M, n)n≥0
forms an Ω-spectrum in D/KA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 240: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/240.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Of course, these Eilenberg–MacLane objects always exist, and areunique up to equivalence.
We use them to define Andre–Quillen cohomology groups: for anyn ≥ 1 and X ∈ D/KA
, we define
HnA(X ,M) = [X ,KA(M, n)]D/KA
.
In fact, by the long exact sequence in homotopy we have thatΩKA(M, n) ' KA(M, n − 1) in D/KA
.
So, setting KA(M, 0) = ΩKA(M, 1) (which of course has typeKMoA), we have that
KAM = KA(M, n)n≥0
forms an Ω-spectrum in D/KA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 241: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/241.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We then define Andre–Quillen cohomology spaces
H nA (X ,M) = homD/KA
(X ,KA(M, n)).
By definition, these have π0H n = Hn.
Since mapping an object into an Ω-spectrum yields an Ω-spectrum,these collect into an Andre–Quillen cohomology Ω-spectrum
HA(X ,M) = H nA (X ,M)n≥0
in S∗. (In particular, ΩH n 'H n−1, as we claimed earlier.)
By construction, π−iH ∼= H i for i ≥ 0 and πjH = 0 for j > 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 242: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/242.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We then define Andre–Quillen cohomology spaces
H nA (X ,M) = homD/KA
(X ,KA(M, n)).
By definition, these have π0H n = Hn.
Since mapping an object into an Ω-spectrum yields an Ω-spectrum,these collect into an Andre–Quillen cohomology Ω-spectrum
HA(X ,M) = H nA (X ,M)n≥0
in S∗. (In particular, ΩH n 'H n−1, as we claimed earlier.)
By construction, π−iH ∼= H i for i ≥ 0 and πjH = 0 for j > 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 243: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/243.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We then define Andre–Quillen cohomology spaces
H nA (X ,M) = homD/KA
(X ,KA(M, n)).
By definition, these have π0H n = Hn.
Since mapping an object into an Ω-spectrum yields an Ω-spectrum,
these collect into an Andre–Quillen cohomology Ω-spectrum
HA(X ,M) = H nA (X ,M)n≥0
in S∗. (In particular, ΩH n 'H n−1, as we claimed earlier.)
By construction, π−iH ∼= H i for i ≥ 0 and πjH = 0 for j > 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 244: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/244.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We then define Andre–Quillen cohomology spaces
H nA (X ,M) = homD/KA
(X ,KA(M, n)).
By definition, these have π0H n = Hn.
Since mapping an object into an Ω-spectrum yields an Ω-spectrum,these collect into an Andre–Quillen cohomology Ω-spectrum
HA(X ,M) = H nA (X ,M)n≥0
in S∗.
(In particular, ΩH n 'H n−1, as we claimed earlier.)
By construction, π−iH ∼= H i for i ≥ 0 and πjH = 0 for j > 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 245: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/245.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We then define Andre–Quillen cohomology spaces
H nA (X ,M) = homD/KA
(X ,KA(M, n)).
By definition, these have π0H n = Hn.
Since mapping an object into an Ω-spectrum yields an Ω-spectrum,these collect into an Andre–Quillen cohomology Ω-spectrum
HA(X ,M) = H nA (X ,M)n≥0
in S∗. (In particular, ΩH n 'H n−1, as we claimed earlier.)
By construction, π−iH ∼= H i for i ≥ 0 and πjH = 0 for j > 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 246: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/246.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We then define Andre–Quillen cohomology spaces
H nA (X ,M) = homD/KA
(X ,KA(M, n)).
By definition, these have π0H n = Hn.
Since mapping an object into an Ω-spectrum yields an Ω-spectrum,these collect into an Andre–Quillen cohomology Ω-spectrum
HA(X ,M) = H nA (X ,M)n≥0
in S∗. (In particular, ΩH n 'H n−1, as we claimed earlier.)
By construction, π−iH ∼= H i for i ≥ 0 and πjH = 0 for j > 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 247: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/247.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We can define functorial Postnikov towers
Ylim−→ · · · → P1Y → P0Y
in D.
If we set A = π0Y ∈ A and M = πnY ∈ ModA(A), then we havea pullback square
PnY KA
Pn−1Y KA(M, n + 1),
where the map on the right is an isomorphism on π0(∗).
This nth k-invariant map Pn−1Y → KA(M, n + 1) can beconstructed functorially, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 248: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/248.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We can define functorial Postnikov towers
Ylim−→ · · · → P1Y → P0Y
in D.
If we set A = π0Y ∈ A and M = πnY ∈ ModA(A),
then we havea pullback square
PnY KA
Pn−1Y KA(M, n + 1),
where the map on the right is an isomorphism on π0(∗).
This nth k-invariant map Pn−1Y → KA(M, n + 1) can beconstructed functorially, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 249: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/249.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We can define functorial Postnikov towers
Ylim−→ · · · → P1Y → P0Y
in D.
If we set A = π0Y ∈ A and M = πnY ∈ ModA(A), then we havea pullback square
PnY KA
Pn−1Y KA(M, n + 1),
where the map on the right is an isomorphism on π0(∗).
This nth k-invariant map Pn−1Y → KA(M, n + 1) can beconstructed functorially, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 250: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/250.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We can define functorial Postnikov towers
Ylim−→ · · · → P1Y → P0Y
in D.
If we set A = π0Y ∈ A and M = πnY ∈ ModA(A), then we havea pullback square
PnY KA
Pn−1Y KA(M, n + 1),
where the map on the right is an isomorphism on π0(∗).
This nth k-invariant map Pn−1Y → KA(M, n + 1) can beconstructed functorially, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 251: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/251.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We can define functorial Postnikov towers
Ylim−→ · · · → P1Y → P0Y
in D.
If we set A = π0Y ∈ A and M = πnY ∈ ModA(A), then we havea pullback square
PnY KA
Pn−1Y KA(M, n + 1),
where the map on the right is an isomorphism on π0(∗).
This nth k-invariant map Pn−1Y → KA(M, n + 1)
can beconstructed functorially, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 252: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/252.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
We can define functorial Postnikov towers
Ylim−→ · · · → P1Y → P0Y
in D.
If we set A = π0Y ∈ A and M = πnY ∈ ModA(A), then we havea pullback square
PnY KA
Pn−1Y KA(M, n + 1),
where the map on the right is an isomorphism on π0(∗).
This nth k-invariant map Pn−1Y → KA(M, n + 1) can beconstructed functorially, too.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 253: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/253.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
But here the algebraic and topological stories diverge:
πiπ∗KtopA∼=
A, i = 0ΩA, i = 20, i 6= 0, 2
and
πiπ∗KtopA (M, n) ∼= πiπ∗K
topA ×
M, i = nΩM, i = n + 20, i 6= n, n + 2.
In particular, π∗KtopA 6' K alg
A and π∗KtopA (M, n) 6' K alg
A (M, n).
However, there is nevertheless a universal morphism
π∗KtopA (M, n)→ K alg
A (M, n),
which for any Y ∈ sCE2 yields an equivalence
hom(sCE2 )
/KtopA
(Y ,K topA (M, n))
∼−→ homsA/K
algA
(π∗Y ,KalgA (M, n)) = H n
A (π∗Y ,M).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 254: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/254.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
But here the algebraic and topological stories diverge:
πiπ∗KtopA∼=
A, i = 0ΩA, i = 20, i 6= 0, 2
and
πiπ∗KtopA (M, n) ∼= πiπ∗K
topA ×
M, i = nΩM, i = n + 20, i 6= n, n + 2.
In particular, π∗KtopA 6' K alg
A and π∗KtopA (M, n) 6' K alg
A (M, n).
However, there is nevertheless a universal morphism
π∗KtopA (M, n)→ K alg
A (M, n),
which for any Y ∈ sCE2 yields an equivalence
hom(sCE2 )
/KtopA
(Y ,K topA (M, n))
∼−→ homsA/K
algA
(π∗Y ,KalgA (M, n)) = H n
A (π∗Y ,M).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 255: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/255.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
But here the algebraic and topological stories diverge:
πiπ∗KtopA∼=
A, i = 0ΩA, i = 20, i 6= 0, 2
and
πiπ∗KtopA (M, n) ∼= πiπ∗K
topA ×
M, i = nΩM, i = n + 20, i 6= n, n + 2.
In particular, π∗KtopA 6' K alg
A and π∗KtopA (M, n) 6' K alg
A (M, n).
However, there is nevertheless a universal morphism
π∗KtopA (M, n)→ K alg
A (M, n),
which for any Y ∈ sCE2 yields an equivalence
hom(sCE2 )
/KtopA
(Y ,K topA (M, n))
∼−→ homsA/K
algA
(π∗Y ,KalgA (M, n)) = H n
A (π∗Y ,M).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 256: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/256.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
But here the algebraic and topological stories diverge:
πiπ∗KtopA∼=
A, i = 0ΩA, i = 20, i 6= 0, 2
and
πiπ∗KtopA (M, n) ∼= πiπ∗K
topA ×
M, i = nΩM, i = n + 20, i 6= n, n + 2.
In particular, π∗KtopA 6' K alg
A and π∗KtopA (M, n) 6' K alg
A (M, n).
However, there is nevertheless a universal morphism
π∗KtopA (M, n)→ K alg
A (M, n),
which for any Y ∈ sCE2 yields an equivalence
hom(sCE2 )
/KtopA
(Y ,K topA (M, n))
∼−→ homsA/K
algA
(π∗Y ,KalgA (M, n)) = H n
A (π∗Y ,M).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 257: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/257.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
But here the algebraic and topological stories diverge:
πiπ∗KtopA∼=
A, i = 0ΩA, i = 20, i 6= 0, 2
and
πiπ∗KtopA (M, n) ∼= πiπ∗K
topA ×
M, i = nΩM, i = n + 20, i 6= n, n + 2.
In particular, π∗KtopA 6' K alg
A and π∗KtopA (M, n) 6' K alg
A (M, n).
However, there is nevertheless a universal morphism
π∗KtopA (M, n)→ K alg
A (M, n),
which for any Y ∈ sCE2 yields an equivalence
hom(sCE2 )
/KtopA
(Y ,K topA (M, n))
∼−→ homsA/K
algA
(π∗Y ,KalgA (M, n)) = H n
A (π∗Y ,M).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 258: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/258.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
But here the algebraic and topological stories diverge:
πiπ∗KtopA∼=
A, i = 0ΩA, i = 20, i 6= 0, 2
and
πiπ∗KtopA (M, n) ∼= πiπ∗K
topA ×
M, i = nΩM, i = n + 20, i 6= n, n + 2.
In particular, π∗KtopA 6' K alg
A and π∗KtopA (M, n) 6' K alg
A (M, n).
However, there is nevertheless a universal morphism
π∗KtopA (M, n)→ K alg
A (M, n),
which for any Y ∈ sCE2 yields an equivalence
hom(sCE2 )
/KtopA
(Y ,K topA (M, n))
∼−→ homsA/K
algA
(π∗Y ,KalgA (M, n)) = H n
A (π∗Y ,M).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 259: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/259.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, the object K topA (M, n) ∈ (sCE2)/K top
Asimultaneously:
acts as the target of the functorial nth k-invariant map, and
represents the functor H nA (π∗−,M).
From here, routine manipulations yield our desired pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
...and that’s how you get based spaces from Π-algebras!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 260: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/260.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, the object K topA (M, n) ∈ (sCE2)/K top
Asimultaneously:
acts as the target of the functorial nth k-invariant map,
and
represents the functor H nA (π∗−,M).
From here, routine manipulations yield our desired pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
...and that’s how you get based spaces from Π-algebras!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 261: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/261.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, the object K topA (M, n) ∈ (sCE2)/K top
Asimultaneously:
acts as the target of the functorial nth k-invariant map, and
represents the functor H nA (π∗−,M).
From here, routine manipulations yield our desired pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
...and that’s how you get based spaces from Π-algebras!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 262: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/262.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, the object K topA (M, n) ∈ (sCE2)/K top
Asimultaneously:
acts as the target of the functorial nth k-invariant map, and
represents the functor H nA (π∗−,M).
From here, routine manipulations yield our desired pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
...and that’s how you get based spaces from Π-algebras!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 263: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/263.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
So, the object K topA (M, n) ∈ (sCE2)/K top
Asimultaneously:
acts as the target of the functorial nth k-invariant map, and
represents the functor H nA (π∗−,M).
From here, routine manipulations yield our desired pullback square
Mn(A) BAut(A,ΩnA)
Mn−1(A) H n+2(A,ΩnA).
...and that’s how you get based spaces from Π-algebras!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 264: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/264.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 265: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/265.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that
agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 266: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/266.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1]
and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 267: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/267.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 268: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/268.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉).
If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 269: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/269.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 270: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/270.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 271: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/271.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 272: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/272.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Categorical generalitiesThe big pictureThe E2-model structureThe spiral exact sequenceObstruction theory
Before moving on, we briefly mention a rather surprisinginterpretation of the objects Y ∈Mn(A) given byBlanc–Dwyer–Goerss.
From the spectral sequence, we can see that |Y | ∈ Top∗ hashomotopy groups that agree with those encoded in A ∈ Π-alg inthe range [1, n + 1] and vanish elsewhere.
We might write |Y | ∈M (A〈1,n+1〉). If Y lifts to an ∞-stage
Y ∈M∞(A), then |Y | ' |Y |〈1,n+1〉.
However, if we write P ilwY for the levelwise i th cotruncation of Y ,
then |P ilwY | ∈M (A〈i ,n+i〉)!
