nLab
bimodule

Contents

Idea

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

Definition

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

In this language a CC-DD bimodule for VV-categories CC and DD is a VV-functor

C opDV. 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 opC^{op} from a VV-category CC requires that VV be symmetric (or at least braided) monoidal. It’s possible to define CC-DD bimodules without recourse to C opC^{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 VV (with \otimes cocontinuous in both arguments; biclosedness would ensure that), and consider smallness assumptions on the objects CC, DD, etc. —Todd.

Examples

  • Let V=SetV = Set and let C=DC = D. Then the hom functor C(,):C op×CSetC(-, -):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 VV) between an object of CC and an object of DD.

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

  • Let V=VectV = Vect and let C=BA 1C = \mathbf{B}A_1 and D=BA 2D = \mathbf{B}A_2 be two one-object VectVect-enriched categories, whose endomorphism vector spaces are hence algebras. Then a CC-DD bimodule is a vector space VV with an action of A 1A_1 on the left and and action of A 2A_2 on the right.

Properties

The 1-category of bimodules and intertwiners

Definition

For RR a commutative ring, write BMod RBMod_R for the category whose

  • objects are triples (A,B,N)(A,B,N) where AA and BB are RR-algebras and where NN is an AA-BB-bimodule;

  • morphisms are triples (f,g,ϕ)(f,g, \phi) consisting of two algebra homomorphisms f:AAf \colon A \to A' and B:BBB \colon B \to B' and an intertwiner of AA-BB'-bimdules ϕ:NgfN\phi \colon N \cdot g \to f \cdot N'. This we may depict as a

    A N B f ϕ g A N B. \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 RBMod_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 RR be a commutative ring and consider bimodules over RR-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

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 RR-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 CatCat

(X 1 0 1X 0)Cat Δ 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=Alg R X_0 = Alg_R

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

X 1=BMod R X_1 = BMod_R

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

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

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

References

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.

Revised on October 24, 2014 08:45:09 by Urs Schreiber (185.26.182.28)