symmetric monoidal (∞,1)-category of spectra
A comodule is to a comonoid as a module is to a monoid. Where a module is equipped with an action, a comodule is dually equipped with a coaction.
Given a comonoid with comultiplication and counit in a monoidal category , and an object in , a left -coaction is
a morphism
which is
coassociative i.e. (for nonstrict use the canonical isomorphism to compare the sides)
and counital i.e. (in this formula, is identified with ).
In some monoidal categories, e.g. of (super)vector spaces, and of Hilbert spaces, one often says (left/right) corepresentation instead of (left/right) coaction.
Although corings are comonoids in the monoidal category of bimodules, the comodules over corings are not defined as bimodules with a coaction but as modules with a coaction on the same side. Let be a -algebra and an -coring, then a left -comodule is just a left -module , rather than an -bimodule in general. However the left -coaction as a left -module map can still be defined by the same equations (but for -module maps), namely and are still well defined as left -modules since is an -bimodule.
In the case when the coring is moreover a left -bialgebroid, each left -comodule , which is by definition a left -module, carries also a unique right -module structure such that the left -coaction is a right -module map as well. It follows moreover that the two actions make an -bimodule and the -coaction factors through the Takeuchi product .
Let be a commutative ring and let . Then one has the following properties:
See Wischnewsky.
Properties of the category of comodules over a coalgebra are studied in
On local presentability of categoris of coalgebras and their comodules:
Last revised on September 12, 2024 at 14:55:56. See the history of this page for a list of all contributions to it.