That is, in some sense, Y “knows” how to thread together anyn + 1 consecutive homotopy groups encoded in A ∈ Π-alg.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 273: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/273.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
3. From Blanc–Dwyer–Goerss toGoerss–Hopkins
Aaron Mazel-Gee GHOsT for∞-categories
![Page 274: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/274.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Over the course of more than 15 years, Goerss–Hopkins wrote atleast six hefty papers on obstruction theory that spanned morethan 500 pages.
These culminated in the 126-page tome Moduli problems forstructured ring spectra, which to this day remains unpublished.
Recall their two further desiderata:
They want to be able to use E∗ instead of π∗.
They’re interested in the moduli space of algebras over anoperad.
Together, these make for a nearly endless supply of new twists andsubtleties.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 275: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/275.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Over the course of more than 15 years, Goerss–Hopkins wrote atleast six hefty papers on obstruction theory that spanned morethan 500 pages.
These culminated in the 126-page tome Moduli problems forstructured ring spectra, which to this day remains unpublished.
Recall their two further desiderata:
They want to be able to use E∗ instead of π∗.
They’re interested in the moduli space of algebras over anoperad.
Together, these make for a nearly endless supply of new twists andsubtleties.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 276: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/276.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Over the course of more than 15 years, Goerss–Hopkins wrote atleast six hefty papers on obstruction theory that spanned morethan 500 pages.
These culminated in the 126-page tome Moduli problems forstructured ring spectra, which to this day remains unpublished.
Recall their two further desiderata:
They want to be able to use E∗ instead of π∗.
They’re interested in the moduli space of algebras over anoperad.
Together, these make for a nearly endless supply of new twists andsubtleties.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 277: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/277.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Over the course of more than 15 years, Goerss–Hopkins wrote atleast six hefty papers on obstruction theory that spanned morethan 500 pages.
These culminated in the 126-page tome Moduli problems forstructured ring spectra, which to this day remains unpublished.
Recall their two further desiderata:
They want to be able to use E∗ instead of π∗.
They’re interested in the moduli space of algebras over anoperad.
Together, these make for a nearly endless supply of new twists andsubtleties.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 278: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/278.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Over the course of more than 15 years, Goerss–Hopkins wrote atleast six hefty papers on obstruction theory that spanned morethan 500 pages.
These culminated in the 126-page tome Moduli problems forstructured ring spectra, which to this day remains unpublished.
Recall their two further desiderata:
They want to be able to use E∗ instead of π∗.
They’re interested in the moduli space of algebras over anoperad.
Together, these make for a nearly endless supply of new twists andsubtleties.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 279: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/279.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Over the course of more than 15 years, Goerss–Hopkins wrote atleast six hefty papers on obstruction theory that spanned morethan 500 pages.
These culminated in the 126-page tome Moduli problems forstructured ring spectra, which to this day remains unpublished.
Recall their two further desiderata:
They want to be able to use E∗ instead of π∗.
They’re interested in the moduli space of algebras over anoperad.
Together, these make for a nearly endless supply of new twists andsubtleties.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 280: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/280.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Over the course of more than 15 years, Goerss–Hopkins wrote atleast six hefty papers on obstruction theory that spanned morethan 500 pages.
These culminated in the 126-page tome Moduli problems forstructured ring spectra, which to this day remains unpublished.
Recall their two further desiderata:
They want to be able to use E∗ instead of π∗.
They’re interested in the moduli space of algebras over anoperad.
Together, these make for a nearly endless supply of new twists andsubtleties.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 281: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/281.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
The key ideas
In the Blanc–Dwyer–Goerss story, we’re taking resolutions ofobjects C by G-free objects of sC.
The important point is that these are sent to GA-free objects of sAby the functor π∗ : sC→ sA.
So first of all, in the Goerss–Hopkins story, we need to be able tosay the same of our functor E∗ : sC→ sA. (Of course, now wetake A = ModE∗ , or even A = ComodE∗E .)
To achieve this, we enlarge our G using this One Weird Old Trick.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 282: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/282.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
The key ideas
In the Blanc–Dwyer–Goerss story, we’re taking resolutions ofobjects C by G-free objects of sC.
The important point is that these are sent to GA-free objects of sAby the functor π∗ : sC→ sA.
So first of all, in the Goerss–Hopkins story, we need to be able tosay the same of our functor E∗ : sC→ sA. (Of course, now wetake A = ModE∗ , or even A = ComodE∗E .)
To achieve this, we enlarge our G using this One Weird Old Trick.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 283: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/283.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
The key ideas
In the Blanc–Dwyer–Goerss story, we’re taking resolutions ofobjects C by G-free objects of sC.
The important point is that these are sent to GA-free objects of sAby the functor π∗ : sC→ sA.
So first of all, in the Goerss–Hopkins story, we need to be able tosay the same of our functor E∗ : sC→ sA. (Of course, now wetake A = ModE∗ , or even A = ComodE∗E .)
To achieve this, we enlarge our G using this One Weird Old Trick.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 284: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/284.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
The key ideas
In the Blanc–Dwyer–Goerss story, we’re taking resolutions ofobjects C by G-free objects of sC.
The important point is that these are sent to GA-free objects of sAby the functor π∗ : sC→ sA.
So first of all, in the Goerss–Hopkins story, we need to be able tosay the same of our functor E∗ : sC→ sA.
(Of course, now wetake A = ModE∗ , or even A = ComodE∗E .)
To achieve this, we enlarge our G using this One Weird Old Trick.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 285: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/285.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
The key ideas
In the Blanc–Dwyer–Goerss story, we’re taking resolutions ofobjects C by G-free objects of sC.
The important point is that these are sent to GA-free objects of sAby the functor π∗ : sC→ sA.
So first of all, in the Goerss–Hopkins story, we need to be able tosay the same of our functor E∗ : sC→ sA. (Of course, now wetake A = ModE∗ , or even A = ComodE∗E .)
To achieve this, we enlarge our G using this One Weird Old Trick.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 286: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/286.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
The key ideas
In the Blanc–Dwyer–Goerss story, we’re taking resolutions ofobjects C by G-free objects of sC.
The important point is that these are sent to GA-free objects of sAby the functor π∗ : sC→ sA.
So first of all, in the Goerss–Hopkins story, we need to be able tosay the same of our functor E∗ : sC→ sA. (Of course, now wetake A = ModE∗ , or even A = ComodE∗E .)
To achieve this, we enlarge our G using this One Weird Old Trick.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 287: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/287.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if
it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE , the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 288: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/288.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE , the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 289: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/289.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE , the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 290: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/290.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE ,
the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 291: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/291.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE , the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 292: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/292.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE , the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 293: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/293.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE , the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 294: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/294.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
We say that a homotopy ring spectrum E satisfies Adams’scondition if it can be written as a filtered colimit E ' colimEα ofdualizable spectra Eα, such that:
E∗DEα ∈ ModE∗ is projective, and
for every M ∈ ModE , the Kunneth map
[DEα,M]C homModE∗ (E∗DEα,M∗)
(DEα → M) π∗(E ⊗ DEα → E ⊗M → M)
is an isomorphism.
This is the condition given by Adams in his blue book to ensurethat E∗ enjoys a Kunneth spectral sequence.
This also implies that
E∗X = π∗(E ⊗ X ) ∼= colim[Sβ ⊗ DEα,X ]CSβ∈G.
So, we assume that E satisfies Adams’s condition.Aaron Mazel-Gee GHOsT for∞-categories
![Page 295: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/295.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, we define the set GEC ⊂ C to be the closure of G ∪ ΣβDEα
under finite coproducts.
This defines the E 2GEC
-model structure on
sC.
Moreover, we setGA = E∗G
EC ⊂ A,
which similarly defines the E 2GA
-model structure on sA.
Then, the functorE∗ : sCE2
GEC
→ sAE2GA
takes GEC -free objects to GA-free objects!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 296: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/296.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, we define the set GEC ⊂ C to be the closure of G ∪ ΣβDEα
under finite coproducts. This defines the E 2GEC
-model structure on
sC.
Moreover, we setGA = E∗G
EC ⊂ A,
which similarly defines the E 2GA
-model structure on sA.
Then, the functorE∗ : sCE2
GEC
→ sAE2GA
takes GEC -free objects to GA-free objects!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 297: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/297.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, we define the set GEC ⊂ C to be the closure of G ∪ ΣβDEα
under finite coproducts. This defines the E 2GEC
-model structure on
sC.
Moreover, we setGA = E∗G
EC ⊂ A,
which similarly defines the E 2GA
-model structure on sA.
Then, the functorE∗ : sCE2
GEC
→ sAE2GA
takes GEC -free objects to GA-free objects!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 298: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/298.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, we define the set GEC ⊂ C to be the closure of G ∪ ΣβDEα
under finite coproducts. This defines the E 2GEC
-model structure on
sC.
Moreover, we setGA = E∗G
EC ⊂ A,
which similarly defines the E 2GA
-model structure on sA.
Then, the functorE∗ : sCE2
GEC
→ sAE2GA
takes GEC -free objects to GA-free objects!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 299: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/299.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, we define the set GEC ⊂ C to be the closure of G ∪ ΣβDEα
under finite coproducts. This defines the E 2GEC
-model structure on
sC.
Moreover, we setGA = E∗G
EC ⊂ A,
which similarly defines the E 2GA
-model structure on sA.
Then, the functorE∗ : sCE2
GEC
→ sAE2GA
takes GEC -free objects to GA-free objects!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 300: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/300.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free, and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O). This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 301: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/301.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free, and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O). This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 302: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/302.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free, and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O). This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 303: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/303.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free,
and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O). This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 304: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/304.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free, and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O). This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 305: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/305.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free, and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O). This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 306: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/306.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free, and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O).
This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 307: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/307.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Now, let’s try to bring the operad into the mix.
If O ∈ Op is our operad of interest, we want to resolve objects ofAlgO(C) by some sort of algebras in sC whose E -homology is stillcofibrant in an appropriate sense.
The other key insight, then, is that we should resolve O by asimplicial operad T ∈ sOp such that:
the operads Tn are all Σ-free, and
the E∗Tn(k) are all projective E∗-modules.
This gives us an adjunction AlgT (sC) AlgO(C).
(Note that when O is an A∞-operad, we can simply takeT = const(O). This is the source of the comparative simplicity ofthe Hopkins–Miller obstruction theory.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 308: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/308.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective: at leastwhen Y ∈ sC is GE
C -free, we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 309: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/309.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective: at leastwhen Y ∈ sC is GE
C -free, we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 310: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/310.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective: at leastwhen Y ∈ sC is GE
C -free, we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 311: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/311.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective: at leastwhen Y ∈ sC is GE
C -free, we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 312: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/312.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective: at leastwhen Y ∈ sC is GE
C -free, we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 313: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/313.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective:
at leastwhen Y ∈ sC is GE
C -free, we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 314: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/314.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective: at leastwhen Y ∈ sC is GE
C -free,
we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 315: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/315.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Then, for any X ∈ C with E∗X projective,
E∗TnX ∼= E∗
∐k≥0
(Tn(k)⊗ X⊗k
)Σk
∼= ⊕k≥0
(E∗Tn(k)⊗E∗ (E∗X )⊗k
)Σk
.
(Recall that the Kunneth spectral sequence takes Tor as input.)
So, when X is a Tn-algebra, then E∗X becomes an E∗Tn-algebra!
These collect into a functor
E∗ : AlgT (sC)→ AlgE∗T (sA)
that preserves cofibrancy.
And this is why we want the E∗Tn(k) to be projective: at leastwhen Y ∈ sC is GE
C -free, we want E∗TY ∈ AlgE∗T (sA) to forgetdown to a cofibrant object of sA.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 316: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/316.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Let’s put this all together.
We know already that any object of C admits a GEC -free resolution.
Also, we can always resolve any algebra by free algebras.
So, we can now resolve any O-algebra X ∈ AlgO(C) by a freeT -algebra Y ∈ AlgT (sC) with E∗Y ∈ AlgE∗T (sA) cofibrant!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 317: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/317.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Let’s put this all together.
We know already that any object of C admits a GEC -free resolution.
Also, we can always resolve any algebra by free algebras.
So, we can now resolve any O-algebra X ∈ AlgO(C) by a freeT -algebra Y ∈ AlgT (sC) with E∗Y ∈ AlgE∗T (sA) cofibrant!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 318: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/318.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Let’s put this all together.
We know already that any object of C admits a GEC -free resolution.
Also, we can always resolve any algebra by free algebras.
So, we can now resolve any O-algebra X ∈ AlgO(C) by a freeT -algebra Y ∈ AlgT (sC) with E∗Y ∈ AlgE∗T (sA) cofibrant!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 319: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/319.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Let’s put this all together.
We know already that any object of C admits a GEC -free resolution.
Also, we can always resolve any algebra by free algebras.
So, we can now resolve any O-algebra X ∈ AlgO(C) by a freeT -algebra Y ∈ AlgT (sC) with E∗Y ∈ AlgE∗T (sA) cofibrant!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 320: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/320.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
However, note that we are now using the functor [P,−]CP∈GEC
to
determined our equivalences in sCE2
GEC
.
In general, this will have strictly fewer equivalences than thosedetermined by taking E∗ levelwise: we’ve decomposedE ' colimEα more or less arbitrarily.
So, we must further localize sCE2
GEC
and AlgT (sC)E2
GEC
so that π∗E∗
creates equivalences.
Of course, we could go on for days about all of the...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 321: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/321.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
However, note that we are now using the functor [P,−]CP∈GEC
to
determined our equivalences in sCE2
GEC
.
