# nLab comodule

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 $C\otimes \left(C\otimes M\right)\cong \left(C\otimes C\right)\otimes M$ to compare the sides) $\left({\Delta }_{C}\otimes {\mathrm{id}}_{M}\right)\circ \rho =\left({\mathrm{id}}_{C}\otimes \rho \right)\circ \rho :M\to C\otimes C\otimes M$

• and counital i.e. $\left(ϵ\otimes {\mathrm{id}}_{M}\right)\circ \rho ={\mathrm{id}}_{M}$ (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.

Revised on November 12, 2012 02:19:23 by Zoran Škoda (193.55.36.32)