nLab
comodule

Idea

A comodule is to a comonoid as a module is to a monoid

Definition

Given a comonoid $C$ with comultiplication ${\Delta }_C:C\to C\otimes C$ and counit $ϵ:C\to 1$ in a monoidal category $ℳ$, and an object $M$ in $ℳ$, a left $C$-coaction is

• a morphism $\rho :M\to C\otimes M$

• which is

  • coassociative i.e. (for $ℳ$ 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. $(ϵ\otimes {\mathrm{id}}_M)\circ \rho ={\mathrm{id}}_M$ (in this formula, $1\otimes M$ 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.