Coproduct

A coproduct in category theory is a general construction that captures the idea of a “sum” of objects. Given a family of objects {A_i} in a category, a coproduct is an object ⨿A_i together with morphisms i_i: A_i → ⨿A_i for each i, such that for any object X and any family of morphisms f_i: A_i → X, there exists a unique morphism u: ⨿A_i → X with f_i = u ∘ i_i for all i. This universal property is dual to that of a product, where arrows go in the opposite direction.

Examples help illustrate the concept. In the category Set, the coproduct of a family of sets is their disjoint union, with the inclusions i_i being the natural injections. Given functions f_i: A_i → X, there is a unique function from the disjoint union to X that extends each f_i. In the category of groups, the coproduct is the free product of groups. In abelian groups, the coproduct is the direct sum. In the category of commutative rings with unity, the coproduct is the tensor product over the integers. In Top, the coproduct is again the disjoint union topology.

Coproducts are examples of colimits: they are the colimit of a discrete diagram. They provide a flexible way to construct objects that unify several inputs via universal properties, and they interact with limits in ways that reveal dualities across mathematical structures.