Download - Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019
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Props in Network Theory
John BaezSYCO4, Chapman University
22 May 2019
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We have left the Holocene and entered a new epoch, theAnthropocene, in which the biosphere is rapidly changing due tohuman activities.
THE EARTH’S POPULATION, IN BILLIONS
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Climate change is not an isolated ‘problem’ of the sort routinely‘solved’ by existing human institutions. It is part of a shift from theexponential growth phase of human impact on the biosphere to a new,uncharted phase.
I About 1/4 of all chemical energy produced by plants is now usedby humans.
I Humans now take more nitrogen from the atmosphere andconvert it into nitrates than all other processes combined.
I 8-9 times as much phosphorus is flowing into oceans than thenatural background rate.
I The rate of species going extinct is 100-1000 times the usualbackground rate.
I Populations of ocean fish have declined 90% since 1950.
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Climate change is not an isolated ‘problem’ of the sort routinely‘solved’ by existing human institutions. It is part of a shift from theexponential growth phase of human impact on the biosphere to a new,uncharted phase.
I About 1/4 of all chemical energy produced by plants is now usedby humans.
I Humans now take more nitrogen from the atmosphere andconvert it into nitrates than all other processes combined.
I 8-9 times as much phosphorus is flowing into oceans than thenatural background rate.
I The rate of species going extinct is 100-1000 times the usualbackground rate.
I Populations of ocean fish have declined 90% since 1950.
![Page 6: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/6.jpg)
Climate change is not an isolated ‘problem’ of the sort routinely‘solved’ by existing human institutions. It is part of a shift from theexponential growth phase of human impact on the biosphere to a new,uncharted phase.
I About 1/4 of all chemical energy produced by plants is now usedby humans.
I Humans now take more nitrogen from the atmosphere andconvert it into nitrates than all other processes combined.
I 8-9 times as much phosphorus is flowing into oceans than thenatural background rate.
I The rate of species going extinct is 100-1000 times the usualbackground rate.
I Populations of ocean fish have declined 90% since 1950.
![Page 7: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/7.jpg)
Climate change is not an isolated ‘problem’ of the sort routinely‘solved’ by existing human institutions. It is part of a shift from theexponential growth phase of human impact on the biosphere to a new,uncharted phase.
I About 1/4 of all chemical energy produced by plants is now usedby humans.
I Humans now take more nitrogen from the atmosphere andconvert it into nitrates than all other processes combined.
I 8-9 times as much phosphorus is flowing into oceans than thenatural background rate.
I The rate of species going extinct is 100-1000 times the usualbackground rate.
I Populations of ocean fish have declined 90% since 1950.
![Page 8: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/8.jpg)
Climate change is not an isolated ‘problem’ of the sort routinely‘solved’ by existing human institutions. It is part of a shift from theexponential growth phase of human impact on the biosphere to a new,uncharted phase.
I About 1/4 of all chemical energy produced by plants is now usedby humans.
I Humans now take more nitrogen from the atmosphere andconvert it into nitrates than all other processes combined.
I 8-9 times as much phosphorus is flowing into oceans than thenatural background rate.
I The rate of species going extinct is 100-1000 times the usualbackground rate.
I Populations of ocean fish have declined 90% since 1950.
![Page 9: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/9.jpg)
Climate change is not an isolated ‘problem’ of the sort routinely‘solved’ by existing human institutions. It is part of a shift from theexponential growth phase of human impact on the biosphere to a new,uncharted phase.
I About 1/4 of all chemical energy produced by plants is now usedby humans.
I Humans now take more nitrogen from the atmosphere andconvert it into nitrates than all other processes combined.
I 8-9 times as much phosphorus is flowing into oceans than thenatural background rate.
I The rate of species going extinct is 100-1000 times the usualbackground rate.
I Populations of ocean fish have declined 90% since 1950.
![Page 10: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/10.jpg)
So, we can expect that in this century, scientists, engineers andmathematicians will be increasingly focused on biology, ecology andcomplex networked systems — just as the last century was dominatedby physics.
What can category theorists contribute?
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One thing category theorists can do: understand networks.
We need a good general theory of these. It will require categorytheory.
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One thing category theorists can do: understand networks.
We need a good general theory of these. It will require categorytheory.
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To understand ecosystems, ultimately will be to understandnetworks. —B. C. Patten and M. Witkamp
I believe biology proceeds at a higher level of abstraction thanphysics, so it calls for new mathematics.
