Hypergraph categories as cospan algebras Brendan Fong, with David - - PowerPoint PPT Presentation
Hypergraph categories as cospan algebras Brendan Fong, with David Spivak Category Theory 2018 University of Azores 10 July 2018 Bonchi, Sobocinski, Zanasi: A categorical semantics of signal flow graphs Baez, Pollard: A compositional framework
Brendan Fong, with David Spivak
Category Theory 2018 University of Azores 10 July 2018
Baez, Pollard: A compositional framework for reaction networks Rosebrugh, Sabadini, Walters: Calculating colimits compositionally Bonchi, Sobocinski, Zanasi: A categorical semantics of signal flow graphs Spivak: The operad of wiring diagrams
Abstractly, how do we construct this? f g h
f g h
f g h (1⊗f⊗ ⊗1);( ⊗1⊗ ⊗1);( ⊗g⊗ );( ⊗h).
f g h
f g h
f g h
f g h f h g
A special commutative Frobenius monoid on X is
µ∶ X ⊗ X → X η∶ I → X δ∶ X → X ⊗ X ǫ∶ X → I
= = = = = = = =
A special commutative Frobenius monoid on X is
µ∶ X ⊗ X → X η∶ I → X δ∶ X → X ⊗ X ǫ∶ X → I
=
A hypergraph category is a symmetric monoidal category in which each object X is equipped with a Frobenius structure in a way compatible with the monoidal product.
A hypergraph category is a symmetric monoidal category in which each object X is equipped with a Frobenius structure in a way compatible with the monoidal product. This means that the Frobenius structure on I is (ρ−1
I ,idI,ρI,idI) and
for all X,Y , the Frobenius structure on X ⊗ Y is
X ⊗ Y X ⊗ Y X ⊗ Y X Y X Y X Y = X ⊗ Y X Y = X ⊗ Y X ⊗ Y X ⊗ Y X Y X Y X Y = X ⊗ Y X Y =
A hypergraph category is a symmetric monoidal category in which each object X is equipped with a Frobenius structure in a way compatible with the monoidal product. This means that the Frobenius structure on I is (ρ−1
I ,idI,ρI,idI) and
for all X,Y , the Frobenius structure on X ⊗ Y is
X ⊗ Y X ⊗ Y X ⊗ Y X Y X Y X Y = X ⊗ Y X Y = X ⊗ Y X ⊗ Y X ⊗ Y X Y X Y X Y = X ⊗ Y X Y =
A hypergraph functor is a strong symmetric monoidal functor (F,ϕ) such that if (µX,ηX,δX,ǫX) is the Frobenius structure on X, then (ϕX,X;FµX, ϕI;FηX, FδX;ϕ−1
X,X, FǫX;ϕ−1 I ) is the Frobenius
structure on FX.
Let Hyp be the 2-category with
morphisms: hypergraph functors 2-morphisms: monoidal natural transformations. Let HypOF be the full sub-2-category of objectwise-free hyper- graph categories. Theorem (Coherence for hypergraph categories) HypOF and Hyp are 2-equivalent.
Abstractly, how do we construct this? f g h
f g h 3 1 2 6 3 5 4 3 4 1 3 2 2 2 1 1
A N B
1 2 3 4 1 2 3 1 2 3 f g h 1 2 3 4 5 6
Define CospanΛ = ∐λ∈Λ Cospan(FinSet). CospanΛ is the symmetric monoidal category with
morphisms: cospans over Λ. X N Y Λ
t f1 s u f2
monoidal product: disjoint union ⊕
x y y z x y z x z
Define CospanΛ = ∐λ∈Λ Cospan(FinSet). CospanΛ is the symmetric monoidal category with
morphisms: cospans over Λ. X N Y Λ
t f1 s u f2
monoidal product: disjoint union ⊕
f g
X Y Z x y y z x y z x z
f g
Let CospanAlg be the category with
Λ A∶(CospanΛ,⊕) → (Set,×) morphisms: monoidal natural transformations Λ List(Λ′)
f
CospanΛ Set CospanΛ′
Cospanf A
⇓α
A′
Theorem HypOF and CospanAlg are (1-)equivalent. Proof sketch:
Lemma There is a Grothendieck fibration Gens∶HypOF → SetList sending an objectwise-free hypergraph category to its set of generating
This implies HypOF ≅ ∫
Λ∈SetList
HypOF(Λ) Note also CospanAlg = ∫
Λ∈SetList
Lax(CospanΛ,Set)
Lemma CospanΛ is the free hypergraph category over Λ (ie. with ob- jects generated by Λ). That is, there is an adjunction SetList HypOF
Cospan−
Given a hypergraph category H over Λ, we can construct a cospan algebra AH∶CospanΛ
Frob
H(I,−)
Lemma Hypergraph categories are self dual compact closed. Given a cospan algebra A over Λ, we may define a hypergraph category HA over Λ with homsets HA(X,Y ) = A(X ⊕ Y ).
The remaining structure is defined by certain cospans.
f g
X Y Z
composition f g
X Y Z W
monoidal product identity braiding (co)multiplication (co)unit
Theorem (Coherence for hypergraph categories) HypOF and Hyp are 2-equivalent. Theorem HypOF and CospanAlg are (1-)equivalent.