In general, this will have strictly fewer equivalences than thosedetermined by taking E∗ levelwise:
we’ve decomposedE ' colimEα more or less arbitrarily.
So, we must further localize sCE2
GEC
and AlgT (sC)E2
GEC
so that π∗E∗
creates equivalences.
Of course, we could go on for days about all of the...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 322: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/322.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
However, note that we are now using the functor [P,−]CP∈GEC
to
determined our equivalences in sCE2
GEC
.
In general, this will have strictly fewer equivalences than thosedetermined by taking E∗ levelwise: we’ve decomposedE ' colimEα more or less arbitrarily.
So, we must further localize sCE2
GEC
and AlgT (sC)E2
GEC
so that π∗E∗
creates equivalences.
Of course, we could go on for days about all of the...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 323: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/323.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
However, note that we are now using the functor [P,−]CP∈GEC
to
determined our equivalences in sCE2
GEC
.
In general, this will have strictly fewer equivalences than thosedetermined by taking E∗ levelwise: we’ve decomposedE ' colimEα more or less arbitrarily.
So, we must further localize sCE2
GEC
and AlgT (sC)E2
GEC
so that π∗E∗
creates equivalences.
Of course, we could go on for days about all of the...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 324: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/324.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
However, note that we are now using the functor [P,−]CP∈GEC
to
determined our equivalences in sCE2
GEC
.
In general, this will have strictly fewer equivalences than thosedetermined by taking E∗ levelwise: we’ve decomposedE ' colimEα more or less arbitrarily.
So, we must further localize sCE2
GEC
and AlgT (sC)E2
GEC
so that π∗E∗
creates equivalences.
Of course, we could go on for days about all of the...
Aaron Mazel-Gee GHOsT for∞-categories
![Page 325: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/325.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Complications
But let’s not.
Especially because the next section is so much cleaner!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 326: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/326.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Complications
But let’s not.
Especially because the next section is so much cleaner!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 327: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/327.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
The key ideasComplications
Complications
But let’s not.
Especially because the next section is so much cleaner!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 328: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/328.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
4. To ∞-categories and beyond!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 329: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/329.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The ∞-categorical GHOsT story, while still complicated, isboatloads simpler than the original one.
The original GHOsT story is mired in an obscene amount ofmodel-categorical technicalities.
For instance, for various point-set reasons involving NDR-pairs andwanting to compute homotopy quotients by group actions usingon-the-nose quotients, Goerss–Hopkins choose to work withsymmetric spectra in topological spaces with the positive modelstructure.
And then, they also need to fuss around with semi-modelstructures in order to perform the Bousfield E -localizations we justmentioned.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 330: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/330.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The ∞-categorical GHOsT story, while still complicated, isboatloads simpler than the original one.
The original GHOsT story is mired in an obscene amount ofmodel-categorical technicalities.
For instance, for various point-set reasons involving NDR-pairs andwanting to compute homotopy quotients by group actions usingon-the-nose quotients, Goerss–Hopkins choose to work withsymmetric spectra in topological spaces with the positive modelstructure.
And then, they also need to fuss around with semi-modelstructures in order to perform the Bousfield E -localizations we justmentioned.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 331: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/331.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The ∞-categorical GHOsT story, while still complicated, isboatloads simpler than the original one.
The original GHOsT story is mired in an obscene amount ofmodel-categorical technicalities.
For instance, for various point-set reasons involving NDR-pairs andwanting to compute homotopy quotients by group actions usingon-the-nose quotients, Goerss–Hopkins choose to work withsymmetric spectra in topological spaces with the positive modelstructure.
And then, they also need to fuss around with semi-modelstructures in order to perform the Bousfield E -localizations we justmentioned.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 332: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/332.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The ∞-categorical GHOsT story, while still complicated, isboatloads simpler than the original one.
The original GHOsT story is mired in an obscene amount ofmodel-categorical technicalities.
For instance, for various point-set reasons involving NDR-pairs andwanting to compute homotopy quotients by group actions usingon-the-nose quotients,
Goerss–Hopkins choose to work withsymmetric spectra in topological spaces with the positive modelstructure.
And then, they also need to fuss around with semi-modelstructures in order to perform the Bousfield E -localizations we justmentioned.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 333: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/333.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The ∞-categorical GHOsT story, while still complicated, isboatloads simpler than the original one.
The original GHOsT story is mired in an obscene amount ofmodel-categorical technicalities.
For instance, for various point-set reasons involving NDR-pairs andwanting to compute homotopy quotients by group actions usingon-the-nose quotients, Goerss–Hopkins choose to work withsymmetric spectra in topological spaces with the positive modelstructure.
And then, they also need to fuss around with semi-modelstructures in order to perform the Bousfield E -localizations we justmentioned.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 334: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/334.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The ∞-categorical GHOsT story, while still complicated, isboatloads simpler than the original one.
The original GHOsT story is mired in an obscene amount ofmodel-categorical technicalities.
For instance, for various point-set reasons involving NDR-pairs andwanting to compute homotopy quotients by group actions usingon-the-nose quotients, Goerss–Hopkins choose to work withsymmetric spectra in topological spaces with the positive modelstructure.
And then, they also need to fuss around with semi-modelstructures in order to perform the Bousfield E -localizations we justmentioned.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 335: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/335.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, these sorts of technicalities all vanish when we pass to∞-categories!
For mere mortals, this simplification is more or less essential if onewants to generalize the obstruction theory to other homotopytheories besides that of spectra.
But perhaps most importantly, when we pass to ∞-categories, theunderlying mathematical ideas – which are extremely beautiful! –come through a lot more clearly.
So: let’s get involved!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 336: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/336.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, these sorts of technicalities all vanish when we pass to∞-categories!
For mere mortals, this simplification is more or less essential if onewants to generalize the obstruction theory to other homotopytheories besides that of spectra.
But perhaps most importantly, when we pass to ∞-categories, theunderlying mathematical ideas – which are extremely beautiful! –come through a lot more clearly.
So: let’s get involved!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 337: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/337.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, these sorts of technicalities all vanish when we pass to∞-categories!
For mere mortals, this simplification is more or less essential if onewants to generalize the obstruction theory to other homotopytheories besides that of spectra.
But perhaps most importantly, when we pass to ∞-categories, theunderlying mathematical ideas – which are extremely beautiful! –come through a lot more clearly.
So: let’s get involved!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 338: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/338.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, these sorts of technicalities all vanish when we pass to∞-categories!
For mere mortals, this simplification is more or less essential if onewants to generalize the obstruction theory to other homotopytheories besides that of spectra.
But perhaps most importantly, when we pass to ∞-categories, theunderlying mathematical ideas – which are extremely beautiful! –come through a lot more clearly.
So: let’s get involved!
Aaron Mazel-Gee GHOsT for∞-categories
![Page 339: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/339.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
∞-categorical generalities
Let C be a presentable ∞-category.
This is equivalent to saying that C admits a set of stronggenerators, i.e. a set G ⊂ C such that the Yoneda embedding
C → P(G) = Fun(Gop, S)
is fully faithful.
We once again assume that G is closed under finite coproducts andsuspensions, and that its objects are cobased.
As before, we use this to define the functor π∗ : C→ PδΣ(G) by
(π∗X )(Sβ) = [Sβ ,X ]C.
(Recall that PδΣ denotes the category of discrete,product-preserving presheaves.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 340: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/340.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
∞-categorical generalitiesLet C be a presentable ∞-category.
This is equivalent to saying that C admits a set of stronggenerators, i.e. a set G ⊂ C such that the Yoneda embedding
C → P(G) = Fun(Gop, S)
is fully faithful.
We once again assume that G is closed under finite coproducts andsuspensions, and that its objects are cobased.
As before, we use this to define the functor π∗ : C→ PδΣ(G) by
(π∗X )(Sβ) = [Sβ ,X ]C.
(Recall that PδΣ denotes the category of discrete,product-preserving presheaves.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 341: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/341.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
∞-categorical generalitiesLet C be a presentable ∞-category.
This is equivalent to saying that C admits a set of stronggenerators,
i.e. a set G ⊂ C such that the Yoneda embedding
C → P(G) = Fun(Gop, S)
is fully faithful.
We once again assume that G is closed under finite coproducts andsuspensions, and that its objects are cobased.
As before, we use this to define the functor π∗ : C→ PδΣ(G) by
(π∗X )(Sβ) = [Sβ ,X ]C.
(Recall that PδΣ denotes the category of discrete,product-preserving presheaves.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 342: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/342.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
∞-categorical generalitiesLet C be a presentable ∞-category.
This is equivalent to saying that C admits a set of stronggenerators, i.e. a set G ⊂ C such that the Yoneda embedding
C → P(G) = Fun(Gop, S)
is fully faithful.
We once again assume that G is closed under finite coproducts andsuspensions, and that its objects are cobased.
As before, we use this to define the functor π∗ : C→ PδΣ(G) by
(π∗X )(Sβ) = [Sβ ,X ]C.
(Recall that PδΣ denotes the category of discrete,product-preserving presheaves.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 343: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/343.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
∞-categorical generalitiesLet C be a presentable ∞-category.
This is equivalent to saying that C admits a set of stronggenerators, i.e. a set G ⊂ C such that the Yoneda embedding
C → P(G) = Fun(Gop, S)
is fully faithful.
We once again assume that G is closed under finite coproducts andsuspensions, and that its objects are cobased.
As before, we use this to define the functor π∗ : C→ PδΣ(G) by
(π∗X )(Sβ) = [Sβ ,X ]C.
(Recall that PδΣ denotes the category of discrete,product-preserving presheaves.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 344: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/344.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
∞-categorical generalitiesLet C be a presentable ∞-category.
This is equivalent to saying that C admits a set of stronggenerators, i.e. a set G ⊂ C such that the Yoneda embedding
C → P(G) = Fun(Gop, S)
is fully faithful.
We once again assume that G is closed under finite coproducts andsuspensions, and that its objects are cobased.
As before, we use this to define the functor π∗ : C→ PδΣ(G) by
(π∗X )(Sβ) = [Sβ ,X ]C.
(Recall that PδΣ denotes the category of discrete,product-preserving presheaves.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 345: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/345.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
∞-categorical generalitiesLet C be a presentable ∞-category.
This is equivalent to saying that C admits a set of stronggenerators, i.e. a set G ⊂ C such that the Yoneda embedding
C → P(G) = Fun(Gop, S)
is fully faithful.
We once again assume that G is closed under finite coproducts andsuspensions, and that its objects are cobased.
As before, we use this to define the functor π∗ : C→ PδΣ(G) by
(π∗X )(Sβ) = [Sβ ,X ]C.
(Recall that PδΣ denotes the category of discrete,product-preserving presheaves.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 346: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/346.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We want to talk about homology.
So, we assume that C has a monoidal structure ⊗ (that commuteswith colimits in the first variable), and we assume that we have ahomotopy monoid object E satisfying a “left” Adams’s condition.
We write A for the target of the functor E∗ : C→ A given by
E∗X = π∗(E ⊗ X ) = [Sβ ,E ⊗ X ]CSβ∈G.
If we define GEC and GA as before, then we have an embedding
A → PδΣ(GA).
In fact, we have an identification A ' ShvδΣ(GA) with the categoryof sheaves for the topology generated by the epimorphisms.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 347: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/347.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We want to talk about homology.
So, we assume that C has a monoidal structure ⊗
(that commuteswith colimits in the first variable), and we assume that we have ahomotopy monoid object E satisfying a “left” Adams’s condition.
We write A for the target of the functor E∗ : C→ A given by
E∗X = π∗(E ⊗ X ) = [Sβ ,E ⊗ X ]CSβ∈G.
If we define GEC and GA as before, then we have an embedding
A → PδΣ(GA).
In fact, we have an identification A ' ShvδΣ(GA) with the categoryof sheaves for the topology generated by the epimorphisms.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 348: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/348.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We want to talk about homology.
So, we assume that C has a monoidal structure ⊗ (that commuteswith colimits in the first variable),
and we assume that we have ahomotopy monoid object E satisfying a “left” Adams’s condition.
We write A for the target of the functor E∗ : C→ A given by
E∗X = π∗(E ⊗ X ) = [Sβ ,E ⊗ X ]CSβ∈G.
If we define GEC and GA as before, then we have an embedding
A → PδΣ(GA).
In fact, we have an identification A ' ShvδΣ(GA) with the categoryof sheaves for the topology generated by the epimorphisms.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 349: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/349.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We want to talk about homology.
So, we assume that C has a monoidal structure ⊗ (that commuteswith colimits in the first variable), and we assume that we have ahomotopy monoid object E satisfying a “left” Adams’s condition.
We write A for the target of the functor E∗ : C→ A given by
E∗X = π∗(E ⊗ X ) = [Sβ ,E ⊗ X ]CSβ∈G.
If we define GEC and GA as before, then we have an embedding
A → PδΣ(GA).
In fact, we have an identification A ' ShvδΣ(GA) with the categoryof sheaves for the topology generated by the epimorphisms.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 350: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/350.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We want to talk about homology.
So, we assume that C has a monoidal structure ⊗ (that commuteswith colimits in the first variable), and we assume that we have ahomotopy monoid object E satisfying a “left” Adams’s condition.
We write A for the target of the functor E∗ : C→ A given by
E∗X = π∗(E ⊗ X ) = [Sβ ,E ⊗ X ]CSβ∈G.
If we define GEC and GA as before, then we have an embedding
A → PδΣ(GA).
In fact, we have an identification A ' ShvδΣ(GA) with the categoryof sheaves for the topology generated by the epimorphisms.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 351: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/351.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We want to talk about homology.
