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 $C$ with comultiplication $\Delta_C: C\to C\otimes C$ and counit $\epsilon:C\to \mathbf{1}$ in a monoidal category $\mathcal{M}$, and an object $M$ in $\mathcal{M}$, a left $C$-coaction is
a morphism $\rho: M\to C\otimes M$
which is
coassociative i.e. (for $\mathcal{M}$ nonstrict use the canonical isomorphism $C\otimes (C\otimes M)\cong (C\otimes C)\otimes M$ to compare the sides) $(\Delta_C\otimes\mathrm{id}_M)\circ\rho = (\mathrm{id}_C\otimes\rho)\circ\rho: M\to C\otimes C\otimes M$
and counital i.e. $(\epsilon\otimes \mathrm{id}_M)\circ\rho = \mathrm{id}_M$ (in this formula, $\mathbf{1}\otimes M$ is identified with $M$).
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 $k$ be a commutative ring and let $\mathcal{M} = Mod(k)$. 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