nLab bimodule object

Contents

Context

Category theory

Enriched category theory

Categorical algebra

Contents

Idea

The notion of bimodule makes sense internal to a monoidal category or a monoidal (infinity,1)-category.

Definition

As a functor into a closed monoidal category

Let VV be a closed monoidal category. Recall that for CC a category enriched over VV, a CC-module object in VV is a VV-functor ρ:CV\rho : C \to V. We think of the objects ρ(a)\rho(a) for aObj(C)a \in Obj(C) as the objects on which CC acts, and of ρ(C(a,b))\rho(C(a,b)) as the action of CC on these objects.

In this language a CC-DD bimodule object in VV for VV-enriched categories CC and DD is a VV-functor

C opDV. C^{op} \otimes D \to V \,.

Such a functor is also called a profunctor or distributor. Bimodule objects in VV can be thought of as a kind of generalized hom, giving a set of morphisms (or object of VV) between an object of CC and an object of DD.

Some points are in order. Strictly speaking, the construction of C opC^{op} from a VV-category CC requires that VV be symmetric (or at least braided) monoidal. It’s possible to define CC-DD-bimodule objects in VV without recourse to C opC^{op}, but then either that should be spelled out, or one should include a symmetry. (If the former is chosen, then closedness one on side might not be the best choice of assumption, in view of the next remark; a more natural choice might be biclosed monoidal.)

Second: bimodule objects are not that much good unless you can compose them; for that one should add some cocompleteness assumptions to VV (with \otimes cocontinuous in both arguments; biclosedness would ensure that), and consider smallness assumptions on the objects CC, DD, etc. —Todd.

Examples

  • Let V=SetV = Set and let C=DC = D. Then the hom functor C(,):C op×CSetC(-, -):C^{op} \times C \to Set is a bimodule.

  • Let C^=Set C op\hat{C} = Set^{C^{op}}; the objects of C^\hat{C} are “generating functions” that assign to each object of CC a set. Every bimodule f:D op×CSetf:D^op \times C \to Set can be curried to give a Kleisli arrow f˜:CD^\tilde{f}:C \to \hat{D}. Composition of these arrows corresponds to convolution of the generating functions.

    Todd: I am not sure what is trying to be said with regard to “convolution”. I know about Day convolution, but this is not the same thing.

    Also, with regard to “Kleisli arrow”: I understand the intent, but one should proceed with caution since there is no global monad CC^C \mapsto \hat{C} to which Kleisli would refer. Again there are size issues that need attending to.

  • Let V=VectV = Vect and let C=BA 1C = \mathbf{B}A_1 and D=BA 2D = \mathbf{B}A_2 be two one-object VectVect-enriched categories, whose endomorphism vector spaces are hence algebras. Then a CC-DD-bimodule is a vector space VV with an action of A 1A_1 on the left and and action of A 2A_2 on the right.

See also

Last revised on May 25, 2022 at 11:20:08. See the history of this page for a list of all contributions to it.