# nLab bimodule

Contents

### Context

#### Algebra

higher algebra

universal algebra

# 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 $C$-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

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

Such a functor is also called a profunctor or distributor.

Some points are in order. Strictly speaking, the construction of $C^{op}$ 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 $C^{op}$, 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 $V = Set$ 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 1-category of bimodules and intertwiners

###### Definition

For $R$ a commutative ring, write $BMod_R$ 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 $(f,g, \phi)$ consisting of two algebra homomorphisms $f \colon A \to A'$ and $B \colon B \to B'$ and an intertwiner of $A$-$B'$-bimdules $\phi \colon N \cdot g \to f \cdot N'$. This we may depict as a

$\array{ A &\stackrel{N}{\to}& B \\ {}^{\mathllap{f}}\downarrow &\Downarrow_{\phi}& \downarrow^{\mathrlap{g}} \\ A' &\stackrel{N'}{\to}& B' } \,.$
###### Remark

As this notation suggests, $BMod_R$ 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

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 $Mod_A$;

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

$(-)\otimes_{A_1} N \;\colon\; Mod_{A_1} \to Mod_{A_2}$
• 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 $Cat$

$\left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0 \right) \in Cat^{\Delta^{op}}$

which satisfies the Segal conditions. Here

$X_0 = Alg_R$

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

$X_1 = BMod_R$

is the category of def. , 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 $f \colon A \to A'$ and $g \colon B \to B'$ and an intertwiner $N \cdot (g) \to (f) \cdot N'$.

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

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

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

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

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