Coalgebras

Coalgebras are algebraic structures dual to algebras. Over a field k, a coalgebra is a k-vector space C equipped with linear maps Δ: C → C ⊗ C and ε: C → k, called comultiplication and counit. These maps satisfy coassociativity (Δ ⊗ id_C) ∘ Δ = (id_C ⊗ Δ) ∘ Δ and counit laws (ε ⊗ id_C) ∘ Δ = id_C = (id_C ⊗ ε) ∘ Δ. Using Sweedler notation, Δ(c) = ∑ c_(1) ⊗ c_(2).

Examples include: the free vector space on a set X, with Δ(x) = x ⊗ x and ε(x) = 1 for all basis elements x ∈ X; extend linearly to all of k[X]. Another standard example is the space C = Fun(G, k) of all functions G → k on a finite group G, with (Δf)(g,h) = f(gh) and ε(f) = f(e). This coalgebra is dual to the group algebra k[G] when G is finite.

Finite-dimensional duality connects algebras and coalgebras: the linear dual A* of an algebra A becomes a coalgebra with Δ and ε given by the transpose of multiplication and unit; conversely the finite dual of a coalgebra C has an algebra structure. A coalgebra also has comodules, which generalize modules: a right C-comodule M has a structure map ρ: M → M ⊗ C satisfying coassociativity and counit conditions.

Coalgebras appear in representation theory, algebraic groups, quantum groups, and the theory of Hopf algebras and tensor categories. They provide a framework for studying co-operations and morphisms that preserve them.