So, we assume that C has a monoidal structure ⊗ (that commuteswith colimits in the first variable), and we assume that we have ahomotopy monoid object E satisfying a “left” Adams’s condition.
We write A for the target of the functor E∗ : C→ A given by
E∗X = π∗(E ⊗ X ) = [Sβ ,E ⊗ X ]CSβ∈G.
If we define GEC and GA as before, then we have an embedding
A → PδΣ(GA).
In fact, we have an identification A ' ShvδΣ(GA) with the categoryof sheaves for the topology generated by the epimorphisms.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 352: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/352.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We want to talk about homology.
So, we assume that C has a monoidal structure ⊗ (that commuteswith colimits in the first variable), and we assume that we have ahomotopy monoid object E satisfying a “left” Adams’s condition.
We write A for the target of the functor E∗ : C→ A given by
E∗X = π∗(E ⊗ X ) = [Sβ ,E ⊗ X ]CSβ∈G.
If we define GEC and GA as before, then we have an embedding
A → PδΣ(GA).
In fact, we have an identification A ' ShvδΣ(GA) with the categoryof sheaves for the topology generated by the epimorphisms.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 353: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/353.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, it turns out that the PΣ construction –
that is,product-preserving presheaves of spaces – is the underlying∞-category of the E 2-model structure!
This makes a lot of our constructions nearly tautological.
It should make this more believable that the PΣ construction isalso the free cocompletion for sifted colimits.
The main examples of sifted colimits are geometric realizations andfiltered colimits; in fact, these generate all sifted colimits, in thesense that the functor
s(Ind(D)) PΣ(D)
Y(d 7→ | homlw
D(d ,Y )|)
is essentially surjective.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 354: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/354.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, it turns out that the PΣ construction – that is,product-preserving presheaves of spaces –
is the underlying∞-category of the E 2-model structure!
This makes a lot of our constructions nearly tautological.
It should make this more believable that the PΣ construction isalso the free cocompletion for sifted colimits.
The main examples of sifted colimits are geometric realizations andfiltered colimits; in fact, these generate all sifted colimits, in thesense that the functor
s(Ind(D)) PΣ(D)
Y(d 7→ | homlw
D(d ,Y )|)
is essentially surjective.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 355: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/355.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, it turns out that the PΣ construction – that is,product-preserving presheaves of spaces – is the underlying∞-category of the E 2-model structure!
This makes a lot of our constructions nearly tautological.
It should make this more believable that the PΣ construction isalso the free cocompletion for sifted colimits.
The main examples of sifted colimits are geometric realizations andfiltered colimits; in fact, these generate all sifted colimits, in thesense that the functor
s(Ind(D)) PΣ(D)
Y(d 7→ | homlw
D(d ,Y )|)
is essentially surjective.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 356: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/356.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, it turns out that the PΣ construction – that is,product-preserving presheaves of spaces – is the underlying∞-category of the E 2-model structure!
This makes a lot of our constructions nearly tautological.
It should make this more believable that the PΣ construction isalso the free cocompletion for sifted colimits.
The main examples of sifted colimits are geometric realizations andfiltered colimits; in fact, these generate all sifted colimits, in thesense that the functor
s(Ind(D)) PΣ(D)
Y(d 7→ | homlw
D(d ,Y )|)
is essentially surjective.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 357: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/357.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, it turns out that the PΣ construction – that is,product-preserving presheaves of spaces – is the underlying∞-category of the E 2-model structure!
This makes a lot of our constructions nearly tautological.
It should make this more believable that the PΣ construction isalso the free cocompletion for sifted colimits.
The main examples of sifted colimits are geometric realizations andfiltered colimits; in fact, these generate all sifted colimits, in thesense that the functor
s(Ind(D)) PΣ(D)
Y(d 7→ | homlw
D(d ,Y )|)
is essentially surjective.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 358: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/358.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, it turns out that the PΣ construction – that is,product-preserving presheaves of spaces – is the underlying∞-category of the E 2-model structure!
This makes a lot of our constructions nearly tautological.
It should make this more believable that the PΣ construction isalso the free cocompletion for sifted colimits.
The main examples of sifted colimits are geometric realizations andfiltered colimits;
in fact, these generate all sifted colimits, in thesense that the functor
s(Ind(D)) PΣ(D)
Y(d 7→ | homlw
D(d ,Y )|)
is essentially surjective.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 359: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/359.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, it turns out that the PΣ construction – that is,product-preserving presheaves of spaces – is the underlying∞-category of the E 2-model structure!
This makes a lot of our constructions nearly tautological.
It should make this more believable that the PΣ construction isalso the free cocompletion for sifted colimits.
The main examples of sifted colimits are geometric realizations andfiltered colimits; in fact, these generate all sifted colimits, in thesense that the functor
s(Ind(D)) PΣ(D)
Y(d 7→ | homlw
D(d ,Y )|)
is essentially surjective.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 360: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/360.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The PΣ construction is also called the nonabelian derived∞-category construction.
When D is an Ind-complete abelian category with enoughprojectives, there is a natural equivalence
D−≥0(D)∼−→ PΣ(Dproj).
In any case, combining all this with what we’ve already seen, weare now ready for our first look at...
The Diagram.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 361: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/361.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The PΣ construction is also called the nonabelian derived∞-category construction.
When D is an Ind-complete abelian category with enoughprojectives,
there is a natural equivalence
D−≥0(D)∼−→ PΣ(Dproj).
In any case, combining all this with what we’ve already seen, weare now ready for our first look at...
The Diagram.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 362: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/362.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The PΣ construction is also called the nonabelian derived∞-category construction.
When D is an Ind-complete abelian category with enoughprojectives, there is a natural equivalence
D−≥0(D)∼−→ PΣ(Dproj).
In any case, combining all this with what we’ve already seen, weare now ready for our first look at...
The Diagram.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 363: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/363.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The PΣ construction is also called the nonabelian derived∞-category construction.
When D is an Ind-complete abelian category with enoughprojectives, there is a natural equivalence
D−≥0(D)∼−→ PΣ(Dproj).
In any case, combining all this with what we’ve already seen, weare now ready for our first look at...
The Diagram.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 364: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/364.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
The PΣ construction is also called the nonabelian derived∞-category construction.
When D is an Ind-complete abelian category with enoughprojectives, there is a natural equivalence
D−≥0(D)∼−→ PΣ(Dproj).
In any case, combining all this with what we’ve already seen, weare now ready for our first look at...
The Diagram.Aaron Mazel-Gee GHOsT for∞-categories
![Page 365: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/365.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
THE DIAGRAM.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 366: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/366.jpg)
C
AlgO(C)
PΣ(GEC )
sCE2
GEC
PΣ(T (GEC ))
AlgT (sC)E2
GEC
LE(AlgO(C))AlgO(LE(C)) AlgΦ(A) PδΣ(Φ(GA))
LE(AlgT (sC)) PΣ(TE(Aproj))
AlgT (sC)π∗E∗ AlgTE(sA)E2
GAAlgTE
(sA)Quillen
∼
π 0
LE(C) A PδΣ(GA)
LE(sC) PΣ(GA)
sCπ∗E∗ sAE2GA
sAQuillen
Mod0(A)
PδΣ(GEC )
PδΣ(T (GEC ))
PΣ(Aproj)
π0,∗ π≥1
Modπ0,∗(PδΣ(GEC ))
π≥
1,∗
⊥ ⊥
d⊥
Q⊥ Q⊥
⊥ ⊥
d⊥ d⊥
Q⊥ Q⊥
⊥ ⊥ ⊥ ⊥
d⊥ d⊥ d⊥
Q⊥
Q⊥
Q⊥
Q⊥
⊥
![Page 367: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/367.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We saw previously that in order to get the obstruction theory upand running, we needed to make a bunch of constructions andcomputations using the Reedy model structure to deducestatements about the E 2-model structure.
These remain necessary in the ∞-categorical setting.
We know that the ∞-categories sC and sA underlie the respectiveReedy model categories, and we know that the ∞-categories givenby the PΣ construction underlie the various E 2-model categories.
But how are these related?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 368: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/368.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We saw previously that in order to get the obstruction theory upand running, we needed to make a bunch of constructions andcomputations using the Reedy model structure to deducestatements about the E 2-model structure.
These remain necessary in the ∞-categorical setting.
We know that the ∞-categories sC and sA underlie the respectiveReedy model categories, and we know that the ∞-categories givenby the PΣ construction underlie the various E 2-model categories.
But how are these related?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 369: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/369.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We saw previously that in order to get the obstruction theory upand running, we needed to make a bunch of constructions andcomputations using the Reedy model structure to deducestatements about the E 2-model structure.
These remain necessary in the ∞-categorical setting.
We know that the ∞-categories sC and sA underlie the respectiveReedy model categories, and we know that the ∞-categories givenby the PΣ construction underlie the various E 2-model categories.
But how are these related?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 370: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/370.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We saw previously that in order to get the obstruction theory upand running, we needed to make a bunch of constructions andcomputations using the Reedy model structure to deducestatements about the E 2-model structure.
These remain necessary in the ∞-categorical setting.
We know that the ∞-categories sC and sA underlie the respectiveReedy model categories,
and we know that the ∞-categories givenby the PΣ construction underlie the various E 2-model categories.
But how are these related?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 371: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/371.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We saw previously that in order to get the obstruction theory upand running, we needed to make a bunch of constructions andcomputations using the Reedy model structure to deducestatements about the E 2-model structure.
These remain necessary in the ∞-categorical setting.
We know that the ∞-categories sC and sA underlie the respectiveReedy model categories, and we know that the ∞-categories givenby the PΣ construction underlie the various E 2-model categories.
But how are these related?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 372: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/372.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We saw previously that in order to get the obstruction theory upand running, we needed to make a bunch of constructions andcomputations using the Reedy model structure to deducestatements about the E 2-model structure.
These remain necessary in the ∞-categorical setting.
We know that the ∞-categories sC and sA underlie the respectiveReedy model categories, and we know that the ∞-categories givenby the PΣ construction underlie the various E 2-model categories.
But how are these related?
Aaron Mazel-Gee GHOsT for∞-categories
![Page 373: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/373.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S, and dually that F and W ∩ F are preserved undercotensoring with S. (These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 374: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/374.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S, and dually that F and W ∩ F are preserved undercotensoring with S. (These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 375: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/375.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S, and dually that F and W ∩ F are preserved undercotensoring with S. (These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 376: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/376.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S, and dually that F and W ∩ F are preserved undercotensoring with S. (These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 377: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/377.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S, and dually that F and W ∩ F are preserved undercotensoring with S. (These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 378: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/378.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S,
and dually that F and W ∩ F are preserved undercotensoring with S. (These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 379: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/379.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S, and dually that F and W ∩ F are preserved undercotensoring with S.
(These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 380: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/380.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Model ∞-categories
A model structure (W,C,F) on a 1-category D allows us to makecomputations in D[W−1].
A model structure on an ∞-category does exactly the same thing.
The axioms are nearly identical.
Some care must be taken with the lifting axiom, which now assertsthat a certain map of spaces is an effective epimorphism (that is,an epimorphism on π0).
We also assume that C and W ∩ C are preserved under tensoringwith S, and dually that F and W ∩ F are preserved undercotensoring with S. (These are “continuous” versions of theanalogous facts for model 1-categories when S is replaced by Set.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 381: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/381.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In the case that D is actually a 1-category, this recovers theclassical definition of a model structure on D.
In fact, if D is a model ∞-category and ho(D) has whateverco/limits one would like to require (though this is quite rare!), thena model structure on D descends to a model structure on ho(D).
(The two-out-of-three, retract, and factorization axioms in D areeven verifiable in ho(D), and the lifting axiom descends as well.)
On the other hand, since in general there is a complicated interplaybetween co/limits in D and co/limits in ho(D), we are nervousabout drawing any conclusions about the relationship betweenthese two model ∞-categories.
(Note that the functor D→ ho(D) is the unit of the adjunctionCat∞ Cat1, but it is not itself an adjoint.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 382: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/382.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In the case that D is actually a 1-category, this recovers theclassical definition of a model structure on D.
In fact, if D is a model ∞-category and ho(D) has whateverco/limits one would like to require (though this is quite rare!),
thena model structure on D descends to a model structure on ho(D).
(The two-out-of-three, retract, and factorization axioms in D areeven verifiable in ho(D), and the lifting axiom descends as well.)
On the other hand, since in general there is a complicated interplaybetween co/limits in D and co/limits in ho(D), we are nervousabout drawing any conclusions about the relationship betweenthese two model ∞-categories.
(Note that the functor D→ ho(D) is the unit of the adjunctionCat∞ Cat1, but it is not itself an adjoint.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 383: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/383.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In the case that D is actually a 1-category, this recovers theclassical definition of a model structure on D.
In fact, if D is a model ∞-category and ho(D) has whateverco/limits one would like to require (though this is quite rare!), thena model structure on D descends to a model structure on ho(D).
(The two-out-of-three, retract, and factorization axioms in D areeven verifiable in ho(D), and the lifting axiom descends as well.)
On the other hand, since in general there is a complicated interplaybetween co/limits in D and co/limits in ho(D), we are nervousabout drawing any conclusions about the relationship betweenthese two model ∞-categories.
(Note that the functor D→ ho(D) is the unit of the adjunctionCat∞ Cat1, but it is not itself an adjoint.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 384: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/384.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In the case that D is actually a 1-category, this recovers theclassical definition of a model structure on D.
