Biclosed bicategories: localisation of convolution

BICLOSED BICA TEGORIES: LOCALISA TION O F CONV OLUTION BRIAN J. D A Y Abstract. W e give a summary (withou t proofs) of the main results i n the author’s thesis en titled “Construction of biclosed categories” (Universit y of New South W ales, Australia, 1970). This summary is repri n ted directly from Report 81-0030 of the Sc hool of M athematics and Ph ysics, Macquarie Univ er- sity , April 1981. In particular, it give s sufficien t conditions for existence of an extension of a (pro) monoidal c ategory structu re along a giv en dense functo r to a cocomplete category . The t w o basic pro cedures used in the proof turn out to be sp ecial cases of the final result, the tw o respective dense functors then being the Y oneda embedding follow ed by a lo calisation. The final resul t has a standard univ ersal property based on l eft Kan extension of (pro)mon oidal functors along the giv en dense functo r, ho we ve r this prop er t y is not stated explicitly here. 1. Introduction The aim of this ar ticle is to recor d, for future r e ference, so me of the elemen- tary formulas arising in t he theor y of biclosed bicatego r ies. These for mulas a re given by tw o outstanding building blocks in the theory , na mely conv olution [2 ] and reflection [3]. When combined, these pr o cesses lead to an extension theo rem for probicatego ries (which is the name we give to the collection of structure functors of the basic biclosed bicatego ries of functors (or “presheaves”) into the gr ound category ). All categor ical algebra shall be r elative to a fixed complete and co complete sym- metric monoidal closed gro und ca teg ory V = ( V , ⊗ , I , [ − , − ] , . . . ). 2. Probica tegories and convolution A pr obic ate gory structure on a family { A xy : x, y ∈ ob( A ) } of small categories is essentially a biclosed bicatego ry structure o n the co llection { [ A xy , V ] : x, y ∈ ob( A ) } , each of the compo sition functors ◦ : [ A y z , V ] ⊗ [ A xy , V ] / / [ A xz , V ] being deter mined, to within isomo r phism, by a structu r e functor: P xy z : A op y z ⊗ A op xy ⊗ A xz / / V . A bicategory is a particular instance of a probicategory fo r whic h there exist functor s ◦ : A y z ⊗ A xy / / A xz such that P xy z ( A, B , C ) ∼ = A xz ( A ◦ B , C ) . Date : Octob er 26, 2018. 1 2 BRIAN J. DA Y A biclosed probicateg ory is a pro bicategor y for which there exis t functors − / − : A xz ⊗ A op xy / / A y z −\− : A op y z ⊗ A xz / / A xy such that P xy z ( A, B , C ) ∼ = A xy ( B , A \ C ) ∼ = A y z ( A, C / B ) A manifold probicategory is a n indexing of Prof . Thus a set ob( A ) is given and, for each x ∈ ob( A ) , a catego r y A x . W e take A xy = A op x ⊗ A y and P xy z (( A, A ′ ) , ( B , B ′ ) , ( C, C ′ )) = A x ( C, B ) ⊗ A y ( B ′ , A ) ⊗ A z ( A ′ , C ′ ) . A probica tegory may (or may not) hav e an identity J ; we shall here include the ident ity conditions thro ug hout. One c a n alw a ys conv olve a probicatego ry A = ( A , P, J, . . . ) with a co complete monoidal ca teg ory B = ( B , ⊗ , I , . . . ) for which − ⊗ B a nd B ⊗ − b oth pr eserve colimits for all ob jects B ∈ B . The bicategory compos itio n and iden tities on the conv olution [ A , B ] are given b y: F ◦ G = Z AA ′ P ( A, A ′ , − ) ∗ ( F A ⊗ GA ′ ) I = J ∗ I . 