Comodule

Remarks

Example: $C$ is a right comodule over itself with $\rho = \Delta$; clauses 1 and 2 are coassociativity and the counit law. Example: let $C = K[S]$ with $\Delta(s) = s \otimes s$ and $\varepsilon(s) = 1$, and let $M = \bigoplus_{s \in S} M_s$ be a vector space graded by $S$. Then $\rho(m) = m \otimes s$ for $m \in M_s$ is a coaction: both sides of clause 1 give $m \otimes s \otimes s$, and clause 2 gives $\varepsilon(s) m = m$. Every right $K[S]$-comodule arises in this way. Non-example: for $M \ne 0$, the zero map $\rho = 0$ satisfies clause 1 but fails clause 2. A morphism of comodules is a linear map $f \colon M \to N$ with $\rho_N \circ f = (f \otimes \id_C) \circ \rho_M$.