Ccomodule

A C-comodule is a algebraic structure that encodes a way a vector space can carry a coaction of a coalgebra C over a field k. Let C be a coalgebra with comultiplication Δ: C → C ⊗ C and counit ε: C → k. A left C-comodule consists of a k-vector space M and a k-linear map ρ: M → C ⊗ M called the coaction, satisfying the coassociativity condition (Δ ⊗ id_M) ∘ ρ = (id_C ⊗ ρ) ∘ ρ and the counit condition (ε ⊗ id_M) ∘ ρ = id_M. A right C-comodule uses a coaction ρ: M → M ⊗ C with the dual axioms (ρ ⊗ id_C) ∘ ρ = (id_M ⊗ Δ) ∘ ρ and (id_M ⊗ ε) ∘ ρ = id_M.

Morphisms between left C-comodules (M, ρ_M) and (N, ρ_N) are linear maps f: M → N that preserve the coaction, i.e., ρ_N ∘ f = (id_C ⊗ f) ∘ ρ_M. For right comodules, the compatibility condition is ρ_N ∘ f = (f ⊗ id_C) ∘ ρ_M.

Examples include C itself with the coaction ρ = Δ, turning C into a comodule over itself, and more generally, comodules arising from subcoalgebras or group-like elements. In finite dimensions, comodules over C correspond to modules over the finite-dimensional algebra C^* (the dual), yielding a duality between corepresentations of C and representations of C^*. The category of comodules is abelian, with subcomodules, quotients, and extensions mirroring the module-theoretic theory.