A Double Approach to Variation and Enrichment for Bicategories - PowerPoint PPT Presentation

A Double Approach to Variation and Enrichment for Bicategories Susan Niefield (with J.R.B. Cockett and R.J. Wood) June 2012

� ev � CT95 Moncat / V mod Mon V Kelly: works for bicategories (N) 1997 Generalize to bicategories? (NW) Relate to Cat / B ≃ Fun ( B , S pan ) ≃ Fun N ( B , P rof )? 2005 (1) Fun ( B , S ) ≃ Fun N ( B , Mod S ) (CNW) (2) LDF / B ≃ Fun ( B co , S pan ) ≃ Fun N ( B co , P rof ) (1) 2011 ▲ ax ( ❇ , ❙ ) double category for nice ❇ and ❙ (Par´ e) ▲ ax (( ❱ B ) op , ❙ pan ) ≃ ❈ at / / B Note: Fun ( B co , S pan ) ≃ H ▲ ax (( ❱ B ) op , ❙ pan ) Idea: (1) for double categories and vertical structure for (2)

� � � � � � � � � Double Categories π 2 d 0 � ❉ 0 Weak category objects ❉ 1 × ❉ 0 ❉ 1 ❉ 1 × ❉ 0 ❉ 1 ❉ 1 × ❉ 0 ❉ 1 µ � ❉ 1 ❉ 1 ❉ 1 ❉ 1 ❉ 1 ❉ 1 ❉ 0 ❉ 0 in CAT ∆ π 1 d 1 Objects: objects of ❉ 0 � D ′ Horizontal morphisms: morphisms of ❉ 0 , D ¯ Vertical morphism: objects of ❉ 1 , D D • � D ′ D ′ D D Cells: morphisms of ❉ 1 , • • ¯ ¯ ¯ ¯ D ′ D ′ D D Note: V ❉ is a bicategory and H ❉ is a 2-category

Examples ❙ pan : sets, functions, spans, . . . ❈ at : categories, functors, profunctors, . . . ❱ B : vertically B , a bicategory (horizontally discrete) ▲ ax ( ❇ , ❙ ): lax functors, transformations, modules, modulations (horizontal) (CKSW) ▼ od ❉ : monads in V ❉ , homomorphisms, modules, . . . / B ) 1 = ❉ 1 / id • ❉ / / B : ( ❉ / / B ) 0 = ❉ 0 / B , ( ❉ / B ❙ pan \ \ 1 = ❙ pan ∗ , pointed sets

� � � � � � The Double Category ▲ ax ( ❇ , ❙ ) (Par´ e) f Ff � FB ′ � B ′ B ′ FB ′ B B FB FB � ❙ • • b ′ • • Fb ′ F : ❇ b � β �→ Fb � F β horiz functorial, vert lax, . . . lax functor ¯ ¯ ¯ ¯ F ¯ F ¯ F ¯ F ¯ B ′ B ′ B ′ B ′ B B B B ¯ F ¯ f f � F id • B , � b : F ¯ � F (¯ F ◦ B : id • b · Fb b · b ) F b , ¯ FB t B � F ′ B F ′ B FB FB B � F ′ • • t b • F ′ b t : F �→ Fb � b � horiz natural, vert functorial, . . . F ′ ¯ F ′ ¯ transformation F ¯ F ¯ ¯ B B B B B t ¯ B

� � � � � � � � � � � � The Double Category ▲ ax ( ❇ , ❙ ), cont. Fb � F ¯ F ¯ FB FB FB B B f Ff � FB ′ � B ′ • B ′ FB ′ B B FB FB � F � � � � � � � � • • m ¯ b ′ mb ′ � • b � • β • �→ mb � • m β • mb • b m � � � � � � ¯ ¯ ¯ ¯ G ¯ G ¯ G ¯ G ¯ B ′ B ′ B ′ B ′ B B B B G G ¯ G ¯ G ˜ G ˜ G ˜ B B B B B ¯ G ¯ • f f G ¯ module b t B � F ′ B t � F ′ F ′ B F ′ FB FB F F B • • • �→ • µ b • m ′ b m � µ m ′ mb � b � G ′ ¯ G ′ ¯ G ¯ G ¯ ¯ G ′ G ′ G G u � B B B B B u ¯ B modulation Define: ❋ un ( B co , S pan ) = ▲ ax (( ❱ B ) op , ❙ pan )

