Structured cospans John Baez and Kenny Courser University of California, Riverside May 22, 2019 John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 1 / 34
Networks can very often be viewed as sets equipped or ‘decorated’ with extra structure... John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 2 / 34
For example, w 1 / 2 4 y z 2 2 1 x John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 3 / 34
For example, w 1 / 2 4 y z 2 2 1 x H α H 2 O O John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 3 / 34
An open network is a network with prescribed inputs and outputs. John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 4 / 34
An open network is a network with prescribed inputs and outputs. w 1 / 2 4 y inputs z outputs 2 2 1 x John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 4 / 34
An open network is a network with prescribed inputs and outputs. w 1 / 2 4 y inputs z outputs 2 2 1 x a b H 1 2 α H 2 O 4 3 O John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 4 / 34
An easy example to have in mind is the example of open graphs: y e 5 e 4 i o { ⋆ } { ⋆ } e 3 w z e 2 e 1 x John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 5 / 34
An easy example to have in mind is the example of open graphs: y e 5 e 4 i o { ⋆ } { ⋆ } e 3 w z e 2 e 1 x The overall shape of this diagram resembles that of a cospan : c i o a b John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 5 / 34
Brendan Fong has developed a theory of decorated cospans which is well suited for describing ‘open’ networks. John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 6 / 34
Brendan Fong has developed a theory of decorated cospans which is well suited for describing ‘open’ networks. Theorem (B. Fong) Let A be a category with finite colimits and F : A → Set a symmetric lax monoidal functor. Then there exists a category FCospan which has: John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 6 / 34
Brendan Fong has developed a theory of decorated cospans which is well suited for describing ‘open’ networks. Theorem (B. Fong) Let A be a category with finite colimits and F : A → Set a symmetric lax monoidal functor. Then there exists a category FCospan which has: • objects as those of A and • morphisms as isomorphism classes of F-decorated cospans, where an F-decorated cospan is given by a pair: c o i d ∈ F ( c ) a b John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 6 / 34
Theorem (B. Fong continued) Two F-decorated cospans are in the same isomorphism class if the following diagrams commute: c F ( c ) i o d ∼ a b 1 f F ( f ) d ′ i ′ o ′ F ( c ′ ) c ′ John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 7 / 34
Theorem (B. Fong continued) To compose two morphisms: i ′ o ′ i o a 1 c 1 a 2 a 2 c 2 a 3 d 1 ∈ F ( c 1 ) d 2 ∈ F ( c 2 ) we take the pushout in A : c 1 + a 2 c 2 ψ ψ j ′ o ′ ψ ji c 1 + c 2 j ′ j c 1 c 2 i ′ o ′ i o a 1 a 2 a 3 John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 8 / 34
Theorem (B. Fong continued) To compose two morphisms: i ′ o ′ i o a 1 c 1 a 2 a 2 c 2 a 3 d 1 ∈ F ( c 1 ) d 2 ∈ F ( c 2 ) we take the pushout in A : c 1 + a 2 c 2 ψ ψ j ′ o ′ ψ ji c 1 + c 2 j ′ j c 1 c 2 i ′ o ′ i o a 1 a 2 a 3 φ c 1 , c 2 d 1 × d 2 F ( ψ ) d 1 ⊙ d 2 : 1 − − − − → F ( c 1 ) × F ( c 2 ) − − − − → F ( c 1 + c 2 ) − − − − → F ( c 1 + a 2 c 2 ) John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 8 / 34
For example, if we let F : Set → Set be the symmetric lax monoidal functor that assigns to a set N the (large) set of all graph structures having N as its set of vertices: s F ( N ) = { E N } t John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 9 / 34
For an example of this example, if we take N = { n 1 , n 2 , n 3 } to be a three element set, then some elements of the (large) set F ( N ) are given by: John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 10 / 34
For an example of this example, if we take N = { n 1 , n 2 , n 3 } to be a three element set, then some elements of the (large) set F ( N ) are given by: e 1 e 1 n 1 • n 2 n 1 n 2 • • • f 1 e 3 e 2 f 3 f 2 e 3 e 2 • n 3 • n 3 d 1 ∈ F ( N ) d 2 ∈ F ( N ) n 1 n 2 n 1 n 2 • • • • e 1 e 2 e 3 • • n 3 n 3 d 3 ∈ F ( N ) d 4 ∈ F ( N ) John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 10 / 34
One defect of this framework lies in what constitutes an isomorphism class: John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 11 / 34
One defect of this framework lies in what constitutes an isomorphism class: F ( c ) c i o d ∼ a b 1 F ( f ) f i ′ d ′ o ′ F ( c ′ ) c ′ The triangle on the right is in Set and commutes on the nose. John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 11 / 34
One defect of this framework lies in what constitutes an isomorphism class: F ( c ) c i o d ∼ a b 1 F ( f ) f i ′ d ′ o ′ F ( c ′ ) c ′ The triangle on the right is in Set and commutes on the nose. This means that a decoration d ∈ F ( c ) together with a bijection f : c → c ′ determines what the decoration d ′ ∈ F ( c ′ ) must be. John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 11 / 34
In the context of open graphs, the following two open graphs would be in the same isomorphism class: y e 5 e 4 e 3 w z o i e 2 e 1 x ↓ f { ⋆ } { ⋆ } y ′ e 5 e 4 i ′ o ′ w ′ e 3 z ′ e 2 e 1 x ′ John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 12 / 34
But the following two open graphs would not be in the same isomorphism class: y e 5 e 4 e 3 w z o i e 2 e 1 x ↓ f { ⋆ } { ⋆ } y ′ e ′ e ′ 5 4 i ′ o ′ e ′ w ′ z ′ 3 e ′ e ′ x ′ 2 1 John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 13 / 34
One remedy to this is to instead use ‘structured cospans’. John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 14 / 34
One remedy to this is to instead use ‘structured cospans’. Theorem (Baez, C.) Let A be a category with finite coproducts, X a category with finite colimits and L : A → X a finite coproduct preserving functor. Then there exists a category L Csp ( X ) which has: John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 14 / 34
One remedy to this is to instead use ‘structured cospans’. Theorem (Baez, C.) Let A be a category with finite coproducts, X a category with finite colimits and L : A → X a finite coproduct preserving functor. Then there exists a category L Csp ( X ) which has: • objects as those of A and • morphisms as isomorphism classes of structured cospans , where a structured cospan is given by a cospan in X of the form: x i o L ( a ) L ( b ) John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 14 / 34
Theorem (Baez, C. continued) Two structured cospans are in the same isomorphism class if the following diagram commutes: x o i ∼ L ( a ) L ( b ) α i ′ o ′ y John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 15 / 34
Theorem (Baez, C. continued) To compose two morphisms: i ′ o ′ i o L ( a 1 ) L ( a 2 ) L ( a 2 ) y L ( a 3 ) x we take the pushout in X : x + L ( a 2 ) y ψ ψ J ′ o ′ ψ Ji x + y J ′ J y x i ′ o ′ i o L ( a 1 ) L ( a 2 ) L ( a 3 ) John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 16 / 34
In the context of open graphs, we take L : Set → Graph to be the discrete graph functor which assigns to a set N the edgeless graph with vertex set N . ! ∅ N ! John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 17 / 34
In the context of open graphs, we take L : Set → Graph to be the discrete graph functor which assigns to a set N the edgeless graph with vertex set N . ! ∅ N ! Both Set and Graph have finite colimits and L is a left adjoint, so we get the following: John Baez and Kenny Courser (University of California, Riverside) Structured cospans May 22, 2019 17 / 34
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