internalization and categorical algebra
algebra object (associative, Lie, …)
The notion of biaction or bimodule makes sense internal to a monoidal category or a monoidal (infinity,1)-category.
Let be a closed monoidal category. Recall that for a category enriched over , a -module object in 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 object or biaction object in for -enriched categories and is a -functor
Such a functor is also called a profunctor or distributor. Bimodule objects in 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 .
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 --bimodule objects in 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: bimodule objects 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.
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.
Last revised on October 12, 2022 at 12:33:11. See the history of this page for a list of all contributions to it.