coassociativity

Coassociativity is a property of comultiplication in coalgebras and, more generally, of comonoid structures in monoidal categories. For a coalgebra over a field k, with a linear map Δ: C → C ⊗ C called the comultiplication and a counit ε: C → k, the structure is coassociative if the two ways of splitting a element into three tensor factors agree: (Δ ⊗ id_C) ∘ Δ = (id_C ⊗ Δ) ∘ Δ as maps C → C ⊗ C ⊗ C.

The counit axiom also accompanies a coalgebra: (ε ⊗ id_C) ∘ Δ = id_C = (id_C ⊗ ε) ∘ Δ. These two identities ensure that the process of repeatedly decomposing elements via Δ is deterministic, regardless of how the decompositions are nested.

Coassociativity is the dual notion of associativity in algebras. If multiplication is associative, then, under a suitable dualization (often in finite dimensions), the corresponding comultiplication is coassociative. In categorical terms, a comonoid (or coalgebra) is an object equipped with Δ and ε satisfying these coassociativity and counit laws.

Examples include the group algebra k[G], where Δ(g) = g ⊗ g and ε(g) = 1 for each group element g, which is coassociative and counital. Another standard example is the symmetric (polynomial) algebra S(V) with a cobracket defined by Δ(v) = v ⊗ 1 + 1 ⊗ v for v in a vector space V, extended to a coalgebra structure; Δ is coassociative and cocommutative in this setting.

In broader contexts, coassociativity appears in Hopf algebras and in comonoids within various monoidal categories, reflecting a foundational duality with associativity in algebra.