3. Reflection theorem A mo rphism ψ = ( ψ , ˜ ψ , ψ ◦ ) : B / / C of bicateg ories will b e calle d str ong if each of the co mparison ma ps ˜ ψ : ψB ◦ ψ B ′ / / ψ ( B ◦ B ′ ) and ψ ◦ : I / / ψ I is a n isomorphism. Theorem. L et B = ( B xy , ◦ , I , / , \ , . . . ) b e a biclose d bic ate gory and let θ : C / / B b e a family of ful l emb e dd ings θ xy : C xy / / B xy with left adjoi nts ψ = { ψ xy } (we omit θ fr om the notation, and denote t he un it of the adjunction by η : 1 / / ψ : B / / B ). F urther, let A xy ⊂ B xy b e a lo c al ly stro ngly gener ating class of 1- c el ls in B and let D xy ⊂ C xy b e a lo c al ly str ongly c o gener ating class of 1-c el ls in C . Then, in or der that ther e should exist a biclose d bic ate gory structu r e on C for which ψ : B / / C admits enrichment t o a str ong map of biclose d bic a te gorie s, it is sufficient that one of the following p airs of morphisms b e a p air of isomorphisms BICLOSED BICA TEGORIES: LOCALISA TION OF CONVOLUTION 3 for al l ap pr opriately indexe d 1-c el ls A ∈ A , B , B ′ ∈ B , C ∈ C , and D ∈ D : 1. a) η : C /B / / ψ ( C / B ) b) η : B \ C / / ψ ( B \ C ) 2. a) η : D / A / / ψ ( D / A ) b) η : A \ D / / ψ ( A \ D ) 3. a) η \ 1 : ψ B \ C / / B \ C b) 1 /η : C / ψ B / / C /B 4. a) ψ ( η ◦ 1) : ψ ( B ◦ B ′ ) / / ψ ( ψ B ◦ B ′ ) b) ψ (1 ◦ η ) : ψ ( B ′ ◦ B ) / / ψ ( B ′ ◦ ψ B ) 5. a) ψ ( η ◦ 1) : ψ ( B ◦ A ) / / ψ ( ψ B ◦ A ) b) ψ (1 ◦ η ) : ψ ( A ◦ B ) / / ψ ( A ◦ ψ B ) 6. ψ ( η ◦ η ) : ψ ( B ◦ B ′ ) / / ψ ( ψ B ◦ ψ B ′ ) . A sp ecia l a pplication o f the reflec tion theorem is to the lo calisation of a probicat- egory A (cf. [4]). Let Σ b e a set of 2-cells in A and fo rm the lo ca lly Cauch y-dense map: Π : A / / A (Σ − 1 ) . Then the conditions of the reflection t heorem a pplied to the full r eflective em bed- ding: ψ ⊣ [Π , 1] : [ A (Σ − 1 ) , V ] / / [ A , V ] are conditions for A (Σ − 1 ) to carry a probicategory s tructure such that Π b ecomes a “map of probicategor ies”. 4. Extension theorem The following r esult is obtained by lo cally c ompleting the family C with respe c t to a suitable change of V -universe. Theorem. L et A = ( A , P , J, . . . ) b e a pr obic ate go ry and let N xy : A op xy / / C xy b e an ob( A ) × ob( A ) -indexe d family of dense functors. Then C admits a biclose d bic ate gory structu re , extending the structu re of A , if the fol lowing inde xe d c olimits and limits exist in C for al l appr opria tely indexe d 1-c el ls A, A ′ ∈ A and C, C ′ ∈ C : Q xy z ( A, A ′ ) = P xy z ( A, A ′ , X ) ∗ N xz X , I = J xx X ∗ N xx X , C ◦ C ′ = ( C y z ( N y z X , C ) ⊗ C xy ( N xy X ′ , C ′ ) ∗ Q xy z ( X, X ′ ) , H y z ( A, C ) = C xz ( Q xy z ( X, A ) , C ) ∗ N y z X , K xy ( A, C ) = C xz ( Q xy z ( A, X ) , C ) ∗ N xy X , C /C ′ = { C xy ( N xy X , C ′ ) , H y z ( X, C ) } , C \ C ′ = { C y z ( N y z X , C ) , K xy ( X, C ′ ) } , 4 BRIAN J. DA Y and t he fol lowing induc e d natur al tr ansformations C xz ( Q xy z ( A ′ , A ) , C ) ∼ = / / C y z ( N y z A ′ , H y z ( A, C )) C xz ( Q xy z ( A, A ′ ) , C ) ∼ = / / C xy ( N xy A ′ , K xy ( A, C )) ar e isomorphi sms (wher e H y z ( − , − ) : A xy ⊗ C op xz / / C y z K xy ( − , − ) : A op y z ⊗ C xz / / C xy , and − / − : C xz ⊗ C op xy / / C y z −\− : C op y z ⊗ C xz / / C xy ) . References [1] J. B´ enabou, Introduction to bicategories, Lecture N otes in Math. 47 Springer (1967) pp 1–77. [2] B. Day , On closed categories of f unctors, Rep orts of the Mi dw est Category Seminar IV, Lecture Notes in M athematics, V ol. 137 (Springer 1970), 1–38. [3] B. Day , A reflection theorem for closed categories, J. Pure Appl. Algebra 2 (1972), 1–11. [4] B. Day , Note on monoidal lo calisation, Bull. A ustral. Math. Soc. 8 (1973), 1–16. Centre of Australian Ca tegor y Theor y, M acquarie Un iversity, NSW, 2109 , Aus- tralia