applied topology - lectures nine semester ii,...

Applied Topology

Lectures NINE

Semester II, 2009-10

Graham EllisNUI Galway, Ireland

Section 9

Categories and functors

The language of categories and functors is needed for concisestatements in topology.


A category C consists of:

The axioms for a category are:



◮ a collection Ob(C) of things called objects,





◮ for each pair of objects A,B ∈ Ob(C) a (possibly empty)collection MorC(A,B) of things called morphisms,





◮ for each pair of objects A,B ∈ Ob(C) a (possibly empty)collection MorC(A,B) of things called morphisms,

◮ a function

MorC(B ,C ) × MorC(A,B) −→ MorC(A,C ),

(f , g) 7→ f ◦ g

called composition which is defined for each triple of objectsA,B ,C ∈ Ob(C).


1. Associativity: The equation

(f ◦ g) ◦ h = f ◦ (g ◦ h)

holds for all morphisms f , g , h for which (f ◦ g) ◦ h is defined.

1. Associativity: The equation

(f ◦ g) ◦ h = f ◦ (g ◦ h)

holds for all morphisms f , g , h for which (f ◦ g) ◦ h is defined.

2. Identity: For every object B ∈ Ob(C) there is a morphism1B ∈ MorC(B ,B) such that 1B ◦ f = f and g ◦ 1B = g for anymorphisms f ∈ MorC(A,B), g ∈ MorC(B ,C ).

EXAMPLE:We have the category VecR whose objects are the vectors spacesover R, and whose morphisms are the R-linear homomorphisms.


EXAMPLE:We have the category ChnR whose objects are the chain complexesover R, and whose morphisms are the chain maps.


EXAMPLE:We have the category ChnR whose objects are the chain complexesover R, and whose morphisms are the chain maps.

EXAMPLE:Any group G can be considered as a category G with one object ∗and with MorG(∗, ∗) = G .

A Functor F : C → D between two categories C,D consists offunctions

F : Ob(C) → Ob(D),

F : MorC(A,B) → MorD(F(A),F(B))

satisfyingF(f ◦ g) = F(f ) ◦ F(g),

F(1A) = 1F(A)

for all composable morphisms f , g and all identity morphisms 1A.

EXAMPLE/EXERCISEFor each n ≥ 0, homology provides a functor

Hn : ChnR → VecR

from the category of chain complexes to the category of vectorspaces.


Hn : ChnR → VecR


EXAMPLEAny group homomorphism φ : G → G ′ can be regarded as afunctor Φ: G → G′ between one-object categories.


Hn : ChnR → VecR


EXAMPLEAny group homomorphism φ : G → G ′ can be regarded as afunctor Φ: G → G′ between one-object categories.

EXAMPLEWe have the category whose objects are (small) categories andwhose morphisms are functors.

Some basic functors in topology

over K

Chain Complexes

Simplicial

Cubical

Haussdorfftopologicalspaces complexes

complexes

Vector Spaces

Abeliangroups

if K isa field

if K is the integers

CW spaces

cell spaces)(also called

over K

Precise descriptions need to be given. Categories and functorswithin the box can be represented on a computer and the efficiencyof their representations needs to be considered.

Some basic functors in topology

over K

Chain Complexes

Simplicial

Cubical

Haussdorfftopologicalspaces complexes

complexes

Vector Spaces

Abeliangroups

if K isa field

if K is the integers

CW spaces

cell spaces)(also called

over K

Precise descriptions need to be given. Categories and functorswithin the box can be represented on a computer, and theefficiency their representations needs to be considered.

The functor

C∗ : Simp → Chn

Simplicial complexes

A simplicial complex consists of an ordered set V of vertices and aset K of finite nonempty subsets of V called simplexes such that:

1. {v} ∈ K for any v ∈ V .

2. Any nonempty subset of a simplex is a simplex.

Simplicial complexes

A simplicial complex consists of an ordered set V of vertices and aset K of finite nonempty subsets of V called simplexes such that:

1. {v} ∈ K for any v ∈ V .

2. Any nonempty subset of a simplex is a simplex.

A simplex σ containing exactly k + 1 vertices will be called ann-simplex. If σ′ ⊆ σ then σ′ will be called a face of σ.

EXAMPLEFor any set V the set of all finite nonempty subsets of V is asimplicial complex.


EXAMPLEFor any set A and any collection V = {vλ ⊂ A} of subsets of A,the set N(V ) of all subsets of V with non-empty intersection is asimplicial complex called the nerve of V .


EXAMPLEFor any set A and any collection V = {vλ ⊂ A} of subsets of A,the set N(V ) of all subsets of V with non-empty intersection is asimplicial complex called the nerve of V .

