On the bicategory of operads and analytic functors Nicola Gambino - PowerPoint PPT Presentation

On the bicategory of operads and analytic functors Nicola Gambino University of Leeds Joint work with Andr´ e Joyal Cambridge, Category Theory 2014 1

Reference N. Gambino and A. Joyal On operads, bimodules and analytic functors ArXiv, 2014 2

Main result Let V be a symmetric monoidal closed presentable category. Theorem. The bicategory Opd V that has ◮ 0-cells = operads (= symmetric many-sorted V -operads) ◮ 1-cells = operad bimodules ◮ 2-cells = operad bimodule maps is cartesian closed. Note. For operads A and B , we have Alg ( B A ) = Opd V [ A, B ] . Alg ( A ⊓ B ) = Alg ( A ) × Alg ( B ) , 3

Plan of the talk 1. Symmetric sequences and operads 2. Bicategories of bimodules 3. A universal property of the bimodule construction 4. Proof of the main theorem 4

1. Symmetric sequences and operads 5

Single-sorted symmetric sequences Let S be the category of finite cardinals and permutations. Definition. A single-sorted symmetric sequence is a functor F : S → V F [ n ] n �→ For F : S → V , we define the single-sorted analytic functor F ♯ : V → V by letting F [ n ] ⊗ Σ n T n . � F ♯ ( T ) = n ∈ N 6

Single-sorted operads Recall that: 1. The functor category [ S , V ] admits a monoidal structure such that G ♯ ◦ F ♯ , ∼ ( G ◦ F ) ♯ = I ♯ ∼ Id V = 2. Monoids in [ S , V ] are exactly single-sorted operads. See [Kelly 1972] and [Joyal 1984]. 7

Symmetric sequences For a set X , let S ( X ) be the category with ◮ objects: ( x 1 , . . . , x n ), where n ∈ N , x i ∈ X for 1 ≤ i ≤ n ◮ morphisms: σ : ( x 1 , . . . , x n ) → ( x ′ 1 , . . . , x ′ n ) is σ ∈ Σ n such that x ′ i = x σ ( i ) . Definition. Let X and Y be sets. A symmetric sequence indexed by X and Y is a functor S ( X ) op × Y F : → V ( x 1 , . . . , x n , y ) �→ F [ x 1 , . . . , x n ; y ] Note. For X = Y = 1 we get single-sorted symmetric sequences. 8

Analytic functors Let F : S ( X ) op × Y → V be a symmetric sequence. We define the analytic functor F ♯ : V X → V Y by letting � ( x 1 ,...,x n ) ∈ S ( X ) F ♯ ( T, y ) = F [ x 1 , . . . , x n ; y ] ⊗ T ( x 1 ) ⊗ . . . T ( x n ) for T ∈ V X , y ∈ Y . Note. For X = Y = 1, we get single-sorted analytic functors. 9

The bicategory of symmetric sequences The bicategory Sym V has ◮ 0-cells = sets ◮ 1-cells = symmetric sequences, i.e. F : S ( X ) op × Y → V ◮ 2-cells = natural transformations. Note. Composition and identities in Sym V are defined so that ( G ◦ F ) ♯ ∼ = G ♯ ◦ F ♯ (Id X ) ♯ ∼ = Id V X 10

Monads in a bicategory Let E be a bicategory. Recall that a monad on X ∈ E consists of ◮ A : X → X ◮ µ : A ◦ A ⇒ A ◮ η : 1 X ⇒ A subject to associativity and unit axioms. Examples. ◮ monads in Ab = monoids in Ab = commutative rings ◮ monads in Mat V = small V -categories ◮ monads in Sym V = (symmetric, many-sorted) V -operads 11

An analogy Mat V Sym V Matrix Symmetric sequence F : S ( X ) op × Y → V F : X × Y → V Linear functor Analytic functor Category Operad Bimodule/profunctor/distributor Operad bimodule 12

Categorical symmetric sequences The bicategory CatSym V has ◮ 0-cells = small V -categories ◮ 1-cells = V -functors F : S ( X ) op ⊗ Y → V , where S ( X ) = free symmetric monoidal V -category on X . ◮ 2-cells = V -natural transformations Note. We have Sym V ⊆ CatSym V . 13

