symmetric monoidal (∞,1)-category of spectra
Let be a closed monoidal category. Recall that for a category enriched over , a -module is a -functor . We think of the objects for as the objects on which acts, and of as the action of on these objects.
In this language a - bimodule for -categories and is a -functor
Such a functor is also called a profunctor or distributor.
Some points are in order. Strictly speaking, the construction of from a -category requires that be symmetric (or at least braided) monoidal. It’s possible to define - bimodules without recourse to , 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 (with cocontinuous in both arguments; biclosedness would ensure that), and consider smallness assumptions on the objects , , etc. —Todd.
Let and let . Then the hom functor is a bimodule. Bimodules can be thought of as a kind of generalized hom, giving a set of morphisms (or object of ) between an object of and an object of .
Let ; the objects of are “generating functions” that assign to each object of a set. Every bimodule can be curried to give a Kleisli arrow . 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 to which Kleisli would refer. Again there are size issues that need attending to.
Let and let and be two one-object -enriched categories, whose endomorphism vector spaces are hence algebras. Then a - bimodule is a vector space with an action of on the left and and action of on the right.
As this notation suggests, is naturally the vertical category of a pseudo double category whose horizontal composition is given by tensor product of bimodules. spring
Let be a commutative ring and consider bimodules over -algebras.
There is a 2-category whose
This is the Eilenberg-Watts theorem.
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 -algebras, bimodules and intertwiners. See also at 2-ring.
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
which satisfies the Segal conditions. Here
is the category of def. 1, whose objects are pairs consisting of two algebras and and an - bimodule between them, and whose morphisms are pairs consisting of two algebra homomorphisms and and an intertwiner .
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
For more on that see at (∞,1)-bimodule.