comodule

A module is a vector space acted on by an algebra: the algebra reaches in and moves vectors around. A comodule is the mirror image: instead of being acted on, the space coacts onto itself tensored with a coalgebra, recording 'where each vector could come from.' If a module is how a symmetry pushes a space, a comodule is how a space resolves itself into pieces labelled by a coalgebra.

Formally, a right comodule over a coalgebra C is a vector space M with a linear coaction ρ : M -> M ⊗ C satisfying coassociativity (ρ ⊗ id) ∘ ρ = (id ⊗ Δ) ∘ ρ and counitality (id ⊗ ε) ∘ ρ = id (after M ⊗ k ≅ M). One uses a Sweedler-style notation ρ(m) = Σ m(0) ⊗ m(1), with the (0)-leg in M and the (1)-leg in C.

Comodules are dual to modules and often closer to geometry. A comodule over the coordinate Hopf algebra O(G) of an algebraic group G is precisely a rational (algebraic) representation of G, whereas modules over O(G) are far more rigid. So the comodule language is the natural home for representations of affine group schemes and, after deformation, for representations of quantum groups.