nLab comodule

Idea

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.

Definition

Over comonoids

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.

Over corings

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 $C\times_A M$.

The category of comodules

Let $k$ be a commutative ring and let $\mathcal{M} = Mod(k)$. Then one has the following properties:

• The category of comodules over a $k$-coalgebra is comonadic over $Mod(k)$.
• It is cocomplete and complete.
• It has a cogenerator.

See Wischnewsky.

Examples

• The category of linear representations of an affine group is equivalent to the category of comodules over a Hopf algebra.

Related concepts

• dg-comodule

• comodule spectrum

References

• Tomasz Brzeziński, Robert Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

Properties of the category of comodules over a coalgebra are studied in

• Manfred Wischnewsky, On linear representations of affine groups, I, Pacific Journal of Mathematics 61, No. 2, 1975, Project Euclid.

On local presentability of categories of coalgebras and their comodules:

• Hans-Eberhard Porst, On corings and comodules, Archivum Mathematicum 42 4 (2006) 419-425 [dmlcz:108017, pdf]