nForum - Discussion Feed ("adjunctions" of bimodules) 2022-01-19T02:46:34-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Théo de Oliveira S. comments on ""adjunctions" of bimodules" (93721) https://nforum.ncatlab.org/discussion/13152/?Focus=93721#Comment_93721 2021-07-07T17:26:09-04:00 2022-01-19T02:46:33-05:00 Théo de Oliveira S. https://nforum.ncatlab.org/account/2018/ Thank you so much, Mike! This is exactly what I was looking for!

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

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Mike Shulman comments on ""adjunctions" of bimodules" (93714) https://nforum.ncatlab.org/discussion/13152/?Focus=93714#Comment_93714 2021-07-07T11:43:36-04:00 2022-01-19T02:46:33-05:00 Mike Shulman https://nforum.ncatlab.org/account/3/ The same proof works in any closed bicategory.

The same proof works in any closed bicategory.

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Mike Shulman comments on ""adjunctions" of bimodules" (93712) https://nforum.ncatlab.org/discussion/13152/?Focus=93712#Comment_93712 2021-07-07T11:39:54-04:00 2022-01-19T02:46:34-05:00 Mike Shulman https://nforum.ncatlab.org/account/3/ There is a 2-variable adjunction, expressed as usual in terms of hom sets: Hom A,C(M&boxtimes; BN,S)&cong;Hom A,B(M,Hom C R(N,S))&cong;Hom B,C(N,Hom A L(M,S)) Hom_{A,C}(M \boxtimes_B N, ...

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} ]]>
Théo de Oliveira S. comments on ""adjunctions" of bimodules" (93697) https://nforum.ncatlab.org/discussion/13152/?Focus=93697#Comment_93697 2021-07-07T05:24:27-04:00 2022-01-19T02:46:34-05:00 Théo de Oliveira S. https://nforum.ncatlab.org/account/2018/ Is there a natural categorical way to capture the “adjunction” involving bimodules in a monoidal category? That is: given a biclosed monoidal category ...

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?

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