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To understand ecosystems, ultimately will be to understandnetworks. —B. C. Patten and M. Witkamp
I believe biology proceeds at a higher level of abstraction thanphysics, so it calls for new mathematics.
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Back in the 1950’s, Howard Odum introduced an Energy SystemsLanguage for ecology:
Maybe we are finally ready to develop these ideas.
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The dream: each different kind of network or open system should be amorphism in a different symmetric monoidal category.
Some examples:
I ResCirc, where morphisms are circuits of resistors with inputsand outputs:
3
1
4
These, and many variants, are important in electrical engineering.
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The dream: each different kind of network or open system should be amorphism in a different symmetric monoidal category.
Some examples:
I ResCirc, where morphisms are circuits of resistors with inputsand outputs:
3
1
4
These, and many variants, are important in electrical engineering.
![Page 18: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/18.jpg)
I Markov, where morphisms are open Markov processes:
2.1
5.3
0.6
3
These help us model stochastic processes: technically, they describecontinuous-time finite-state Markov chains with inflows and outflows.
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I RxNet, where morphisms are open reaction networks withrates:
0.6 1.7
Also known as open Petri nets with rates, these are used inchemistry, population biology, epidemiology etc. to describe changingpopulations of interacting entities.
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All these examples can be seen as props: strict symmetric monoidalcategories whose objects are natural numbers, with addition as tensorproduct.
A morphism f : 4→ 3 in a prop can be drawn this way:
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Circ
FinCospan
FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPs
Brandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes JB & Blake Pollard,A compositional framework
for reaction networks
![Page 22: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/22.jpg)
Circ
FinCospan FinCorel
LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPs
Brandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes JB & Blake Pollard,A compositional framework
for reaction networks
![Page 23: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/23.jpg)
Circ FinCospan FinCorel
LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes JB & Blake Pollard,A compositional framework
for reaction networks
![Page 24: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/24.jpg)
Circ FinCospan FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes JB & Blake Pollard,A compositional framework
for reaction networks
![Page 25: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/25.jpg)
Circ FinCospan FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes JB & Blake Pollard,A compositional framework
for reaction networks
![Page 26: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/26.jpg)
Circ FinCospan FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes JB & Blake Pollard,A compositional framework
for reaction networks
![Page 27: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/27.jpg)
Circ FinCospan FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov
RxNet Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes
JB & Blake Pollard,A compositional framework
for reaction networks
![Page 28: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/28.jpg)
Circ FinCospan FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet
Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes
JB & Blake Pollard,A compositional framework
for reaction networks
![Page 29: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/29.jpg)
Circ FinCospan FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes
JB & Blake Pollard,A compositional framework
for reaction networks
![Page 30: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/30.jpg)
Circ FinCospan FinCorel LagRel
LinRel
SemiAlgRel
ResCirc
Markov RxNet Dynam
Steve Lack,
Composing PROPsBrandon Coya & Brendan Fong,
Corelations are the prop for extraspecial
commutative Frobenius monoids
R. Rosebrugh, N. Sabadini & R. F. C. Walters
Generic commutative separable algebras and cospans
of graphs
JB & Brendan Fong,
A compositional framework for passive
linear circuits
JB, Brandon Coya & Franciscus Rebro,
Props in network theory
Filippo Bonchi, Pawel Sobocinski &Fabio Zanasi,
Interacting Hopf algebras
JB & Jason Erbele,
Categories in control
JB & Brendan Fong,
A compositional framework for passive
linear circuitsJB, Brendan Fong & Blake Pollard,
A compositional framework for Markov
processes
JB & Blake Pollard,A compositional framework
for reaction networks
![Page 31: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/31.jpg)
Let’s look at a little piece of this picture:
Circ FinCospan FinCorel LagRelG H K
The composite sends any circuit made just of purely conductive wires
f : m→ n
to the linear relation
KHG( f ) ⊆ R2m ⊕ R2n
that this circuit establishes between the potentials and currents at itsinputs and outputs.
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In the prop Circ, a morphism looks like this:
We can use such a morphism to describe an electrical circuit made ofpurely conductive wires.
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In the prop FinCospan, a morphism looks like this:
We can use such a morphism to say which inputs and outputs lie inwhich connected component of our circuit.
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In the prop FinCorel, a morphism looks like this:
Here a morphism f : m→ n is a corelation: a partition of the setm + n. We can use such a morphism to say which inputs and outputsare connected to which others by wires.
