Contents

Idea

A bimodule is a module in two compatible ways over two algebras.

Definition

Let $V$ be a closed monoidal category. Recall that for $C$ a category enriched over $V$, a $V$-module 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 for $V$-categories $C$ and $D$ is a $V$-functor

Such a functor is also called a profunctor or distributor.

Examples

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

• 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.

Properties

The 2-category of algebras and bimodules

Let $R$ be a commutative ring and consider bimodules over $R$-algebras.

This is the Eilenberg-Watts theorem.

The $(\infty,2)$-category of $\infty$-algebras and $\infty$-bimodules

Let $\mathcal{C}$ be monoidal (∞,1)-category such that

1. it admits geometric realization of simplicial objects in an (∞,1)-category;

2. the tensor product $\otimes \colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$ preserves this geometric realization separately in each argument.

Then there is an (∞,2)-category $Mod(\mathcal{C})$ which given as an (∞,1)-category object internal to (∞,1)Cat has

• $(\infty,1)$-category of objects

  $$Mod(\mathcal{C})_{[0]} \simeq Alg(\mathcal{C})$$

  the A-∞ algebras in $\mathcal{C}$;

• $(\infty,1)$-category of morphisms

  $$Mod(\mathcal{C})_{[1]} \simeq BMod(\mathcal{C})$$

  the $\infty$-bimodules in $\mathcal{C}$.

This is (Lurie, def. 4.3.6.10, remark 4.3.6.11).

Related concepts

• 2-module, 2-ring

References

Bimodules in homotopy theory/higher algebra are discussed in section 4.3 of

• Jacob Lurie, Higher Algebra

Specifically the homotopy theory of A-infinity bimodules? is discussed in

• Volodymyr Lyubashenko, Oleksandr Manzyuk, A-infinity-bimodules and Serre A-infinity-functors (arXiv:math/0701165)

and section 5.4.1 of

• Boris Tsygan, Noncommutative calculus and operads in Guillermo Cortinas (ed.) Topics in Noncommutative geometry, Clay Mathematics Proceedings volume 16