nLab bimodule object

Contents

category theory

Applications

Enriched category theory

enriched category theory

Extra stuff, structure, property

Homotopical enrichment

Categorical algebra

internalization and categorical algebra

universal algebra

categorical semantics

Contents

Idea

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

Definition

As a functor into a closed monoidal category

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

In this language a $C$-$D$ bimodule object or biaction object in $V$ for $V$-enriched categories $C$ and $D$ is a $V$-functor

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

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

Some points are in order. Strictly speaking, the construction of $C^{op}$ from a $V$-category $C$ requires that $V$ be symmetric (or at least braided) monoidal. It’s possible to define $C$-$D$-bimodule objects in $V$ without recourse to $C^{op}$, 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 $V$ (with $\otimes$ cocontinuous in both arguments; biclosedness would ensure that), and consider smallness assumptions on the objects $C$, $D$, etc. —Todd.

Examples

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

• Let $\hat{C} = Set^{C^{op}}$; the objects of $\hat{C}$ are “generating functions” that assign to each object of $C$ a set. Every bimodule $f:D^op \times C \to Set$ can be curried to give a Kleisli arrow $\tilde{f}:C \to \hat{D}$. 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 $C \mapsto \hat{C}$ to which Kleisli would refer. Again there are size issues that need attending to.

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