In fact, if D is a model ∞-category and ho(D) has whateverco/limits one would like to require (though this is quite rare!), thena model structure on D descends to a model structure on ho(D).
(The two-out-of-three, retract, and factorization axioms in D areeven verifiable in ho(D), and the lifting axiom descends as well.)
On the other hand, since in general there is a complicated interplaybetween co/limits in D and co/limits in ho(D), we are nervousabout drawing any conclusions about the relationship betweenthese two model ∞-categories.
(Note that the functor D→ ho(D) is the unit of the adjunctionCat∞ Cat1, but it is not itself an adjoint.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 385: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/385.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In the case that D is actually a 1-category, this recovers theclassical definition of a model structure on D.
In fact, if D is a model ∞-category and ho(D) has whateverco/limits one would like to require (though this is quite rare!), thena model structure on D descends to a model structure on ho(D).
(The two-out-of-three, retract, and factorization axioms in D areeven verifiable in ho(D), and the lifting axiom descends as well.)
On the other hand, since in general there is a complicated interplaybetween co/limits in D and co/limits in ho(D),
we are nervousabout drawing any conclusions about the relationship betweenthese two model ∞-categories.
(Note that the functor D→ ho(D) is the unit of the adjunctionCat∞ Cat1, but it is not itself an adjoint.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 386: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/386.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In the case that D is actually a 1-category, this recovers theclassical definition of a model structure on D.
In fact, if D is a model ∞-category and ho(D) has whateverco/limits one would like to require (though this is quite rare!), thena model structure on D descends to a model structure on ho(D).
(The two-out-of-three, retract, and factorization axioms in D areeven verifiable in ho(D), and the lifting axiom descends as well.)
On the other hand, since in general there is a complicated interplaybetween co/limits in D and co/limits in ho(D), we are nervousabout drawing any conclusions about the relationship betweenthese two model ∞-categories.
(Note that the functor D→ ho(D) is the unit of the adjunctionCat∞ Cat1, but it is not itself an adjoint.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 387: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/387.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In the case that D is actually a 1-category, this recovers theclassical definition of a model structure on D.
In fact, if D is a model ∞-category and ho(D) has whateverco/limits one would like to require (though this is quite rare!), thena model structure on D descends to a model structure on ho(D).
(The two-out-of-three, retract, and factorization axioms in D areeven verifiable in ho(D), and the lifting axiom descends as well.)
On the other hand, since in general there is a complicated interplaybetween co/limits in D and co/limits in ho(D), we are nervousabout drawing any conclusions about the relationship betweenthese two model ∞-categories.
(Note that the functor D→ ho(D) is the unit of the adjunctionCat∞ Cat1, but it is not itself an adjoint.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 388: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/388.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We can make all the same constructions as in the classical case.
For instance, any ∞-category D can be given the trivial modelstructure, in which we set W = D' and C = F = D. We can usethis to define a Reedy model structure on sD. (Of course, this willjust be the trivial model structure on sD.)
More interestingly, given a small full subcategory of D that’sclosed under finite coproducts, we can use it to equip sD with anE 2-model structure!
These are both a lot cleaner than their classical counterparts,because everything is already homotopically well-behaved in the∞-category D.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 389: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/389.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We can make all the same constructions as in the classical case.
For instance, any ∞-category D can be given the trivial modelstructure, in which we set W = D' and C = F = D.
We can usethis to define a Reedy model structure on sD. (Of course, this willjust be the trivial model structure on sD.)
More interestingly, given a small full subcategory of D that’sclosed under finite coproducts, we can use it to equip sD with anE 2-model structure!
These are both a lot cleaner than their classical counterparts,because everything is already homotopically well-behaved in the∞-category D.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 390: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/390.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We can make all the same constructions as in the classical case.
For instance, any ∞-category D can be given the trivial modelstructure, in which we set W = D' and C = F = D. We can usethis to define a Reedy model structure on sD.
(Of course, this willjust be the trivial model structure on sD.)
More interestingly, given a small full subcategory of D that’sclosed under finite coproducts, we can use it to equip sD with anE 2-model structure!
These are both a lot cleaner than their classical counterparts,because everything is already homotopically well-behaved in the∞-category D.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 391: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/391.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We can make all the same constructions as in the classical case.
For instance, any ∞-category D can be given the trivial modelstructure, in which we set W = D' and C = F = D. We can usethis to define a Reedy model structure on sD. (Of course, this willjust be the trivial model structure on sD.)
More interestingly, given a small full subcategory of D that’sclosed under finite coproducts, we can use it to equip sD with anE 2-model structure!
These are both a lot cleaner than their classical counterparts,because everything is already homotopically well-behaved in the∞-category D.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 392: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/392.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We can make all the same constructions as in the classical case.
For instance, any ∞-category D can be given the trivial modelstructure, in which we set W = D' and C = F = D. We can usethis to define a Reedy model structure on sD. (Of course, this willjust be the trivial model structure on sD.)
More interestingly, given a small full subcategory of D that’sclosed under finite coproducts,
we can use it to equip sD with anE 2-model structure!
These are both a lot cleaner than their classical counterparts,because everything is already homotopically well-behaved in the∞-category D.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 393: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/393.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We can make all the same constructions as in the classical case.
For instance, any ∞-category D can be given the trivial modelstructure, in which we set W = D' and C = F = D. We can usethis to define a Reedy model structure on sD. (Of course, this willjust be the trivial model structure on sD.)
More interestingly, given a small full subcategory of D that’sclosed under finite coproducts, we can use it to equip sD with anE 2-model structure!
These are both a lot cleaner than their classical counterparts,because everything is already homotopically well-behaved in the∞-category D.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 394: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/394.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We can make all the same constructions as in the classical case.
For instance, any ∞-category D can be given the trivial modelstructure, in which we set W = D' and C = F = D. We can usethis to define a Reedy model structure on sD. (Of course, this willjust be the trivial model structure on sD.)
More interestingly, given a small full subcategory of D that’sclosed under finite coproducts, we can use it to equip sD with anE 2-model structure!
These are both a lot cleaner than their classical counterparts,because everything is already homotopically well-behaved in the∞-category D.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 395: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/395.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In model 1-categories, one simply quotients by relations to obtaina set of “homotopy classes of maps”.
Of course, this goes against the core thesis of higher categorytheory.
Instead, we will remember the relations, and we will build theminto a space of “homotopy classes of maps”.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 396: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/396.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In model 1-categories, one simply quotients by relations to obtaina set of “homotopy classes of maps”.
Of course, this goes against the core thesis of higher categorytheory.
Instead, we will remember the relations, and we will build theminto a space of “homotopy classes of maps”.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 397: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/397.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In model 1-categories, one simply quotients by relations to obtaina set of “homotopy classes of maps”.
Of course, this goes against the core thesis of higher categorytheory.
Instead, we will remember the relations,
and we will build theminto a space of “homotopy classes of maps”.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 398: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/398.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In model 1-categories, one simply quotients by relations to obtaina set of “homotopy classes of maps”.
Of course, this goes against the core thesis of higher categorytheory.
Instead, we will remember the relations, and we will build theminto a space of “homotopy classes of maps”.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 399: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/399.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that classically, the homotopy relations are defined usingcylinders and paths.
Given an object d ∈ D, a cylinder is a factorization
d t d cyl(d)≈−→ d
of the fold map, and a path is a factorization
d≈−→ path(d) d × d
of the diagonal map.
If D is now a model ∞-category, then a cylinder object will be acertain cosimplicial object cyl•(d) ∈ cD, and a path object will bea certain simplicial object path•(d) ∈ sD.
These should be thought of as a “cofibrant W-cohypercover” anda “fibrant W-hypercover”, respectively. They also very closelyresemble Dwyer–Kan’s co/simplicial resolutions.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 400: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/400.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that classically, the homotopy relations are defined usingcylinders and paths.
Given an object d ∈ D, a cylinder is a factorization
d t d cyl(d)≈−→ d
of the fold map,
and a path is a factorization
d≈−→ path(d) d × d
of the diagonal map.
If D is now a model ∞-category, then a cylinder object will be acertain cosimplicial object cyl•(d) ∈ cD, and a path object will bea certain simplicial object path•(d) ∈ sD.
These should be thought of as a “cofibrant W-cohypercover” anda “fibrant W-hypercover”, respectively. They also very closelyresemble Dwyer–Kan’s co/simplicial resolutions.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 401: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/401.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that classically, the homotopy relations are defined usingcylinders and paths.
Given an object d ∈ D, a cylinder is a factorization
d t d cyl(d)≈−→ d
of the fold map, and a path is a factorization
d≈−→ path(d) d × d
of the diagonal map.
If D is now a model ∞-category, then a cylinder object will be acertain cosimplicial object cyl•(d) ∈ cD, and a path object will bea certain simplicial object path•(d) ∈ sD.
These should be thought of as a “cofibrant W-cohypercover” anda “fibrant W-hypercover”, respectively. They also very closelyresemble Dwyer–Kan’s co/simplicial resolutions.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 402: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/402.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that classically, the homotopy relations are defined usingcylinders and paths.
Given an object d ∈ D, a cylinder is a factorization
d t d cyl(d)≈−→ d
of the fold map, and a path is a factorization
d≈−→ path(d) d × d
of the diagonal map.
If D is now a model ∞-category, then a cylinder object will be acertain cosimplicial object cyl•(d) ∈ cD,
and a path object will bea certain simplicial object path•(d) ∈ sD.
These should be thought of as a “cofibrant W-cohypercover” anda “fibrant W-hypercover”, respectively. They also very closelyresemble Dwyer–Kan’s co/simplicial resolutions.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 403: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/403.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that classically, the homotopy relations are defined usingcylinders and paths.
Given an object d ∈ D, a cylinder is a factorization
d t d cyl(d)≈−→ d
of the fold map, and a path is a factorization
d≈−→ path(d) d × d
of the diagonal map.
If D is now a model ∞-category, then a cylinder object will be acertain cosimplicial object cyl•(d) ∈ cD, and a path object will bea certain simplicial object path•(d) ∈ sD.
These should be thought of as a “cofibrant W-cohypercover” anda “fibrant W-hypercover”, respectively. They also very closelyresemble Dwyer–Kan’s co/simplicial resolutions.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 404: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/404.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that classically, the homotopy relations are defined usingcylinders and paths.
Given an object d ∈ D, a cylinder is a factorization
d t d cyl(d)≈−→ d
of the fold map, and a path is a factorization
d≈−→ path(d) d × d
of the diagonal map.
If D is now a model ∞-category, then a cylinder object will be acertain cosimplicial object cyl•(d) ∈ cD, and a path object will bea certain simplicial object path•(d) ∈ sD.
These should be thought of as a “cofibrant W-cohypercover” anda “fibrant W-hypercover”, respectively.
They also very closelyresemble Dwyer–Kan’s co/simplicial resolutions.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 405: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/405.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that classically, the homotopy relations are defined usingcylinders and paths.
Given an object d ∈ D, a cylinder is a factorization
d t d cyl(d)≈−→ d
of the fold map, and a path is a factorization
d≈−→ path(d) d × d
of the diagonal map.
If D is now a model ∞-category, then a cylinder object will be acertain cosimplicial object cyl•(d) ∈ cD, and a path object will bea certain simplicial object path•(d) ∈ sD.
These should be thought of as a “cofibrant W-cohypercover” anda “fibrant W-hypercover”, respectively. They also very closelyresemble Dwyer–Kan’s co/simplicial resolutions.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 406: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/406.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, suppose D is a model ∞-category,
d1, d2 ∈ D, and that wehave chosen some cyl•(d1) and path•(d2).
Then, we define the space of left homotopy classes of maps to be
homl∼D(d1, d2) =
∣∣∣homlwD(cyl•(d1), d2)
∣∣∣ ,and the space of right homotopy classes of maps to be
homr∼D(d1, d2) =
∣∣∣homlwD(d1, path•(d2))
∣∣∣ .(One of the keys to passing from 1-topos theory to ∞-topos theoryis replacing quotients by equivalence relations with geometricrealizations of simplicial objects.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 407: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/407.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, suppose D is a model ∞-category, d1, d2 ∈ D, and that wehave chosen some cyl•(d1) and path•(d2).
Then, we define the space of left homotopy classes of maps to be
homl∼D(d1, d2) =
∣∣∣homlwD(cyl•(d1), d2)
∣∣∣ ,and the space of right homotopy classes of maps to be
homr∼D(d1, d2) =
∣∣∣homlwD(d1, path•(d2))
∣∣∣ .(One of the keys to passing from 1-topos theory to ∞-topos theoryis replacing quotients by equivalence relations with geometricrealizations of simplicial objects.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 408: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/408.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, suppose D is a model ∞-category, d1, d2 ∈ D, and that wehave chosen some cyl•(d1) and path•(d2).
Then, we define the space of left homotopy classes of maps to be
homl∼D(d1, d2) =
∣∣∣homlwD(cyl•(d1), d2)
∣∣∣ ,
and the space of right homotopy classes of maps to be
homr∼D(d1, d2) =
∣∣∣homlwD(d1, path•(d2))
∣∣∣ .(One of the keys to passing from 1-topos theory to ∞-topos theoryis replacing quotients by equivalence relations with geometricrealizations of simplicial objects.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 409: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/409.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, suppose D is a model ∞-category, d1, d2 ∈ D, and that wehave chosen some cyl•(d1) and path•(d2).
Then, we define the space of left homotopy classes of maps to be
homl∼D(d1, d2) =
∣∣∣homlwD(cyl•(d1), d2)
∣∣∣ ,and the space of right homotopy classes of maps to be
homr∼D(d1, d2) =
∣∣∣homlwD(d1, path•(d2))
∣∣∣ .
