Coalgebra

A coalgebra over a field or ring is a linear space C equipped with a comultiplication Δ: C→C⊗C and a counit ε: C→k satisfying coassociativity ((Δ⊗id)∘Δ=(id⊗Δ)∘Δ) and counitality ((ε⊗id)∘Δ=id=(id⊗ε)∘Δ). It is the categorical dual notion to an algebra when arrows are reversed.