Showing changes from revision #19 to #20: Added | Removed | Changed

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 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.

This is the Eilenberg-Watts theorem.

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

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. 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 $f \colon A \to A'$ 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

We discuss the generalization of the notion of bimodules to homotopy theory, hence the generalization from category theory to (∞,1)-category theory. (Lurie, section 4.3).

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

1. it admits geometric realization of simplicial objects in an (∞,1)-category (hence a left adjoint (∞,1)-functor ${\vert-\vert} \colon \mathcal{C}^{\Delta^{op}} \to \mathcal{C}$ to the constant simplicial object functor), true notably when $\mathcal{C}$ is a presentable (∞,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 and ∞-algebra homomorphisms in $\mathcal{C}$;

• $(\infty,1)$-category of morphisms

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

  the $\infty$-bimodules and bimodule homomorphisms (intertwiners) in $\mathcal{C}$

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

Morover, the horizontal composition of bimodules in this (∞,2)-category is indeed the relative tensor product

This is (Lurie, lemma 4.3.6.9 (3)).

Here are some steps in the construction:

(Lurie, notation 4.3.4.15)

This is (Lurie, cor. 4.3.6.2) specified to the case of (Lurie, lemma 4.3.6.9).

(…)

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