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In the prop LagRel, a morphism L : m→ n is a Lagrangian linearrelation
L ⊆ R2m ⊕ R2n
that is, a linear subspace of dimension m + n such that
ω(v,w) = 0 for all v,w ∈ L.
Here ω is a well-known bilinear form on R2m ⊕ R2n, called a“symplectic structure”.
Remarkably, any circuit made of purely conductive wires establishes alinear relation between the potentials and currents at its inputs and itsoutputs that is Lagrangian!
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A morphism f : 2→ 1 in Circ:
The morphism G( f ) : 2→ 1 in FinCospan:The morphism HG( f ) : 2→ 1 in FinCorel:
(φ1, I1)
(φ2, I2)
(φ3, I3)
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1 + I2 = I3}
The morphism L = KHG( f ) : 2→ 1 in LagRel:
Circ FinCospan FinCorel LagRelG H K
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A morphism f : 2→ 1 in Circ:
The morphism G( f ) : 2→ 1 in FinCospan:
The morphism HG( f ) : 2→ 1 in FinCorel:
(φ1, I1)
(φ2, I2)
(φ3, I3)
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1 + I2 = I3}
The morphism L = KHG( f ) : 2→ 1 in LagRel:
Circ FinCospan
FinCorel LagRel
G
H K
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A morphism f : 2→ 1 in Circ:The morphism G( f ) : 2→ 1 in FinCospan:
The morphism HG( f ) : 2→ 1 in FinCorel:
(φ1, I1)
(φ2, I2)
(φ3, I3)
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1 + I2 = I3}
The morphism L = KHG( f ) : 2→ 1 in LagRel:
Circ FinCospan FinCorel
LagRel
G H
K
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A morphism f : 2→ 1 in Circ:The morphism G( f ) : 2→ 1 in FinCospan:The morphism HG( f ) : 2→ 1 in FinCorel:
(φ1, I1)
(φ2, I2)
(φ3, I3)
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1 + I2 = I3}
The morphism L = KHG( f ) : 2→ 1 in LagRel:
Circ FinCospan FinCorel LagRelG H K
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In working on these issues, three questions come up:
I When is a symmetric monoidal category equivalent to a prop?I What exactly is a map between props?I How can you present a prop using generators and relations?
Answers can be found here:
I John Baez, Brendan Coya and Franciscus Rebro, Props innetwork theory, arXiv:1707.08321.
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We start with the 2-category SymMonCat, where:I objects are symmetric monoidal categories,I morphisms are symmetric monoidal functors,I 2-morphisms are monoidal natural transformations.
We often prefer to think about the category PROP, where:
I object are props: strict symmetric monoidal categories withnatural numbers as objects and addition as tensor product,
I morphisms are strict symmetric monoidal functors sending 1 to 1.
This is evil, but convenient. When can we get away with it?
![Page 42: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/42.jpg)
We start with the 2-category SymMonCat, where:I objects are symmetric monoidal categories,I morphisms are symmetric monoidal functors,I 2-morphisms are monoidal natural transformations.
We often prefer to think about the category PROP, where:
I object are props: strict symmetric monoidal categories withnatural numbers as objects and addition as tensor product,
I morphisms are strict symmetric monoidal functors sending 1 to 1.
This is evil, but convenient. When can we get away with it?
![Page 43: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/43.jpg)
We start with the 2-category SymMonCat, where:I objects are symmetric monoidal categories,I morphisms are symmetric monoidal functors,I 2-morphisms are monoidal natural transformations.
We often prefer to think about the category PROP, where:
I object are props: strict symmetric monoidal categories withnatural numbers as objects and addition as tensor product,
I morphisms are strict symmetric monoidal functors sending 1 to 1.
This is evil, but convenient. When can we get away with it?
![Page 44: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/44.jpg)
Theorem. C ∈ SymMonCat is equivalent to a prop iff there is anobject x ∈ C such that every object of C is isomorphic to
x⊗n = x ⊗ (x ⊗ (x ⊗ · · · ))
for some n ∈ N.
Theorem. Suppose F : C → D is a symmetric monoidal functorbetween props. Then F is isomorphic, in SymMonCat, to a strictsymmetric monoidal functor G : C → D.
If F(1) = 1, G is a morphism of props.
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Theorem. C ∈ SymMonCat is equivalent to a prop iff there is anobject x ∈ C such that every object of C is isomorphic to
x⊗n = x ⊗ (x ⊗ (x ⊗ · · · ))
for some n ∈ N.