(One of the keys to passing from 1-topos theory to ∞-topos theoryis replacing quotients by equivalence relations with geometricrealizations of simplicial objects.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 410: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/410.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, suppose D is a model ∞-category, d1, d2 ∈ D, and that wehave chosen some cyl•(d1) and path•(d2).
Then, we define the space of left homotopy classes of maps to be
homl∼D(d1, d2) =
∣∣∣homlwD(cyl•(d1), d2)
∣∣∣ ,and the space of right homotopy classes of maps to be
homr∼D(d1, d2) =
∣∣∣homlwD(d1, path•(d2))
∣∣∣ .(One of the keys to passing from 1-topos theory to ∞-topos theory
is replacing quotients by equivalence relations with geometricrealizations of simplicial objects.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 411: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/411.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, suppose D is a model ∞-category, d1, d2 ∈ D, and that wehave chosen some cyl•(d1) and path•(d2).
Then, we define the space of left homotopy classes of maps to be
homl∼D(d1, d2) =
∣∣∣homlwD(cyl•(d1), d2)
∣∣∣ ,and the space of right homotopy classes of maps to be
homr∼D(d1, d2) =
∣∣∣homlwD(d1, path•(d2))
∣∣∣ .(One of the keys to passing from 1-topos theory to ∞-topos theoryis replacing quotients by equivalence relations
with geometricrealizations of simplicial objects.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 412: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/412.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Now, suppose D is a model ∞-category, d1, d2 ∈ D, and that wehave chosen some cyl•(d1) and path•(d2).
Then, we define the space of left homotopy classes of maps to be
homl∼D(d1, d2) =
∣∣∣homlwD(cyl•(d1), d2)
∣∣∣ ,and the space of right homotopy classes of maps to be
homr∼D(d1, d2) =
∣∣∣homlwD(d1, path•(d2))
∣∣∣ .(One of the keys to passing from 1-topos theory to ∞-topos theoryis replacing quotients by equivalence relations with geometricrealizations of simplicial objects.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 413: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/413.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, the main result is that when d1 is cofibrant and d2 isfibrant,
then for any cyl•(d1) and path•(d2),
homl∼D(d1, d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ homr∼D(d1, d2)
homD[W−1](d1, d2)
∼
∼
∼
(where ‖−‖ denotes the colimit of a bisimplicial space).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 414: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/414.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, the main result is that when d1 is cofibrant and d2 isfibrant, then for any cyl•(d1) and path•(d2),
homl∼D(d1, d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ homr∼D(d1, d2)
homD[W−1](d1, d2)
∼
∼
∼
(where ‖−‖ denotes the colimit of a bisimplicial space).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 415: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/415.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, the main result is that when d1 is cofibrant and d2 isfibrant, then for any cyl•(d1) and path•(d2),
homl∼D(d1, d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ homr∼D(d1, d2)
homD[W−1](d1, d2)
∼
∼
∼
(where ‖−‖ denotes the colimit of a bisimplicial space).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 416: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/416.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Proving for instance that the right horizontal map
homr∼D(d1, d2) ' hom
r∼D(cyl0(d1), d2)
colimn homr∼D(cyln(d1), d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ =
↓
is an equivalence
amounts to showing that if d≈ d ′ is an acyclic
cofibration, then the induced map
homlwD(d ′, path•(d2))→ homlw
D(d , path•(d2))
in sS yields a weak equivalence of homr∼D-spaces upon geometric
realization. (When d1 is cofibrant, the coface maps of cyl•(d1) areall acyclic cofibrations.)
In fact, this is not an equivalence in sS itself...but it’s anE 2-equivalence! (To prove this, use the lifting axiom for the acyclic
cofibrations Sn ⊗ d≈ Sn ⊗ d ′ against the various fibrations
contained in path•(d2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 417: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/417.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Proving for instance that the right horizontal map
homr∼D(d1, d2) ' hom
r∼D(cyl0(d1), d2)
colimn homr∼D(cyln(d1), d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ =
↓
is an equivalence amounts to showing that if d≈ d ′ is an acyclic
cofibration,
then the induced map
homlwD(d ′, path•(d2))→ homlw
D(d , path•(d2))
in sS yields a weak equivalence of homr∼D-spaces upon geometric
realization. (When d1 is cofibrant, the coface maps of cyl•(d1) areall acyclic cofibrations.)
In fact, this is not an equivalence in sS itself...but it’s anE 2-equivalence! (To prove this, use the lifting axiom for the acyclic
cofibrations Sn ⊗ d≈ Sn ⊗ d ′ against the various fibrations
contained in path•(d2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 418: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/418.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Proving for instance that the right horizontal map
homr∼D(d1, d2) ' hom
r∼D(cyl0(d1), d2)
colimn homr∼D(cyln(d1), d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ =
↓
is an equivalence amounts to showing that if d≈ d ′ is an acyclic
cofibration, then the induced map
homlwD(d ′, path•(d2))→ homlw
D(d , path•(d2))
in sS yields a weak equivalence of homr∼D-spaces upon geometric
realization.
(When d1 is cofibrant, the coface maps of cyl•(d1) areall acyclic cofibrations.)
In fact, this is not an equivalence in sS itself...but it’s anE 2-equivalence! (To prove this, use the lifting axiom for the acyclic
cofibrations Sn ⊗ d≈ Sn ⊗ d ′ against the various fibrations
contained in path•(d2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 419: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/419.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Proving for instance that the right horizontal map
homr∼D(d1, d2) ' hom
r∼D(cyl0(d1), d2)
colimn homr∼D(cyln(d1), d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ =
↓
is an equivalence amounts to showing that if d≈ d ′ is an acyclic
cofibration, then the induced map
homlwD(d ′, path•(d2))→ homlw
D(d , path•(d2))
in sS yields a weak equivalence of homr∼D-spaces upon geometric
realization. (When d1 is cofibrant, the coface maps of cyl•(d1) areall acyclic cofibrations.)
In fact, this is not an equivalence in sS itself...but it’s anE 2-equivalence! (To prove this, use the lifting axiom for the acyclic
cofibrations Sn ⊗ d≈ Sn ⊗ d ′ against the various fibrations
contained in path•(d2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 420: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/420.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Proving for instance that the right horizontal map
homr∼D(d1, d2) ' hom
r∼D(cyl0(d1), d2)
colimn homr∼D(cyln(d1), d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ =
↓
is an equivalence amounts to showing that if d≈ d ′ is an acyclic
cofibration, then the induced map
homlwD(d ′, path•(d2))→ homlw
D(d , path•(d2))
in sS yields a weak equivalence of homr∼D-spaces upon geometric
realization. (When d1 is cofibrant, the coface maps of cyl•(d1) areall acyclic cofibrations.)
In fact, this is not an equivalence in sS itself...
but it’s anE 2-equivalence! (To prove this, use the lifting axiom for the acyclic
cofibrations Sn ⊗ d≈ Sn ⊗ d ′ against the various fibrations
contained in path•(d2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 421: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/421.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Proving for instance that the right horizontal map
homr∼D(d1, d2) ' hom
r∼D(cyl0(d1), d2)
colimn homr∼D(cyln(d1), d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ =
↓
is an equivalence amounts to showing that if d≈ d ′ is an acyclic
cofibration, then the induced map
homlwD(d ′, path•(d2))→ homlw
D(d , path•(d2))
in sS yields a weak equivalence of homr∼D-spaces upon geometric
realization. (When d1 is cofibrant, the coface maps of cyl•(d1) areall acyclic cofibrations.)
In fact, this is not an equivalence in sS itself...but it’s anE 2-equivalence!
(To prove this, use the lifting axiom for the acyclic
cofibrations Sn ⊗ d≈ Sn ⊗ d ′ against the various fibrations
contained in path•(d2).)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 422: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/422.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Proving for instance that the right horizontal map
homr∼D(d1, d2) ' hom
r∼D(cyl0(d1), d2)
colimn homr∼D(cyln(d1), d2)
∥∥homlwD(cyl•(d1), path•(d2))
∥∥ =
↓
is an equivalence amounts to showing that if d≈ d ′ is an acyclic
cofibration, then the induced map
homlwD(d ′, path•(d2))→ homlw
D(d , path•(d2))
in sS yields a weak equivalence of homr∼D-spaces upon geometric
realization. (When d1 is cofibrant, the coface maps of cyl•(d1) areall acyclic cofibrations.)
In fact, this is not an equivalence in sS itself...but it’s anE 2-equivalence! (To prove this, use the lifting axiom for the acyclic
cofibrations Sn ⊗ d≈ Sn ⊗ d ′ against the various fibrations
contained in path•(d2).)Aaron Mazel-Gee GHOsT for∞-categories
![Page 423: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/423.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
On the other hand, to show that the map∥∥∥homlwD(cyl•(d1), path•(d2))
∥∥∥→ homD[W−1](d1, d2)
is an equivalence, we first need to have access to D[W−1].
Our T-shaped diagram is essentially lifted from Dwyer–Kan’soriginal proof that the hammock construction has the correcthomotopy type, only the hammock construction is now replaced bythe hom-space homD[W−1](d1, d2).
In fact, there’s an extremely elegant method for inverting a class ofmaps in an ∞-category – which looks a whole lot like a“high-dimensional hammock construction” – using Rezk’s theoryof complete Segal spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 424: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/424.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
On the other hand, to show that the map∥∥∥homlwD(cyl•(d1), path•(d2))
∥∥∥→ homD[W−1](d1, d2)
is an equivalence, we first need to have access to D[W−1].
Our T-shaped diagram is essentially lifted from Dwyer–Kan’soriginal proof that the hammock construction has the correcthomotopy type,
only the hammock construction is now replaced bythe hom-space homD[W−1](d1, d2).
In fact, there’s an extremely elegant method for inverting a class ofmaps in an ∞-category – which looks a whole lot like a“high-dimensional hammock construction” – using Rezk’s theoryof complete Segal spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 425: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/425.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
On the other hand, to show that the map∥∥∥homlwD(cyl•(d1), path•(d2))
∥∥∥→ homD[W−1](d1, d2)
is an equivalence, we first need to have access to D[W−1].
Our T-shaped diagram is essentially lifted from Dwyer–Kan’soriginal proof that the hammock construction has the correcthomotopy type, only the hammock construction is now replaced bythe hom-space homD[W−1](d1, d2).
In fact, there’s an extremely elegant method for inverting a class ofmaps in an ∞-category – which looks a whole lot like a“high-dimensional hammock construction” – using Rezk’s theoryof complete Segal spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 426: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/426.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
On the other hand, to show that the map∥∥∥homlwD(cyl•(d1), path•(d2))
∥∥∥→ homD[W−1](d1, d2)
is an equivalence, we first need to have access to D[W−1].
Our T-shaped diagram is essentially lifted from Dwyer–Kan’soriginal proof that the hammock construction has the correcthomotopy type, only the hammock construction is now replaced bythe hom-space homD[W−1](d1, d2).
In fact, there’s an extremely elegant method for inverting a class ofmaps in an ∞-category –
which looks a whole lot like a“high-dimensional hammock construction” – using Rezk’s theoryof complete Segal spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 427: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/427.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
On the other hand, to show that the map∥∥∥homlwD(cyl•(d1), path•(d2))
∥∥∥→ homD[W−1](d1, d2)
is an equivalence, we first need to have access to D[W−1].
Our T-shaped diagram is essentially lifted from Dwyer–Kan’soriginal proof that the hammock construction has the correcthomotopy type, only the hammock construction is now replaced bythe hom-space homD[W−1](d1, d2).
In fact, there’s an extremely elegant method for inverting a class ofmaps in an ∞-category – which looks a whole lot like a“high-dimensional hammock construction” –
using Rezk’s theoryof complete Segal spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 428: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/428.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
On the other hand, to show that the map∥∥∥homlwD(cyl•(d1), path•(d2))
∥∥∥→ homD[W−1](d1, d2)
is an equivalence, we first need to have access to D[W−1].
Our T-shaped diagram is essentially lifted from Dwyer–Kan’soriginal proof that the hammock construction has the correcthomotopy type, only the hammock construction is now replaced bythe hom-space homD[W−1](d1, d2).
In fact, there’s an extremely elegant method for inverting a class ofmaps in an ∞-category – which looks a whole lot like a“high-dimensional hammock construction” – using Rezk’s theoryof complete Segal spaces.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 429: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/429.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that the complete Segal space functor CSS : RelCat∞ → sStakes a relative ∞-category (D,W) and returns the simplicialspace CSS(D,W)• given by
CSS(D,W)n =(
Fun([n],D)W)gpd
.
That is, to construct CSS(D,W)n, we take the ∞-category ofchains of n composable arrows in D, pass to its full subcategory onthose natural transformations whose components all lie in W, andthen groupoid-complete the result.
Then, we have the simple formula
homD[W−1](d1, d2) = lim
CSS(D,W)1
pt CSS(D,W)0 × CSS(D,W)0
(δ0, δ1)
(d1, d2)
.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 430: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/430.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that the complete Segal space functor CSS : RelCat∞ → sStakes a relative ∞-category (D,W) and returns the simplicialspace CSS(D,W)• given by
CSS(D,W)n =(
Fun([n],D)W)gpd
.
That is, to construct CSS(D,W)n,
we take the ∞-category ofchains of n composable arrows in D, pass to its full subcategory onthose natural transformations whose components all lie in W, andthen groupoid-complete the result.
