nLab comodule




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 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.

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.



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, Vol. 61, No. 2, 1975, Project Euclid.

On local presentability of categoris of coalgebras and their comodules:

Last revised on November 6, 2023 at 07:48:04. See the history of this page for a list of all contributions to it.