Theorem. Suppose F : C → D is a symmetric monoidal functorbetween props. Then F is isomorphic, in SymMonCat, to a strictsymmetric monoidal functor G : C → D.
If F(1) = 1, G is a morphism of props.
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We all “know” how to describe props using generators and relations.For example, the prop for commutative monoids can be presented withtwo generators:
µ : 2→ 1 ι : 0→ 1
and three relations:
= = =
(associativity) (unitality) (commutativity)
But what are we really doing here?
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There is a forgetful functor from props to signatures:
U : PROP→ SetN×N
A signature just gives a set hom(m, n) for each (m, n) ∈ N × N.
Theorem. The forgetful functor U is monadic, meaning that it has aleft adjoint
F : SetN×N → PROP
and PROP is equivalent to the category of algebras of the resultingmonad UF : SetN×N → SetN×N.
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There is a forgetful functor from props to signatures:
U : PROP→ SetN×N
A signature just gives a set hom(m, n) for each (m, n) ∈ N × N.
Theorem. The forgetful functor U is monadic, meaning that it has aleft adjoint
F : SetN×N → PROP
and PROP is equivalent to the category of algebras of the resultingmonad UF : SetN×N → SetN×N.
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Everything one wants to do with generators and relations follows fromU : PROP→ SetN×N being monadic.
For example:
Corollary. Any prop T is a coequalizer
F(R) // // F(G) // T
for some signatures G, R.
We call elements of G generators and elements of R relations.
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Everything one wants to do with generators and relations follows fromU : PROP→ SetN×N being monadic.
For example:
Corollary. Any prop T is a coequalizer
F(R) //// F(G) // T
for some signatures G, R.
We call elements of G generators and elements of R relations.
![Page 51: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/51.jpg)
Example. The symmetric monoidal category where
I objects are finite setsI morphisms are isomorphism classes of cospans of finite sets:
I the tensor product is disjoint union
is equivalent to a prop, FinCospan.
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Theorem (Lack). The prop FinCospan has generators
and relations:
= = =
associativity unitality commutativity
= = =
coassociativity counitality cocommutativity
= = =
Frobenius law special law
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Thus, for any strict symmetric monoidal category C, there’s a 1-1correspondence between:
I strict symmetric monoidal functors F : FinCospan→ C
andI special commutative Frobenius monoids in C.
We summarize this by saying FinCospan is “the prop for specialcommutative Frobenius monoids”.
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Example. The symmetric monoidal category where:
I objects are finite sets,I morphisms are corelations:
I the tensor product is disjoint union
is equivalent to a prop, FinCorel.
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Theorem (Coya, Fong). The prop FinCorel has the same generatorsas FinCospan:
and all the same relations, together with one more:
=
extra law
Thus, FinCorel is the prop for extraspecial commutative Frobeniusmonoids.
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Theorem (Coya, Fong). The prop FinCorel has the same generatorsas FinCospan:
and all the same relations, together with one more:
=
extra law
Thus, FinCorel is the prop for extraspecial commutative Frobeniusmonoids.
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Example. The symmetric monoidal category where:
I objects are finite sets,I morphisms are circuits made solely of wires:
I the tensor product is disjoint union
is equivalent to a prop, Circ.
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Theorem (Rosebrugh, Sabadani, Walters). The prop Circ has allthe same generators and relations as Cospan, together with oneadditional generator f : 1→ 1.
Thus, Circ is the prop for special commutative Frobenius monoids Xequipped with a morphism f : X → X .
In applications to electrical circuits, this morphism describes a purelyconductive wire:
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Theorem (Rosebrugh, Sabadani, Walters). The prop Circ has allthe same generators and relations as Cospan, together with oneadditional generator f : 1→ 1.
Thus, Circ is the prop for special commutative Frobenius monoids Xequipped with a morphism f : X → X .
In applications to electrical circuits, this morphism describes a purelyconductive wire:
![Page 60: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/60.jpg)
We can now understand these maps of props:
Circ FinCospan FinCorel LagRelG H K
using generators and relations:I Circ is the prop for special commutative Frobenius monoids withendomorphism f .
I FinCospan is the prop for special commutative Frobeniusmonoids. G sends f to the identity.
I FinCorel is the prop for extraspecial commutative Frobeniusmonoids. H does the obvious thing.