Then, we have the simple formula
homD[W−1](d1, d2) = lim
CSS(D,W)1
pt CSS(D,W)0 × CSS(D,W)0
(δ0, δ1)
(d1, d2)
.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 431: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/431.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that the complete Segal space functor CSS : RelCat∞ → sStakes a relative ∞-category (D,W) and returns the simplicialspace CSS(D,W)• given by
CSS(D,W)n =(
Fun([n],D)W)gpd
.
That is, to construct CSS(D,W)n, we take the ∞-category ofchains of n composable arrows in D,
pass to its full subcategory onthose natural transformations whose components all lie in W, andthen groupoid-complete the result.
Then, we have the simple formula
homD[W−1](d1, d2) = lim
CSS(D,W)1
pt CSS(D,W)0 × CSS(D,W)0
(δ0, δ1)
(d1, d2)
.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 432: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/432.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that the complete Segal space functor CSS : RelCat∞ → sStakes a relative ∞-category (D,W) and returns the simplicialspace CSS(D,W)• given by
CSS(D,W)n =(
Fun([n],D)W)gpd
.
That is, to construct CSS(D,W)n, we take the ∞-category ofchains of n composable arrows in D, pass to its full subcategory onthose natural transformations whose components all lie in W,
andthen groupoid-complete the result.
Then, we have the simple formula
homD[W−1](d1, d2) = lim
CSS(D,W)1
pt CSS(D,W)0 × CSS(D,W)0
(δ0, δ1)
(d1, d2)
.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 433: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/433.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that the complete Segal space functor CSS : RelCat∞ → sStakes a relative ∞-category (D,W) and returns the simplicialspace CSS(D,W)• given by
CSS(D,W)n =(
Fun([n],D)W)gpd
.
That is, to construct CSS(D,W)n, we take the ∞-category ofchains of n composable arrows in D, pass to its full subcategory onthose natural transformations whose components all lie in W, andthen groupoid-complete the result.
Then, we have the simple formula
homD[W−1](d1, d2) = lim
CSS(D,W)1
pt CSS(D,W)0 × CSS(D,W)0
(δ0, δ1)
(d1, d2)
.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 434: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/434.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Recall that the complete Segal space functor CSS : RelCat∞ → sStakes a relative ∞-category (D,W) and returns the simplicialspace CSS(D,W)• given by
CSS(D,W)n =(
Fun([n],D)W)gpd
.
That is, to construct CSS(D,W)n, we take the ∞-category ofchains of n composable arrows in D, pass to its full subcategory onthose natural transformations whose components all lie in W, andthen groupoid-complete the result.
Then, we have the simple formula
homD[W−1](d1, d2) = lim
CSS(D,W)1
pt CSS(D,W)0 × CSS(D,W)0
(δ0, δ1)
(d1, d2)
.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 435: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/435.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This construction makes it clear that homD[W−1](d1, d2) is indeedsome sort of “space of zig-zags”, subject to the usual relations.
Then, just as in the classical case, we can use our model structure
axioms to reduce to the space of zig-zags of the form≈→
≈.
That is, the model structure once again buys us a three-arrowcalculus for computing mapping spaces in D[W−1].
From here, we can mimic the analogous classical proof byDwyer–Kan to show that we have an equivalence∥∥∥homlw
D(cyl•(d1), path•(d2))∥∥∥ ∼−→ homD[W−1](d1, d2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 436: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/436.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This construction makes it clear that homD[W−1](d1, d2) is indeedsome sort of “space of zig-zags”, subject to the usual relations.
Then, just as in the classical case, we can use our model structure
axioms to reduce to the space of zig-zags of the form≈→
≈.
That is, the model structure once again buys us a three-arrowcalculus for computing mapping spaces in D[W−1].
From here, we can mimic the analogous classical proof byDwyer–Kan to show that we have an equivalence∥∥∥homlw
D(cyl•(d1), path•(d2))∥∥∥ ∼−→ homD[W−1](d1, d2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 437: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/437.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This construction makes it clear that homD[W−1](d1, d2) is indeedsome sort of “space of zig-zags”, subject to the usual relations.
Then, just as in the classical case, we can use our model structure
axioms to reduce to the space of zig-zags of the form≈→
≈.
That is, the model structure once again buys us a three-arrowcalculus for computing mapping spaces in D[W−1].
From here, we can mimic the analogous classical proof byDwyer–Kan to show that we have an equivalence∥∥∥homlw
D(cyl•(d1), path•(d2))∥∥∥ ∼−→ homD[W−1](d1, d2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 438: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/438.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This construction makes it clear that homD[W−1](d1, d2) is indeedsome sort of “space of zig-zags”, subject to the usual relations.
Then, just as in the classical case, we can use our model structure
axioms to reduce to the space of zig-zags of the form≈→
≈.
That is, the model structure once again buys us a three-arrowcalculus for computing mapping spaces in D[W−1].
From here, we can mimic the analogous classical proof byDwyer–Kan to show that we have an equivalence∥∥∥homlw
D(cyl•(d1), path•(d2))∥∥∥ ∼−→ homD[W−1](d1, d2).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 439: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/439.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory; we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories: that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial : sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S. (This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 440: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/440.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory;
we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories: that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial : sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S. (This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 441: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/441.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory; we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories: that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial : sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S. (This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 442: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/442.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory; we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories:
that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial : sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S. (This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 443: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/443.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory; we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories: that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial : sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S. (This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 444: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/444.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory; we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories: that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial :
sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S. (This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 445: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/445.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory; we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories: that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial : sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S.
(This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 446: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/446.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Of course, there are the notions of a Quillen adjunction of model∞-categories and of its derived adjunction.
As we’ve indicated, these play a pivotal role in setting up an∞-categorical Goerss–Hopkins obstruction theory; we’ll display thediagram again in a moment.
However, we first mention that we will have a need for simplicialmodel ∞-categories: that is, model ∞-categories with a tensoringover sSet which is compatible with the localization sSet→ S.
In fact, sCE2
GEC
is even simplicio-spatial : sC is naturally tensored
over sS, and the E 2GEC
-model structure is compatible with the
colimit functor sS→ S. (This is of course a left localization, andhence determines a model structure on sS where all objects arecofibrant.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 447: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/447.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
And now once again, for our collective viewing pleasure, here isThe Diagram.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 448: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/448.jpg)
C
AlgO(C)
PΣ(GEC )
sCE2
GEC
PΣ(T (GEC ))
AlgT (sC)E2
GEC
LE(AlgO(C))AlgO(LE(C)) AlgΦ(A) PδΣ(Φ(GA))
LE(AlgT (sC)) PΣ(TE(Aproj))
AlgT (sC)π∗E∗ AlgTE(sA)E2
GAAlgTE
(sA)Quillen
∼
π 0
LE(C) A PδΣ(GA)
LE(sC) PΣ(GA)
sCπ∗E∗ sAE2GA
sAQuillen
Mod0(A)
PδΣ(GEC )
PδΣ(T (GEC ))
PΣ(Aproj)
π0,∗ π≥1
Modπ0,∗(PδΣ(GEC ))
π≥
1,∗
⊥ ⊥
d⊥
Q⊥ Q⊥
⊥ ⊥
d⊥ d⊥
Q⊥ Q⊥
⊥ ⊥ ⊥ ⊥
d⊥ d⊥ d⊥
Q⊥
Q⊥
Q⊥
Q⊥
⊥
![Page 449: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/449.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Applications and generalizations
Of course, as we’ve hopefully made clear, this recovers the existingobstruction theories.
But now, this can be used in any presentable ∞-category!
As a sample application, let’s attempt to reimagine theconstruction of tmf .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 450: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/450.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Applications and generalizations
Of course, as we’ve hopefully made clear, this recovers the existingobstruction theories.
But now, this can be used in any presentable ∞-category!
As a sample application, let’s attempt to reimagine theconstruction of tmf .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 451: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/451.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Applications and generalizations
Of course, as we’ve hopefully made clear, this recovers the existingobstruction theories.
But now, this can be used in any presentable ∞-category!
As a sample application, let’s attempt to reimagine theconstruction of tmf .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 452: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/452.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Applications and generalizations
Of course, as we’ve hopefully made clear, this recovers the existingobstruction theories.
But now, this can be used in any presentable ∞-category!
As a sample application, let’s attempt to reimagine theconstruction of tmf .
Aaron Mazel-Gee GHOsT for∞-categories
![Page 453: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/453.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine, and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case. Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory – an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 454: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/454.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine, and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case. Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory – an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 455: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/455.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine,
and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case. Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory – an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 456: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/456.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine, and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case. Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory – an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 457: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/457.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine, and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case. Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory – an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 458: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/458.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine, and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case.
Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory – an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 459: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/459.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine, and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case. Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory –
an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 460: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/460.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
In Behrens’s Notes on the construction of tmf , for p > 2 he
constructs the K (1)-local sheaf OtopK(1) over M
ordell,p “one spectrum
at a time”:
(1) first he constructs the global sections tmfK(1),
(2) then he constructs the sections over any etale affine, and
(3) finally he shows that these patch together in an essentiallyunique way.
(The first step is because the obstructions in the category oftmfK(1)-algebras vanish, but those in the category of S-algebras donot.)
(At p = 2, the obstructions don’t vanish in either case. Instead, heconstructs a handicrafted and non-functorial “cells and disks”obstruction theory – an Eckmann–Hilton dual of GHOsT.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 461: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/461.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 462: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/462.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X ,
the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 463: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/463.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 464: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/464.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure,
inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 465: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/465.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 466: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/466.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff,
it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 467: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/467.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 468: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/468.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:
by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 469: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/469.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This is sufficient, because he shows that:
For any Deligne–Mumford stack X , the inclusioni : Xet,aff →Xet induces a Quillen equivalence
i∗ : Fun(Xet, Sp)Jardine Fun(Xet,aff, Sp)Jardine : i∗
of presheaves of spectra with the Jardine model structure, inwhich the fibrant objects are the sheaves (i.e. hypersheaves).
To construct a sheaf of spectra on Xet,aff, it suffices toconstruct a presheaf of spectra there whose sectionwisehomotopy groups form a quasicoherent sheaf.
This latter condition is automatic for elliptic cohomology theories:by fiat, Otop is locally required to have that π2nOtop ∼= ω⊗n andπ2n+1Otop = 0.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 470: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/470.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff, so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp), do obstruction theory for the stalks, and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 471: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/471.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff,
so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp), do obstruction theory for the stalks, and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 472: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/472.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff, so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp), do obstruction theory for the stalks, and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 473: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/473.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff, so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp), do obstruction theory for the stalks, and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 474: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/474.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff, so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp), do obstruction theory for the stalks, and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 475: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/475.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff, so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp),
do obstruction theory for the stalks, and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 476: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/476.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff, so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp), do obstruction theory for the stalks,
and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 477: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/477.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
So, we can hope to construct OtopK(1) (or maybe even Otop!) in one
swoop by doing obstruction theory in C = Fun(Xet,aff, Sp).
However, taking homotopy or homology will now land not in setsbut in presheaves of sets on Xet,aff, so this requires an enrichedobstruction theory.
This is in the works!
Of course, we make no claims that this will be any computationallyeasier than the existing obstruction-theoretic approach.
Another probably intractable method for constructing tmf wouldbe to put a Jardine model structure on the ∞-categoryFun(Xet, Sp), do obstruction theory for the stalks, and thenfibrantly replace the result.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 478: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/478.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
But really, enriching the obstruction theory should be purely formal.
For instance, we should obtain Andre–Quillen cohomologyΩ-spectra in our enriching ∞-category.
Then, the obstructions will live in Andre–Quillen cohomologygroup objects in its subcategory of discrete objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 479: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/479.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
But really, enriching the obstruction theory should be purely formal.
For instance, we should obtain Andre–Quillen cohomologyΩ-spectra in our enriching ∞-category.
Then, the obstructions will live in Andre–Quillen cohomologygroup objects in its subcategory of discrete objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 480: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/480.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
But really, enriching the obstruction theory should be purely formal.
For instance, we should obtain Andre–Quillen cohomologyΩ-spectra in our enriching ∞-category.
Then, the obstructions will live in Andre–Quillen cohomologygroup objects in its subcategory of discrete objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 481: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/481.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 482: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/482.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces),
there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 483: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/483.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 484: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/484.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra,
andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 485: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/485.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 486: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/486.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra,
and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 487: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/487.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 488: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/488.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 489: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/489.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups,
so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 490: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/490.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
This should have other applications, too.
For instance, if we care about motivic spectra (which are enrichedin motivic spaces), there are two things we can do.
We can do enriched obstruction theory in motivic spectra, andobtain obstructions in “motivic groups”.
Or, we can stop caring about motivic spectra, and restrict ourattention to their colocalization to cellular motivic spectra.
By definition, these are the motivic spectra that can be built out ofthe S i ,j under homotopy colimits.
Here, equivalences are detected by homotopy groups instead of byhomotopy motivic groups, so we can safely ignore the enrichment.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 491: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/491.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Luckily, many motivic spectra of interest are cellular: the sphere,algebraic K -theory, MGL, and indeed any Landweber exact motivicspectrum.
(This a complicated question for Eilenberg–MacLane spectra,though.)
There’s also a cellular colocalization in the equivariant world.
Similarly, the cellular G -spectra are just those that are generatedunder homotopy colimits by the virtual representation spheres.
So, among cellular equivariant spectra, to detect equivalences itsuffices to check RO(G )-graded homotopy groups (instead ofMackey functors).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 492: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/492.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Luckily, many motivic spectra of interest are cellular: the sphere,algebraic K -theory, MGL, and indeed any Landweber exact motivicspectrum.
