nLab
bimodule

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 $a\in \mathrm{Obj}(C)$ 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

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

Such a functor is also called a profunctor or distributor.

Some points are in order. Strictly speaking, the construction of $C^{\mathrm{op}}$ from a $V$-category $C$ requires that $V$ be symmetric (or at least braided) monoidal. It’s possible to define $C$-$D$ bimodules without recourse to $C^{\mathrm{op}}$, 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: bimodules 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 $V=\mathrm{Set}$ and let $C=D$. Then the hom functor $C(-,-):C^{\mathrm{op}}\times C\to \mathrm{Set}$ 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 $\hat{C}={\mathrm{Set}}^{C^{\mathrm{op}}}$; the objects of $\hat{C}$ are “generating functions” that assign to each object of $C$ a set. Every bimodule $f:D^{\mathrm{op}}\times C\to \mathrm{Set}$ 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 $V=\mathrm{Vect}$ and let $C=B{A}_1$ and $D=B{A}_2$ be two one-object $\mathrm{Vect}$-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 1-category of bimodules and intertwiners

Definition

For $R$ a commutative ring, write ${\mathrm{BMod}}_R$ for the category whose

• objects are triples $(A,B,N)$ where $A$ and $B$ are $R$-algebras and where $N$ is an $A$-$B$-bimodule;

• morphisms are triples $(f,g,\varphi )$ consisting of two algebra homomorphisms $f:A\to A\prime$ and $B:B\to B\prime$ and an intertwiner of $A$-$B\prime$-bimdules $\varphi :N\cdot g\to f\cdot N\prime$. This we may depict as a

  $$\begin{array}{ccc}A& \stackrel{N}{\to }& B\\ {}^f\downarrow & {\Downarrow }_{\varphi }& {\downarrow }^g\\ A\prime & \stackrel{N\prime }{\to }& B\prime \end{array}.$$\array{ A &\stackrel{N}{\to}& B \\ {}^{\mathllap{f}}\downarrow &\Downarrow_{\phi}& \downarrow^{\mathrlap{g}} \\ A' &\stackrel{N'}{\to}& B' } \,.

Remark

As this notation suggests, ${\mathrm{BMod}}_R$ is naturally the vertical category of a pseudo double category whose horizontal composition is given by tensor product of bimodules. spring

The 2-category of algebras and bimodules

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

Proposition

There is a 2-category whose

• objects are $R$-algebras;

• 1-morphisms are bimodules;

• 2-morphisms are intertwiners.

The composition of 1-morphisms is given by the tensor product of modules over the middle algebra.

Proposition

There is a 2-functor from the above 2-category of algebras and bimodules to Cat which

• sends an $R$-algebra $A$ to its category of modules ${\mathrm{Mod}}_A$;

• sends a $A_1$-$A_2$-bimodule $N$ to the tensor product functor

  $$(-){\otimes }_{A_1}N:{\mathrm{Mod}}_{A_1}\to {\mathrm{Mod}}_{A_2}$$(-)\otimes_{A_1} N \;\colon\; Mod_{A_1} \to Mod_{A_2}

• sends an intertwiner to the evident natural transformation of the above functors.

Proposition

This construction has as its image precisely the colimit-preserving functors between categories of modules.

This is the Eilenberg-Watts theorem.

Remark

In the context of higher category theory/higher algebra one may interpret this as says that the 2-category of those 2-modules over the given ring which are equivalent to a category of modules is that of $R$-algebras, bimodules and intertwiners. See also at 2-ring.

Remark

The 2-category of algebras and bimodules is an archtypical example for a 2-category with proarrow equipment, hence for a pseudo double category with niche-fillers. Or in the language of internal (infinity,1)-category-theory: it naturally induces the structure of a simplicial object in the (2,1)-category $\mathrm{Cat}$

$$\left(\cdots \overrightarrow{\overrightarrow{\to }}{X}_1\stackrel{\stackrel{{\partial }_1}{\to }}{\underset{{\partial }_0}{\to }}{X}_0\right)\in {\mathrm{Cat}}^{{\Delta }^{\mathrm{op}}}$$\left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0 \right) \in Cat^{\Delta^{op}}

which satisfies the Segal conditions. Here

$$X_0={\mathrm{Alg}}_R$$X_0 = Alg_R

is the category of associative algebras and homomorphisms between them, while

$$X_1={\mathrm{BMod}}_R$$X_1 = BMod_R

is the category of def. 1, whose objects are pairs consisting of two algebras $A$ and $B$ and an $A$-$B$ bimodule $N$ between them, and whose morphisms are pairs consisting of two algebra homomorphisms $f:A\to A\prime$ and $g:B\to B\prime$ and an intertwiner $N\cdot (g)\to (f)\cdot N\prime$.

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

The above has a generalization to (infinity,1)-bimodules. See there for more.

Related concepts

• Hilbert bimodule

• bibundle

• (infinity,1)-bimodule

• 2-module, 2-ring

References

The 2-category of bimodules in its incarnation as a 2-category with proarrow equipment appears as example 2.3 in

• Michael Shulman, Framed bicategories and monoidal fibrations (arXiv:0706.1286)

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

• Jacob Lurie, Higher Algebra

For more on that see at (∞,1)-bimodule.