Theorem 1. The bicategory CatSym V is cartesian closed. Proof. Enriched version of main result in [FGHW 2008]. ◮ Products: X ⊓ Y = def X ⊔ Y , ◮ Exponentials: [ X , Y ] = def S ( X ) op ⊗ Y . 14

2. Bicategories of bimodules 15

Bimodules Let E be a bicategory. Let A : X → X and B : Y → Y be monads in E . Definition. A ( B, A ) -bimodule consists of ◮ M : X → Y ◮ a left B -action λ : B ◦ M ⇒ M ◮ a right A -action ρ : M ◦ A ⇒ M . subject to a commutation condition. Examples. ◮ bimodules in Ab = ring bimodules ◮ bimodules in Mat V = bimodules/profunctors/distributors ◮ bimodules in Sym V = operad bimodules 16

Bicategories with local reflexive coequalizers Definition. We say that a bicategory E has local reflexive coequalizers if (i) the hom-categories E [ X, Y ] have reflexive coequalizers, (ii) the composition functors preserve reflexive coequalizers in each variable. Examples. ◮ ( Ab , ⊗ , Z ) ◮ Mat V ◮ Sym V and CatSym V 17

� The bicategory of bimodules The bicategory Bim( E ) has ◮ 0-cells = ( X, A ), where X ∈ E and A : X → X monad ◮ 1-cells = bimodules ◮ 2-cells = bimodule morphisms Composition: for M : ( X, A ) → ( Y, B ) , N : ( Y, B ) → ( Z, C ), N ◦ B M : ( X, A ) → ( Z, C ) is given by N ◦ λ � N ◦ B M . N ◦ B ◦ M � N ◦ M ρ ◦ M Identities: Id ( X,A ) : ( X, A ) → ( X, A ) is A : X → X . 18

Examples 1. The bicategory of ring bimodules Bim( Ab ) ◮ 0-cells = rings ◮ 1-cells = ring bimodules ◮ 2-cells = bimodule maps 2. The bicategory of bimodules/profunctors/distributors Bim( Mat V ) ◮ 0-cells = small V -categories ◮ 1-cells = V -functors X op ⊗ Y → V ◮ 2-cells = V -natural transformations. 19

3. The bicategory of operads Opd V = def Bim( Sym V ) ◮ 0-cells = V -operads ◮ 1-cells = operad bimodules ◮ 2-cells = operad bimodule maps. Note. The composition operation of Opd V obtained in this way generalizes Rezk’s circle-over construction. Remark. For an operad bimodule F : ( X, A ) → ( Y, B ), we define the analytic functor F ♯ : Alg ( A ) Alg ( B ) → M �→ F ◦ A M These include restriction and extension functors. 20

Cartesian closed bicategories of bimodules Theorem 2. Let E be a bicategory with local reflexive coequalizers. If E is cartesian closed, then so is Bim( E ). Idea. ◮ Products ( X, A ) × ( Y, B ) = ( X × Y, A × B ) ◮ Exponentials � � � � ( X, A ) , ( Y, B ) = [ X, Y ] , [ A, B ] Note. The proof uses a homomorphism Mnd( E ) → Bim( E ) , where Mnd( E ) is Street’s bicategory of monads. 21

3. A universal property of the bimodule construction 22

� � Eilenberg-Moore completions Let E be a bicategory with local reflexive coequalizers. The bicategory Bim( E ) is the Eilenberg-Moore completion of E as a bicategory with local reflexive coequalizers: J E � Bim( E ) E F ♯ F F Note. ◮ This was proved independently by Garner and Shulman, extending work of Carboni, Kasangian and Walters. ◮ Different universal property from the Eilenberg-Moore completion studied by Lack and Street. 23

Theorem 3. The inclusion Bim( Sym V ) ⊆ Bim( CatSym V ) is an equivalence. Idea. Every 0-cell of CatSym V is an Eilenberg-Moore object for a monad in Sym V . 24

4. Proof of the main theorem 25

� � � Theorem. The bicategory Opd V is cartesian closed. Proof. Recall � CatSym V Sym V ❴ ❴ � Bim( CatSym V ) Opd V � Theorem 1 says that CatSym V is cartesian closed. So, by Theorem 2, Bim( CatSym V ) is cartesian closed. But, Theorem 3 says Opd V = Bim( Sym V ) ≃ Bim( CatSym V ) . 26