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  1. Is there a natural categorical way to capture the “adjunction” involving bimodules in a monoidal category? That is: given a biclosed monoidal category (𝒞,,1,[,] 𝒞 L,[,] 𝒞 R)(\mathcal{C},\otimes,1,[-,-]^{\mathrm{L}}_{\mathcal{C}},[-,-]^{\mathrm{R}}_{\mathcal{C}}) which is also bicomplete and whose tensor product respects equalisers and coequalisers, and given also monoids AA, BB, and CC in 𝒞\mathcal{C}, we have functors

    together with isomorphisms

    of (D,A)(D,A)-bimodules and (C,D)(C,D)-bimodules with

    • MM an (A,B)(A,B)-bimodule;
    • NN a (B,C)(B,C)-bimodule;
    • PP a (D,C)(D,C)-bimodule;
    • QQ an (A,D)(A,D)-bimodule.

    Due to this annoying combination of bimodule structures, it seems two-variable adjunctions don’t quite capture this concept. Is there some other notion which does?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2021

    There is a 2-variable adjunction, expressed as usual in terms of hom sets:

    Hom A,C(M BN,S)Hom A,B(M,Hom C R(N,S))Hom B,C(N,Hom A L(M,S)) Hom_{A,C}(M \boxtimes_B N, S) \cong Hom_{A,B}(M, \mathbf{Hom}_C^R(N,S)) \cong Hom_{B,C}(N, \mathbf{Hom}_A^L(M,S))

    for MM an (A,B)(A,B)-bimodule, NN a (B,C)(B,C)-bimodule, and SS an (A,C)(A,C)-bimodule. Your isomorphisms can be recovered from this by the Yoneda lemma together with associativity of \boxtimes:

    Hom D,A(T,Hom C R(M BN,P)) Hom D,C(T A(M BN),P) Hom D,C((T AM) BN,P) Hom D,B(T AM,Hom C R(N,P)) Hom D,A(T,Hom B R(M,Hom C R(N,P))) \begin{aligned} Hom_{D,A}(T, \mathbf{Hom}_C^R(M\boxtimes_B N, P)) &\cong Hom_{D,C}(T\boxtimes_A (M \boxtimes_B N), P)\\ &\cong Hom_{D,C}((T\boxtimes_A M) \boxtimes_B N, P) \\ &\cong Hom_{D,B}(T\boxtimes_A M, \mathbf{Hom}_C^R(N,P)) \\ &\cong Hom_{D,A}(T, \mathbf{Hom}_B^R(M, \mathbf{Hom}_C^R(N,P))) \end{aligned}
    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2021

    The same proof works in any closed bicategory.

  2. Thank you so much, Mike! This is exactly what I was looking for!

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