Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural nforum nlab nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  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!

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)