ILLUSTRATION: Suppose that A is a circle S1 and thatV = {R ,S ,T} is a covering of S1 by three open subsets R ,S ,T

which intersect pairwise but for which R ∩ S ∩ T = ∅.

R

S

T

{R}

{S} {T}

{R,S} {R,T}

{S,T}

Then N(V ) consists of three 0-simplexes and three 1-simplexes.

EXAMPLEThe Rips complex R(S , t) is a simplicial complex arising from ann × n symmetric matrix S = (sij) and t ≥ 0. The vertex set isV = {1, . . . , n} and there is one simplex for each subset σ ⊂ V

satisfying sij ≤ t for all distinct i , j ∈ σ.

EXAMPLEThe Rips complex R(S , t) is a simplicial complex arising from ann × n symmetric matrix S = (sij) and t ≥ 0. The vertex set isV = {1, . . . , n} and there is one simplex for each subset σ ⊂ V

satisfying sij ≤ t for all distinct i , j ∈ σ.

Recall from Lecture 6 that simplicial complexes can be investigatedusing gap.

EXAMPLEFor any set A and any collection S = {sλ ⊆ A} of subsets of A, wecan consider the set V = {vλµ : sλ ( sµ} of proper inclusions withsλ, sµ ∈ S .


A chain in V is a finite subset

{sλ1⊂ sλ2

, sλ2⊂ sλ3

. . . , sλn⊂ sλn+1

}

of V consisting of composable inclusions.



{sλ1⊂ sλ2

, sλ2⊂ sλ3

. . . , sλn⊂ sλn+1

}


The collection of all chains in V is a simplicial complex called theorder complex of S .



{sλ1⊂ sλ2

, sλ2⊂ sλ3

. . . , sλn⊂ sλn+1

}


The collection of all chains in V is a simplicial complex called theorder complex of S .

(This order complex is particularly interesting when A is a finitegroup and S is the collection of elementary abelian p-subgroups forsome prime p. There is an unsolved conjecture about thissimplicial complex due to Daniel Quillen.)

gap commands for Quillen’s complex

We can comstruct the order complex on the set S of elementaryabelian 2-groups of S5 as follows. (The simplicial complex canviewed as a graph with 45 vertices and 60 edges.)

gap> G:=SymmetricGroup(5);;

gap> K:=QuillenComplex(G,2);

Simplicial complex of dimension 1.

gap> K!.nrSimplices(0);

45


60


0

Simplicial maps

A simplicial map f : K → K ′ from a simplicial complex K to asimplicial complex K ′ is a function f : V → V ′ of the vertex setssuch that,

1. for each simplex σ in K , the set

f (σ) = {f (v) : v ∈ σ}

is a simplex in K ′;

2. for each simplex σ in K , and each v , v ′ ∈ σ, if v ≤ v ′ thenf (v) ≤ f (v ′).

.

Simplicial maps

A simplicial map f : K → K ′ from a simplicial complex K to asimplicial complex K ′ is a function f : V → V ′ of the vertex setssuch that,

1. for each simplex σ in K , the set

f (σ) = {f (v) : v ∈ σ}

is a simplex in K ′;

2. for each simplex σ in K , and each v , v ′ ∈ σ, if v ≤ v ′ thenf (v) ≤ f (v ′).

.We let Simp denote the category whose objects are simplicialcomplexes and whose morphisms are simplicial maps.

EXAMPLEConsider the simplicial complex K with vertices VK ={0, 1, 2, 3, 4, 5} and six maximal simplices

{1, 2}, {2, 3}, {3, 0}, {0, 4}, {4, 5}, {1, 5}.

Consider also the simplicial complex L with vertices VL = {a, b, c}and three maximal simplices

{a, b}, {b, c}, {a, c}.

EXAMPLEConsider the simplicial complex K with vertices VK ={0, 1, 2, 3, 4, 5} and six maximal simplices

{1, 2}, {2, 3}, {3, 0}, {0, 4}, {4, 5}, {1, 5}.

Consider also the simplicial complex L with vertices VL = {a, b, c}and three maximal simplices

{a, b}, {b, c}, {a, c}.

{a}

{b}{c}

{a,b}

{b,c}

{a,c}

{1}

{1,2}

{2}

{2,3}

{3}

K

L

{0}

{0,3}{0,4}

{4}

{5}

{4,5}

{1,5}

In the picture the ordering of the vertices is represented usingarrow heads.


There is a simplicial map f : K → L between the two illustratedsimplicial complexes K ,L defined on vertices by

f (1) = a, f (2) = b, f (3) = c , f (0) = a, f (4) = b, f (5) = c .


