Comodule ($(M,\rho )$)

A right comodule over a coalgebra $(C,\Delta ,\epsilon )$ over a commutative ring $k$ is a $k$-module $M$ with a coaction $\rho :M\to M{\otimes }_{k}C$ satisfyingusing the canonical tensor identifications. In Sweedler notation, $\rho (m)=\sum m_{(0)}\otimes m_{(1)}$. The equations are the reversed-arrow counterparts of associativity and the unit law for an algebra module.^[ref]

For the group algebra coalgebra $kG$ with $\Delta (g)=g\otimes g$, a direct sum $M={⨁}_{g\in G}{M}_g$ gives a right comodule by $\rho (m_g)=m_g\otimes g$. Conversely, expanding a coaction in the basis of group elements recovers this direct sum decomposition. Thus comodules over this coalgebra encode a decomposition indexed by $G$, whereas modules over the same group algebra encode a group representation. More generally, the definition extends to comonoids in a monoidal category whenever the relevant tensor action is specified.

A natural underlying transformation of tensor bifunctors on right comodules over a field has components ${\phi }_{M,N}(u\otimes v)=\sum u_{(0)}\otimes v_{(0)}\gamma (u_{(1)},v_{(1)})$. Recover $\gamma =(\epsilon \otimes \epsilon ){\phi }_{H,H}$; colinearity and monoidal axioms impose further equations.

For a transformation from the tensor built using $n$ to the tensor built using $m$, colinearity is $m\ast \gamma =\gamma \ast n$: $\sum m(a_{(1)},b_{(1)})\gamma (a_{(2)},b_{(2)})=\sum \gamma (a_{(1)},b_{(1)})n(a_{(2)},b_{(2)})$. Scalar maps enter this convolution product for coalgebra maps via the common algebra unit.

A comonoid morphism $f:H\to K$ induces a functor $f_{\ast }$ on right comodules, replacing $\rho$ by $(1\otimes f)\rho$ and preserving underlying objects and arrows. For bimonoids, it is strict monoidal exactly when $f$ is also a monoid morphism.

A natural transformation between corestriction functors over a field has components ${\beta }_M(u)=\sum u_{(0)}\alpha (u_{(1)})$, recovered by $\alpha =\epsilon {\beta }_H$. Monoidality forces $\alpha$ to be a unital algebra homomorphism over a field; for a Hopf algebra, $\alpha S$ supplies the inverse.

The structure map of a comodule; a right coaction satisfies $(\rho \otimes 1)\rho =(1\otimes \Delta )\rho$ with coherent parentheses, and $(1\otimes \epsilon )\rho$ is the inverse right unitor.