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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 $(\mathcal{C},\otimes,1,[-,-]^{\mathrm{L}}_{\mathcal{C}},[-,-]^{\mathrm{R}}_{\mathcal{C}})$ which is also bicomplete and whose tensor product respects equalisers and coequalisers, and given also monoids $A$, $B$, and $C$ in $\mathcal{C}$, we have functors

together with isomorphisms

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

• $M$ an $(A,B)$-bimodule;
• $N$ a $(B,C)$-bimodule;
• $P$ a $(D,C)$-bimodule;
• $Q$ an $(A,D)$-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:

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

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

\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!