nLab comodule

Redirected from "co-action".
Contents

Contents

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 CC with comultiplication Δ C:CCC\Delta_C: C\to C\otimes C and counit ϵ:C1\epsilon:C\to \mathbf{1} in a monoidal category \mathcal{M}, and an object MM in \mathcal{M}, a left CC-coaction is

  • a morphism ρ:MCM\rho: M\to C\otimes M

  • which is

    • coassociative i.e. (for \mathcal{M} nonstrict use the canonical isomorphism C(CM)(CC)MC\otimes (C\otimes M)\cong (C\otimes C)\otimes M to compare the sides) (Δ Cid M)ρ=(id Cρ)ρ:MCCM(\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. (ϵid M)ρ=id M(\epsilon\otimes \mathrm{id}_M)\circ\rho = \mathrm{id}_M (in this formula, 1M\mathbf{1}\otimes M is identified with MM).

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 AA be a kk-algebra and CC an AA-coring, then a left CC-comodule is just a left AA-module MM, rather than an AA-bimodule in general. However the left CC-coaction as a left AA-module map MC AMM\to C\otimes_A M can still be defined by the same equations (but for AA-module maps), namely C AMC\otimes_A M and C AC AMC\otimes_A C\otimes_A M are still well defined as left AA-modules since CC is an AA-bimodule.

In the case when the coring CC is moreover a left AA-bialgebroid, each left CC-comodule MM, which is by definition a left AA-module, carries also a unique right AA-module structure such that the left CC-coaction is a right AA-module map as well. It follows moreover that the two actions make MM an AA-bimodule and the CC-coaction factors through the Takeuchi product C× AMC\times_A M.

The category of comodules

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

See Wischnewsky.

Examples

References

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 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.