(This a complicated question for Eilenberg–MacLane spectra,though.)
There’s also a cellular colocalization in the equivariant world.
Similarly, the cellular G -spectra are just those that are generatedunder homotopy colimits by the virtual representation spheres.
So, among cellular equivariant spectra, to detect equivalences itsuffices to check RO(G )-graded homotopy groups (instead ofMackey functors).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 493: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/493.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Luckily, many motivic spectra of interest are cellular: the sphere,algebraic K -theory, MGL, and indeed any Landweber exact motivicspectrum.
(This a complicated question for Eilenberg–MacLane spectra,though.)
There’s also a cellular colocalization in the equivariant world.
Similarly, the cellular G -spectra are just those that are generatedunder homotopy colimits by the virtual representation spheres.
So, among cellular equivariant spectra, to detect equivalences itsuffices to check RO(G )-graded homotopy groups (instead ofMackey functors).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 494: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/494.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Luckily, many motivic spectra of interest are cellular: the sphere,algebraic K -theory, MGL, and indeed any Landweber exact motivicspectrum.
(This a complicated question for Eilenberg–MacLane spectra,though.)
There’s also a cellular colocalization in the equivariant world.
Similarly, the cellular G -spectra are just those that are generatedunder homotopy colimits by the virtual representation spheres.
So, among cellular equivariant spectra, to detect equivalences itsuffices to check RO(G )-graded homotopy groups (instead ofMackey functors).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 495: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/495.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Luckily, many motivic spectra of interest are cellular: the sphere,algebraic K -theory, MGL, and indeed any Landweber exact motivicspectrum.
(This a complicated question for Eilenberg–MacLane spectra,though.)
There’s also a cellular colocalization in the equivariant world.
Similarly, the cellular G -spectra are just those that are generatedunder homotopy colimits by the virtual representation spheres.
So, among cellular equivariant spectra, to detect equivalences itsuffices to check RO(G )-graded homotopy groups (instead ofMackey functors).
Aaron Mazel-Gee GHOsT for∞-categories
![Page 496: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/496.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
As long as we’re talking about equivariant and motivic homotopytheory, we mention one last bit of work in progress.
Note that the categories of G -spaces and G -spectra are not justsymmetric monoidal: they are G -symmetric monoidal.
That is, given a finite G -set T and a G -equivariant functorX : T//G → C into either of these categories, we can form theindexed monoidal product
⊗t∈T Xt ∈ C.
In particular, if X is constant at an object X ∈ C, then thisdeserves to be written X⊗T . (When T has trivial G -action, this isjust X⊗|T |.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 497: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/497.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
As long as we’re talking about equivariant and motivic homotopytheory, we mention one last bit of work in progress.
Note that the categories of G -spaces and G -spectra are not justsymmetric monoidal:
they are G -symmetric monoidal.
That is, given a finite G -set T and a G -equivariant functorX : T//G → C into either of these categories, we can form theindexed monoidal product
⊗t∈T Xt ∈ C.
In particular, if X is constant at an object X ∈ C, then thisdeserves to be written X⊗T . (When T has trivial G -action, this isjust X⊗|T |.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 498: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/498.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
As long as we’re talking about equivariant and motivic homotopytheory, we mention one last bit of work in progress.
Note that the categories of G -spaces and G -spectra are not justsymmetric monoidal: they are G -symmetric monoidal.
That is, given a finite G -set T and a G -equivariant functorX : T//G → C into either of these categories, we can form theindexed monoidal product
⊗t∈T Xt ∈ C.
In particular, if X is constant at an object X ∈ C, then thisdeserves to be written X⊗T . (When T has trivial G -action, this isjust X⊗|T |.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 499: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/499.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
As long as we’re talking about equivariant and motivic homotopytheory, we mention one last bit of work in progress.
Note that the categories of G -spaces and G -spectra are not justsymmetric monoidal: they are G -symmetric monoidal.
That is, given a finite G -set T and a G -equivariant functorX : T//G → C into either of these categories,
we can form theindexed monoidal product
⊗t∈T Xt ∈ C.
In particular, if X is constant at an object X ∈ C, then thisdeserves to be written X⊗T . (When T has trivial G -action, this isjust X⊗|T |.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 500: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/500.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
As long as we’re talking about equivariant and motivic homotopytheory, we mention one last bit of work in progress.
Note that the categories of G -spaces and G -spectra are not justsymmetric monoidal: they are G -symmetric monoidal.
That is, given a finite G -set T and a G -equivariant functorX : T//G → C into either of these categories, we can form theindexed monoidal product
⊗t∈T Xt ∈ C.
In particular, if X is constant at an object X ∈ C, then thisdeserves to be written X⊗T . (When T has trivial G -action, this isjust X⊗|T |.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 501: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/501.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
As long as we’re talking about equivariant and motivic homotopytheory, we mention one last bit of work in progress.
Note that the categories of G -spaces and G -spectra are not justsymmetric monoidal: they are G -symmetric monoidal.
That is, given a finite G -set T and a G -equivariant functorX : T//G → C into either of these categories, we can form theindexed monoidal product
⊗t∈T Xt ∈ C.
In particular, if X is constant at an object X ∈ C, then thisdeserves to be written X⊗T .
(When T has trivial G -action, this isjust X⊗|T |.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 502: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/502.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
As long as we’re talking about equivariant and motivic homotopytheory, we mention one last bit of work in progress.
Note that the categories of G -spaces and G -spectra are not justsymmetric monoidal: they are G -symmetric monoidal.
That is, given a finite G -set T and a G -equivariant functorX : T//G → C into either of these categories, we can form theindexed monoidal product
⊗t∈T Xt ∈ C.
In particular, if X is constant at an object X ∈ C, then thisdeserves to be written X⊗T . (When T has trivial G -action, this isjust X⊗|T |.)
Aaron Mazel-Gee GHOsT for∞-categories
![Page 503: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/503.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors) and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra: that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 504: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/504.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors)
and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra: that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 505: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/505.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors) and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra: that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 506: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/506.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors) and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra: that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 507: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/507.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors) and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra: that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 508: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/508.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors) and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra:
that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 509: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/509.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors) and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra: that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 510: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/510.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, we say that X is a genuine commutative monoid object if itis equipped with compatible maps X⊗T → X for all T .
In homotopy, this is the difference between a Green functor (thatis, a commutative monoid in Mackey functors) and a Tambarafunctor (that is, a G -commutative monoid in Mackey functors).
There is a motivic version, too!
If we are working over a base scheme B and T → B is etale, thenfor any motivic space or spectrum X we can obtain an object X⊗T .
For instance, if B = Spec k , then this must be given byT = SpecA for A an etale k-algebra: that is, a finite product offinite field extensions of k.
This provides for an analogous notion of genuine commutativemotivic objects.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 511: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/511.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 512: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/512.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor
running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 513: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/513.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B
to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 514: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/514.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B).
(When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 515: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/515.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 516: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/516.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects,
etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 517: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/517.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 518: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/518.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads,
andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 519: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/519.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
We hope to generalize Barwick’s theory of operator categories foran arbitrary “set of etale objects” to simultaneously encapsulateboth of these notions.
Moreover, there is an etale realization functor running frommotivic homotopy theory over B to equivariant homotopy theorywith respect to the (profinite) group πet
1 (B). (When B = Spec k ,then we have πet
1 (Spec k) ∼= Gal(ksep/k).)
By functoriality in the set of etale objects, etale realization shouldtake genuine commutative motivic objects to genuine commutativeequivariant objects.
Equivariant genuine symmetric operads are explored extensively inrecent work of Blumberg–Hill on what they call N∞-operads, andwe expect to recover these as a special case.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 520: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/520.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, of course, we hope to extend the obstruction theory to thesegenuine operads!
Of course, this will very much be a “B.Y.O. Dyer–Lashof algebra”sort of party...at least for now.
In addition to putting genuine commutative structures on variousequivariant and motivic spectra, we might hope to one day use thisto construct a motivic spectrum mmf of motivic modular forms: amotivic version of tmf .
We might very well be able to do this just with motivic E∞-rings,but then mmf would only inherit an E∞-structure.
In any case, this is probably still light-years away.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 521: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/521.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, of course, we hope to extend the obstruction theory to thesegenuine operads!
Of course, this will very much be a “B.Y.O. Dyer–Lashof algebra”sort of party...at least for now.
In addition to putting genuine commutative structures on variousequivariant and motivic spectra, we might hope to one day use thisto construct a motivic spectrum mmf of motivic modular forms: amotivic version of tmf .
We might very well be able to do this just with motivic E∞-rings,but then mmf would only inherit an E∞-structure.
In any case, this is probably still light-years away.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 522: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/522.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, of course, we hope to extend the obstruction theory to thesegenuine operads!
Of course, this will very much be a “B.Y.O. Dyer–Lashof algebra”sort of party...at least for now.
In addition to putting genuine commutative structures on variousequivariant and motivic spectra,
we might hope to one day use thisto construct a motivic spectrum mmf of motivic modular forms: amotivic version of tmf .
We might very well be able to do this just with motivic E∞-rings,but then mmf would only inherit an E∞-structure.
In any case, this is probably still light-years away.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 523: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/523.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, of course, we hope to extend the obstruction theory to thesegenuine operads!
Of course, this will very much be a “B.Y.O. Dyer–Lashof algebra”sort of party...at least for now.
In addition to putting genuine commutative structures on variousequivariant and motivic spectra, we might hope to one day use thisto construct a motivic spectrum mmf of motivic modular forms:
amotivic version of tmf .
We might very well be able to do this just with motivic E∞-rings,but then mmf would only inherit an E∞-structure.
In any case, this is probably still light-years away.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 524: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/524.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, of course, we hope to extend the obstruction theory to thesegenuine operads!
Of course, this will very much be a “B.Y.O. Dyer–Lashof algebra”sort of party...at least for now.
In addition to putting genuine commutative structures on variousequivariant and motivic spectra, we might hope to one day use thisto construct a motivic spectrum mmf of motivic modular forms: amotivic version of tmf .
We might very well be able to do this just with motivic E∞-rings,but then mmf would only inherit an E∞-structure.
In any case, this is probably still light-years away.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 525: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/525.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, of course, we hope to extend the obstruction theory to thesegenuine operads!
Of course, this will very much be a “B.Y.O. Dyer–Lashof algebra”sort of party...at least for now.
In addition to putting genuine commutative structures on variousequivariant and motivic spectra, we might hope to one day use thisto construct a motivic spectrum mmf of motivic modular forms: amotivic version of tmf .
We might very well be able to do this just with motivic E∞-rings,but then mmf would only inherit an E∞-structure.
In any case, this is probably still light-years away.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 526: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/526.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
∞-categorical generalitiesThe diagramModel∞-categoriesApplications and generalizations
Then, of course, we hope to extend the obstruction theory to thesegenuine operads!
Of course, this will very much be a “B.Y.O. Dyer–Lashof algebra”sort of party...at least for now.
In addition to putting genuine commutative structures on variousequivariant and motivic spectra, we might hope to one day use thisto construct a motivic spectrum mmf of motivic modular forms: amotivic version of tmf .
We might very well be able to do this just with motivic E∞-rings,but then mmf would only inherit an E∞-structure.
In any case, this is probably still light-years away.
Aaron Mazel-Gee GHOsT for∞-categories
![Page 527: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/527.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Thank you so much for listening!
References:
Goerss–Hopkins, Moduli problems for structured ring spectra.
Blanc–Dwyer–Goerss, The realization space of a Π-algebra: a moduli problem inalgebraic topology.
Dwyer–Kan–Stover, An E2 model category structure for pointed simplicialspaces.
Dwyer–Kan–Stover, The bigraded homotopy groups πi,jX of a pointedsimplicial space X .
Dwyer–Kan, Function complexes in homotopical algebra.
Rezk, Notes on the Hopkins–Miller theorem.
Powell, Existence of E∞-structures.
http://math.berkeley.edu/∼aaron/writing/thursday-cghost-beamer.pdf/thursday-cghost-talk-notes.pdf
/BDG-diagram-beamer.pdf
Aaron Mazel-Gee GHOsT for∞-categories
![Page 528: Every love story is a GHOsT story: Goerss Hopkins ... · GHOsT for 1-categories History Overview Motivation History Goerss{Hopkins obstruction theory is a tool for obtaining E 1-ring](https://reader036.vdocument.in/reader036/viewer/2022081522/5fa3336ba58826582330d54c/html5/thumbnails/528.jpg)
IntroductionBlanc–Dwyer–Goerss obstruction theory
From Blanc–Dwyer–Goerss to Goerss–HopkinsGHOsT for∞-categories
Thank you so much for listening!
References:
Goerss–Hopkins, Moduli problems for structured ring spectra.
Blanc–Dwyer–Goerss, The realization space of a Π-algebra: a moduli problem inalgebraic topology.
Dwyer–Kan–Stover, An E2 model category structure for pointed simplicialspaces.
Dwyer–Kan–Stover, The bigraded homotopy groups πi,jX of a pointedsimplicial space X .
Dwyer–Kan, Function complexes in homotopical algebra.
Rezk, Notes on the Hopkins–Miller theorem.
Powell, Existence of E∞-structures.
http://math.berkeley.edu/∼aaron/writing/thursday-cghost-beamer.pdf/thursday-cghost-talk-notes.pdf
/BDG-diagram-beamer.pdf
Aaron Mazel-Gee GHOsT for∞-categories