There is a simplicial map f : K → L between the two illustratedsimplicial complexes K ,L defined on vertices by

f (1) = a, f (2) = b, f (3) = c , f (0) = a, f (4) = b, f (5) = c .

REMARKIn other examples the ordering of vertices of 2-simplices could berepresented by circular arrows.

{x}

{y}{z}

{x}

{y} {z}

x<y<z

The chain complex of a simplicial complex

For a simplicial complex K with ordered vertex set V we let Cn(K )denote the free abelian group with one basis element σ for eachn-simplex σ ∈ K .



For each n-simplex σ = {v1 < v2 < · · · < vn+1} ⊂ V ,n ≥ 1, wedefine σi to be the (n − 1)-simplex obtained by deleting vi from σ.



For each n-simplex σ = {v1 < v2 < · · · < vn+1} ⊂ V ,n ≥ 1, wedefine σi to be the (n − 1)-simplex obtained by deleting vi from σ.

For n ≥ 1 we define

∂n : Cn(K ) −→ Cn−1(K ) ,

to be the homomorphism which sends the basis element σ to

∂n(σ) =∑

1≤i≤n+1

(−1)iσi .

EXERCISE: ∂n(∂n+1(σ)) = 0 for all basis elements σ.

EXERCISE: ∂n(∂n+1(σ)) = 0 for all basis elements σ.

We denote by C∗(K ) the chain complex

→ Cn(K )∂n→ Cn−1(K ) → . . . → C0(K ) .

Given a simplicial map f : K → K ′ and an n-simplex σ ∈ K , wedefine f (σ) = {f (v) : v ∈ σ}. If f (σ) is an n-simplex in K ′ thenwe set

Cn(f )(σ) = f (σ) ∈ Cn(K′);

otherwise we setCn(f )(σ) = 0 ∈ Cn(K

′).

Given a simplicial map f : K → K ′ and an n-simplex σ ∈ K , wedefine f (σ) = {f (v) : v ∈ σ}. If f (σ) is an n-simplex in K ′ thenwe set

Cn(f )(σ) = f (σ) ∈ Cn(K′);

otherwise we setCn(f )(σ) = 0 ∈ Cn(K

′).

We define the chain map

C∗(f ) : C∗(K ) → C∗(K′)

to be that consisting of the abelian group homomorphisms

Cn(f ) : Cn(K ) → Cn(K′)

defined on generators as above.

It is an exercise to see that C∗ is a chain map.


It is a further exercise to see that we have defined a functor

C∗ : Simp → Chn

from the category of simplicial complexes to the category Chn

whose objects are chain complexes of free abelian groups andwhose morphisms are chain maps.


It is a further exercise to see that we have defined a functor

C∗ : Simp → Chn

from the category of simplicial complexes to the category Chn

whose objects are chain complexes of free abelian groups andwhose morphisms are chain maps.

NOTATION: We write Hn(K ) for Hn(C∗(K )) when K is asimplicial complex.

gap commands for constructing the homology of a simplicial map

gap> K_faces:=[[1,2],[2,3],[3,0],[0,4],[4,5],[5,1]];;

gap> K:=MaximalSimplicesToSimplicialComplex(K_faces);;

gap> L_faces:=[[’a’,’b’],[’b’,’c’],[’c’,’a’]];;

gap> L:=MaximalSimplicesToSimplicialComplex(L_faces);;

gap> g:=function(x);

> if x=1 or x=0 then return ’a’;fi;

> if x=2 or x=4 then return ’b’;fi;

> if x=3 or x=5 then return ’c’;fi;

> end;;

gap> Simp_g:=SimplicialMap(K,L,g);;

gap> Chain_g:=ChainMapOfSimplicialMap(Simp_g);;

gap> Homology(Chain_g,1);

[ f1 ] -> [ f1^2 ]

gap commands for constructing the homology of a simplicial map

gap> K_faces:=[[1,2],[2,3],[3,0],[0,4],[4,5],[5,1]];;

gap> K:=MaximalSimplicesToSimplicialComplex(K_faces);;

gap> L_faces:=[[’a’,’b’],[’b’,’c’],[’c’,’a’]];;

gap> L:=MaximalSimplicesToSimplicialComplex(L_faces);;

gap> g:=function(x);

> if x=1 or x=0 then return ’a’;fi;

> if x=2 or x=4 then return ’b’;fi;

> if x=3 or x=5 then return ’c’;fi;

> end;;

gap> Simp_g:=SimplicialMap(K,L,g);;

gap> Chain_g:=ChainMapOfSimplicialMap(Simp_g);;

gap> Homology(Chain_g,1);

[ f1 ] -> [ f1^2 ]

This ilustrates that the above simplicial map induces

H1(K )×2→ H1(L).

applied topology - lectures nine semester ii,...

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