Compact closed bicategories. Mike Stay University of Auckland and Google

Transcription

1 Compact closed icategories Mike Stay University of uckland and Google

2 Examples of compact closed icategories: Rel sets, relations, and implications Prof categories, profunctors, and natural transformations 2Vect 2-vector spaces, linear functors, transformations nco 2 manifolds, manifolds with oundary, manifolds with corners and of course any compact closed category where we take the 2-morphisms to e identities

3 Compact closed icategories have... a tensor product functor with Stasheff polytopes to govern it a a ((B)C)D a : (B)C (BC) ad π (B(CD)) a a a monoidal unit with maps of Stasheff polytopes to govern it a r : I l B λ l

4 Compact closed icategories have... a raiding with shuffle polytopes and the Breen polytope to govern it a a : B B R C a B

5 Compact closed icategories have... a syllepsis satisfying equations B v B B B B B 1 = 1 v 1 v 1 B B B B

6 Compact closed icategories have... duals for ojects with ends, yanking and the swallowtail law to govern them i i i e e ζ e ζ i i (ζ 1 i ) i e i ( ζ ) i i = i i = e e

7 Theorem (Stay) If T is a 2-category with finite products and pseudo pullacks, then the icategory Span 2 (T) of ojects of T spans in T isomorphism classes of weak maps of spans is compact closed. Theorem (Stay) compact closed icategory satisfies the axioms of a traced monoidal category in a canonical way up to isomorphism.

8 We can present the Lawvere theory of a symmetric monoidal category ut not of a symmetric monoidal closed category ecause of the currying isomorphism etween the contravariant functors hom(x y,z) = hom(x,y z).

9 Conjecture There exists a compact closed icategory Th(SMCC) such that the 2-category hom(th(smcc), Prof) of symmetric monoidal functors (of icategories), symmetric monoidal natural transformations and symmetric monoidal modifications is 2-equivalent to the 2-category SMCC of symmetric monoidal closed categories, symmetric monoidal closed functors and symmetric monoidal closed natural isomorphisms.

10 Useful fact Every profunctor with a right adjoint is of the form hom(,f ) where F is a functor. This means that when presenting Th(SMCC) we can talk aout not only profunctors ut also functors. Given a function symol F : C D that we mean to e interpreted as a functor, we add a function symol F : D C, end rewrites i F : C F F and e F : F F D and yanking equations (F e F ) (i F F) = F (e F F ) (F i F ) = F

11 Presentation of Th(SMCC), where is the monoidal product, J is the monoidal unit, and ( ) is the dual. a sort C function symols : C C C : C C C I: J C I : C J : C C C : C C C

12 Presentation of Th(SMCC), where is the monoidal product, J is the monoidal unit, and ( ) is the dual. rewrites ends for,i, and a: ( C) (C ) assocc,c,c : swapc,c c : curry 1 ( ) l: (I C) leftc r: (C I) rightc formal inverses for a,,c,l,r

13 Presentation of Th(SMCC), where is the monoidal product, J is the monoidal unit, and ( ) is the dual. equations pentagon equation triangle equation hexagon equations yanking equations for each of,i, a,,c,l,r composed with their inverses equal the identity

14 When we interpret the rewrite c : curry 1 ( ) in Prof, we get c : hom(x y,z) hom(x,y z) : C op C op C op

15 Start with the tensor : C C C (x,y) z hom(z,x y), then prime and take its dual to get : C op C op C op (x,y) z hom(x y,z), which is the first half of the currying isomorphism.

16 Now start with the internal hom: Take the dual: Uncurry: : C op C C (y,z) x hom(x,y z), : C op C C op x (y,z) hom(x,y z), curry 1 ( ): C op C op C op (x,y) z hom(x,y z), which is the second half of the currying isomorphism.

17 Just as monoidal categories are the right place to talk aout monoids, compact closed icategories are the right place to talk aout monoidal closed categories, e.g. programming languages.