Coproducts

A coproduct is a categorical construction that serves as the dual notion to a product. In a category C, a family of objects {A_i} has a coproduct if there exists an object ⊔_i A_i together with morphisms ι_i: A_i → ⊔_i A_i called coprojections, such that for any object X and any family of morphisms f_i: A_i → X, there exists a unique morphism f: ⊔_i A_i → X satisfying f ∘ ι_i = f_i for all i. This universal property means that morphisms going out of the coproduct correspond precisely to compatible families of morphisms from each summand.

In the two-object case, the coproduct of A and B is denoted A ⊔ B, with injections ι_A: A → A ⊔ B and ι_B: B → A ⊔ B, and it satisfies the same universal property with pairs of morphisms.

Examples from familiar categories illustrate how coproducts specialize. In Set, the coproduct is the disjoint union: the family {A_i} maps into the union via inclusions, and a function from the union to X corresponds to a family of functions from each A_i to X. In Group, the coproduct is the free product; in the category of abelian groups or R-modules, the coproduct is the direct sum (finite or infinite). In Top, the coproduct is the disjoint union topology. In the category of commutative algebras over a ring, the coproduct is given by the tensor product over the base ring, reflecting a “gluing” that preserves algebraic structure.

Coproducts are a kind of colimit, and they generalize the idea of forming unions of objects while preserving their embedding structure. They are preserved by left adjoint functors and play a central role in constructions that assemble objects from pieces.