The Equivalence ▲ ax ( ❇ , ❙ ) Mon � ▲ ax N ( ❇ , ▼ od ❙ ) � ❙ , define Mon F : ❇ � ▼ od ❙ by Given F : ❇ F id • FB , � B � B , F ◦ B �→ ( FB F id • B ) • B , id • Ff homomorphism, Fb module, F β equivariant (since F is lax) Mon : transformations, modules, modulations �→ same ( Mon ) − 1 is composition with U : ▼ od ❙ � ❙ ▲ ax ( ❇ , ❙ pan ) ≃ ▲ ax N ( ❇ , ❈ at ), ❋ un ( B , S ) ≃ ❋ un N ( B , Mod S ) � V W in Moncat Note: loosely related to ev ⊣ mod � Mod V ✶ W normal in Bicat

� � � � � � � � Variation for Bicategories F � ❙ pan , consider the projection B F P � B , where Given ( ❱ B ) op objects of B F : ( B , x ∈ FB ) 1 1 1 s � x ¯ ( b , s ) x � (¯ morphisms of B F : ( B , x ) B , ¯ x ), with Fb Fb Fb � � � � � � � � � F ¯ F ¯ F ¯ FB FB FB B B B s ′ ( b , s ) � Fb ′ Fb ′ 1 1 � � (¯ (¯ � cells of B F : ( B , x ) ( B , x ) B , ¯ B , ¯ x ) x ) , with β � � F β � s � ( b ′ , s ′ ) Fb Fb � ❙ pan Note: ❱ B F is Par´ e’s “elements of F ” ❊ l F , for F : ( ❱ B ) op

� � � � � � � � � � Local Discrete Fibrations � B is a local discrete fibration (LDF) if A lax functor P : X X ( X , ¯ � B ( PX , P ¯ X ) is a discrete fibration, for all X , ¯ X ) X P � B is an LDF strict functor with small fibers Proposition: B F Proof: ( b , F β s ′ ) (¯ (¯ β B F ( B , x ) ( B , x ) B , ¯ B , ¯ x ) x ) ( b ′ , s ′ ) P b ¯ ¯ B B B B B β b ′ � S pan ∗ B co B co S pan ∗ F F Remark: pb in the category of bicats and lax functors B co B co S pan S pan

� � � � � � � � � � � � � Local Discrete Fibrations, cont. � F ′ : ( ❱ B ) op � ❙ pan A transformation t : F t B � F ′ B F ′ B FB FB B F ′ b • �→ Fb • t b • b � ¯ F ′ ¯ F ′ ¯ F ¯ F ¯ B B B B B t ¯ B � B F ′ over B defined by induces an LDF functor B t : B F ( b , s ) � ( b , t b s ) � (¯ (¯ (¯ (¯ ( B , x ) ( B , x ) B , ¯ B , ¯ x ) x ) �→ ( B , t B x ) ( B , t B x ) B , t ¯ B , t ¯ B ¯ B ¯ x ) x ) β β ( b ′ , s ′ ) ( b ′ , t b ′ s ′ ) since the following diagram commutes when the triangle does by horiz naturality of t t b ′ � F ′ b ′ s ′ Fb ′ Fb ′ Fb ′ Fb ′ F ′ b ′ 1 1 � � � � F ′ � F β � β s � F ′ b F ′ b Fb Fb Fb Fb t b

� � � � � � Local Discrete Fibrations, cont. � ❙ pan is given by a lax functor • G : ( ❱ B ) op A module m : F � ❙ pan s.t. M ( − , 0) = F and M ( − , 1) = G M : ( ❱ ( B × ✷ )) op � B × ✷ , together Thus, m induces an LDF functor ( B × ✷ ) M with a diagram P F � B B F B F B pb LDF � ( − , 0) � B × ✷ ( B × ✷ ) M ( B × ✷ ) M ( B × ✷ ) M ( B × ✷ ) M B × ✷ B × ✷ B × ✷ pb ( − , 1) LDopF B G B G B B P G

