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.
Let be a commutative ring and let . Then one has the following properties:
See Wischnewsky.
The category of linear representations of an affine group is equivalent to the category of comodules over a Hopf algebra.
Properties of the category of comodules over a coalgebra are studied in
Last revised on March 7, 2017 at 19:36:03. See the history of this page for a list of all contributions to it.