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

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Outline I. Hypergraph categories II. Cospan algebras III. The equivalence

I. Hypergraph categories

Abstractly, how do we construct this? g f h

. . . as structured monoidal category g g f h f h ( 1 ⊗ f ⊗ ⊗ 1 ) ; ( ⊗ 1 ⊗ ⊗ 1 ) ; ( ⊗ g ⊗ ) ; ( ⊗ h ) .

. . . as structured monoidal category g g f h f h

. . . as structured monoidal category g g f h f h g f h

A special commutative Frobenius monoid on X is µ ∶ X ⊗ X → X η ∶ I → X δ ∶ X → X ⊗ X ǫ ∶ X → I obeying = = = = = = = =

A special commutative Frobenius monoid on X is µ ∶ X ⊗ X → X η ∶ I → X δ ∶ X → X ⊗ X ǫ ∶ X → I obeying the spider theorem =

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 , id I ,ρ I , id I ) and for all X,Y , the Frobenius structure on X ⊗ Y is X X X ⊗ Y X Y = = X ⊗ Y X ⊗ Y X X ⊗ Y Y Y Y X X X ⊗ Y Y X = = X ⊗ Y X ⊗ Y X ⊗ Y X Y Y 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 , id I ,ρ I , id I ) and for all X,Y , the Frobenius structure on X ⊗ Y is X X X ⊗ Y X Y = = X ⊗ Y X ⊗ Y X X ⊗ Y Y Y Y X X X ⊗ Y Y X = = X ⊗ Y X ⊗ Y X ⊗ Y X Y Y Y A hypergraph functor is a strong symmetric monoidal functor ( F,ϕ ) such that if ( µ X ,η X ,δ X ,ǫ X ) is the Frobenius structure on X , then ( ϕ X,X ; Fµ X , ϕ I ; Fη X , Fδ X ; ϕ − 1 I ) is the Frobenius X,X , Fǫ X ; ϕ − 1 structure on FX .

Let Hyp be the 2-category with objects: hypergraph categories morphisms: hypergraph functors 2-morphisms: monoidal natural transformations. Let Hyp OF be the full sub-2-category of objectwise-free hyper- graph categories. Theorem (Coherence for hypergraph categories) Hyp OF and Hyp are 2-equivalent.

II. Cospan algebras

Abstractly, how do we construct this? g f h

. . . as operad algebra 1 1 4 2 f 2 3 g 4 1 2 3 2 3 1 5 g 2 4 3 1 1 1 2 3 2 5 f 1 h 4 3 h 2 6 3 6 3 A N B

Define Cospan Λ = ∐ λ ∈ Λ Cospan ( FinSet ) . Cospan Λ is the symmetric monoidal category with objects: Λ -typed finite sets t ∶ X → Λ . morphisms: cospans over Λ . f 1 f 2 X N Y s u t Λ monoidal product: disjoint union ⊕ x x x y y y z z z

Define Cospan Λ = ∐ λ ∈ Λ Cospan ( FinSet ) . Cospan Λ is the symmetric monoidal category with objects: Λ -typed finite sets t ∶ X → Λ . morphisms: cospans over Λ . f 1 f 2 X N Y s u t Λ monoidal product: disjoint union ⊕ x x x f y y X Y Z g f y g z z z

Let CospanAlg be the category with objects: lax symmetric monoidal functors A ∶( Cospan Λ , ⊕) � → ( Set , ×) Λ morphisms: monoidal natural transformations Cospan Λ Λ A f Cospan f ⇓ α Set List ( Λ ′ ) Cospan Λ ′ A ′

III. The equivalence

Theorem Hyp OF and CospanAlg are (1-)equivalent. Proof sketch: 1. Work over Λ . 2. Frobenius monoids define cospan algebra. 3. Cospan algebras define homsets of hypergraph categories.

1. Working over Λ Lemma There is a Grothendieck fibration Gens ∶ Hyp OF → Set List sending an objectwise-free hypergraph category to its set of generating objects. This implies Λ ∈ Set List Hyp OF ≅ ∫ Hyp OF ( Λ ) Note also CospanAlg = ∫ Λ ∈ Set List Lax ( Cospan Λ , Set )

2. Frobenius defines cospan algebras Lemma Cospan Λ is the free hypergraph category over Λ (ie. with ob- jects generated by Λ ). That is, there is an adjunction Cospan − Set List Hyp OF � Gens Given a hypergraph category H over Λ , we can construct a cospan algebra A H ∶ Cospan Λ � � → H H( I, − ) Frob � � � � → Set .

3. Cospans define hypergraph structure Lemma Hypergraph categories are self dual compact closed. Given a cospan algebra A over Λ , we may define a hypergraph category H A over Λ with homsets H A ( X,Y ) = A ( X ⊕ Y ) .

3. Cospans define hypergraph structure The remaining structure is defined by certain cospans. X Y f X Y Z g f Z W g composition monoidal product identity braiding (co)multiplication (co)unit

Theorem (Coherence for hypergraph categories) Hyp OF and Hyp are 2-equivalent. Theorem Hyp OF and CospanAlg are (1-)equivalent.