I K sends the extraspecial commutative Frobenius monoid1 ∈ FinCorel to R2 ∈ LagRel, which becomes an extraspecialcommutative Frobenius monoid by ‘duplicating potentials andadding currents’. For example
gets sent to the Lagrangian relation
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1+ I2 = I3} ⊆ R4⊕R2.
![Page 61: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/61.jpg)
We can now understand these maps of props:
Circ FinCospan FinCorel LagRelG H K
using generators and relations:I Circ is the prop for special commutative Frobenius monoids withendomorphism f .
I FinCospan is the prop for special commutative Frobeniusmonoids. G sends f to the identity.
I FinCorel is the prop for extraspecial commutative Frobeniusmonoids. H does the obvious thing.
I K sends the extraspecial commutative Frobenius monoid1 ∈ FinCorel to R2 ∈ LagRel, which becomes an extraspecialcommutative Frobenius monoid by ‘duplicating potentials andadding currents’. For example
gets sent to the Lagrangian relation
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1+ I2 = I3} ⊆ R4⊕R2.
![Page 62: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/62.jpg)
We can now understand these maps of props:
Circ FinCospan FinCorel LagRelG H K
using generators and relations:I Circ is the prop for special commutative Frobenius monoids withendomorphism f .
I FinCospan is the prop for special commutative Frobeniusmonoids. G sends f to the identity.
I FinCorel is the prop for extraspecial commutative Frobeniusmonoids. H does the obvious thing.
I K sends the extraspecial commutative Frobenius monoid1 ∈ FinCorel to R2 ∈ LagRel, which becomes an extraspecialcommutative Frobenius monoid by ‘duplicating potentials andadding currents’. For example
gets sent to the Lagrangian relation
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1+ I2 = I3} ⊆ R4⊕R2.
![Page 63: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/63.jpg)
We can now understand these maps of props:
Circ FinCospan FinCorel LagRelG H K
using generators and relations:I Circ is the prop for special commutative Frobenius monoids withendomorphism f .
I FinCospan is the prop for special commutative Frobeniusmonoids. G sends f to the identity.
I FinCorel is the prop for extraspecial commutative Frobeniusmonoids. H does the obvious thing.
I K sends the extraspecial commutative Frobenius monoid1 ∈ FinCorel to R2 ∈ LagRel, which becomes an extraspecialcommutative Frobenius monoid by ‘duplicating potentials andadding currents’. For example
gets sent to the Lagrangian relation
L = {(φ1, I1, φ2, I2, φ3, I3) : φ1 = φ2 = φ3, I1+ I2 = I3} ⊆ R4⊕R2.
![Page 64: Props in Network TheoryProps in Network Theory John Baez SYCO4, Chapman University 22 May 2019](https://reader034.vdocument.in/reader034/viewer/2022050102/5f41060dd6df5c13dd6f0d84/html5/thumbnails/64.jpg)
This is just the tip of the iceberg. Many fields of science andengineering use networks. A unified theory of networks will:
I reveal and clarify the mathematics underlying these fields,I help integrate these fields,I enhance interoperability of human-designed systems,I focus attention on open systems: systems with inflows andoutflows.
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References
I J. Baez, B. Coya and F. Rebro, Props in network theory.Available as arXiv:1707.08321.
I J. Baez, J. Erbele, Categories in control, Theory Appl. Categ. 30(2015), 836–881. Available athttp://www.tac.mta.ca/tac/volumes/30/24/30-24abs.html.
I J. Baez, B. Fong, A compositional framework for passive linearcircuits. Available at arXiv:1504.05625.
I J. Baez, B. Fong and B. S. Pollard, A compositional frameworkfor Markov processes, Jour. Math. Phys. 57 (2016), 033301.Available at arXiv:1508.06448.
I F. Bonchi, P. Sobociński, F. Zanasi, Interacting Hopf algebras,J. Pure Appl. Alg. 221 (2017), 144–184. Available asarXiv:1403.7048.
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I B. Coya, B. Fong, Corelations are the prop for extraspecialcommutative Frobenius monoids, Theory Appl. Categ. 32(2017), 380–395. Available athttp://www.tac.mta.ca/tac/volumes/32/11/32-11abs.html.
I S. Lack, Composing PROPs, Theory Appl. Categ. 13 (2004),147–163. Available athttp://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html.
I R. Rosebrugh, N. Sabadini, R. F. C. Walters, Genericcommutative separable algebras and cospans of graphs, TheoryAppl. Categ. 15 (2005), 164–177. Available athttp://www.tac.mta.ca/tac/volumes/15/6/15-06abs.html.