Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex 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 foundation 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 homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory 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 nforum nlab 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 sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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).

nLab > nLab General Discussions: [deleted]

Bottom of Page

1 to 4 of 4

  1.  
    • CommentRowNumber1.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJul 7th 2021
    • (edited Apr 11th 2023)

    [deleted]

    • 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.

    • CommentRowNumber4.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJul 7th 2021
    • (edited Apr 11th 2023)

    [deleted]

1 to 4 of 4

  1.