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.
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 $A$ be a $k$-algebra and $C$ an $A$-coring, then a left $C$-comodule is just a left $A$-module $M$, rather than an $A$-bimodule in general. However the left $C$-coaction as a left $A$-module map $M\to C\otimes_A M$ can still be defined by the same equations (but for $A$-module maps), namely $C\otimes_A M$ and $C\otimes_A C\otimes_A M$ are still well defined as left $A$-modules since $C$ is an $A$-bimodule.
In the case when the coring $C$ is moreover a left $A$-bialgebroid, each left $C$-comodule $M$, which is by definition a left $A$-module, carries also a unique right $A$-module structure such that the left $C$-coaction is a right $A$-module map as well. It follows moreover that the two actions make $M$ an $A$-bimodule and the $C$-coaction factors through the Takeuchi product $C\times_A M$.
Let $k$ be a commutative ring and let $\mathcal{M} = Mod(k)$. 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.