� � � � � � � Local Discrete Fibrations, cont. t � F ′ F ′ F F A modulation induces a lax functor m � • µ • m ′ G ′ G ′ u � G G � ( B × ✷ ) M ′ over B × ✷ , and a diagram ( B × ✷ ) M B t � B F ′ B F B F B F ′ pb � ( B × ✷ ) M ′ ( B × ✷ ) M ( B × ✷ ) M ( B × ✷ ) M ( B × ✷ ) M ( B × ✷ ) M ′ ( B × ✷ ) M ′ ( B × ✷ ) M ′ pb B G B G B G ′ B G ′ B u

� � � � � � � � � � � � � � The Double Category ▲ DF / / B P � B LDF functors with small fibers objects: X P � B X X B pb LDF � ( − , 0) H � X ′ X ′ X X � � B × ✷ morphisms: � � ���� M M M M B × ✷ B × ✷ B × ✷ � � P P ′ pb B B LDopF ( − , 1) Y Y B B horizontal Q vertical � X ′ X ′ X X B pb ( − , 0) � M ′ M ′ M ′ M ′ cells: M M M M over B × ✷ B × ✷ pb ( − , 1) Y ′ Y ′ Y Y B

� � � � � � � The Equivalence � ▲ DF / Theorem: B − : ▲ ax (( ❱ B ) op , ❙ pan ) / B is an equivalence � B , define F : ( ❱ B ) op � ❙ pan by Proof (sketch): Given P : X b � ¯ x � ¯ FB = { X | PX = B } and F ( B B ) = { X X | Px = b } with projections FB � d 0 Fb B , and constraints FB F ◦ � F id • d 1 � F ¯ B given � F � F (¯ X , and F ¯ by X �→ id • b × F ¯ B Fb bb ) by � F ( x , ¯ x ) � ˜ ˜ X X X X X � � ����� � � � x x ¯ ¯ ¯ X X P ¯ bb � ˜ ˜ B B B B � � ����� � id B � � ¯ b ¯ ¯ b B B Horizontal and vertical morphisms of ▲ DF / / B give rise to transformations and modules, and cells induce modulations.

A Double Approach to Enrichment for Bicategories Showed LDF / B ≃ ˆ B - Cat , where ˆ 2005 B is the bicategory with (CNW) B ) = Sets B ( B , ¯ | ˆ B| = |B| and ˆ B ( B , ¯ B ) op For F : B ( B , ¯ � Sets and ¯ F : B (¯ B , ˜ � Sets , B ) op B ) op F · F : B ( B , ˜ ¯ � Sets is given for c : B � ˜ B ) op B by � b � ¯ b (¯ Fb × ¯ F ¯ b × B ( c , ¯ F · F )( c ) = bb ) � Sets and the identity on B is ( − , id B ): B ( B , B ) op

� � � � � � � The Double Category ˆ B - ❈ at : Objects ˆ B -categories X , i.e., a set |X| together with a function � |B| , ˆ B -morphisms X [ X , ¯ � P ¯ P : |X| X ]: PX X , and cells X [ ¯ X , ˜ X ] · X [ X , ¯ � X [ X , ˜ � X [ X , X ] s.t. . . . X ] X ], and id PX � B an LDF, define Example: For P : X X [ X , ¯ X ]: B ( PX , P ¯ � Sets X ) op x � ¯ b �→ X [ X , ¯ X ] b = { X X | Px = b } β ∗ x ′ � � X , ¯ ¯ ¯ X X X X X X β ∗ : X [ X , ¯ � X [ X , ¯ x ′ X ] b ′ X ] b b � � PX , P ¯ P ¯ P ¯ B X PX PX X X β b ′

� � � � The Double Category ˆ B - ❈ at : Horizontal Morphisms ˆ � X ′ B -functors H : X X [ X , ¯ X ] H � |X ′ | |X ′ | |X| |X| P ¯ P ¯ PX PX X X s.t. . . . � � � ��� � P � P ′ X ′ [ HX , H ¯ X ] |B| |B| H � X ′ X ′ X X � Example: For � � ���� in ▲ DF / / B , define � � P P ′ B B H b : X [ X , ¯ � X ′ [ HX , H ¯ x � ¯ Hx � H ¯ X ] b X ] b by X X �→ HX X