pushout

A pushout is a construction in category theory that formalizes gluing two objects along a common part. Given a pair of morphisms f: A -> B and g: A -> C in a category, a pushout consists of an object P and morphisms i: B -> P and j: C -> P such that i ∘ f = j ∘ g, and with the following universal property: for any object X with morphisms u: B -> X and v: C -> X satisfying u ∘ f = v ∘ g, there exists a unique w: P -> X with w ∘ i = u and w ∘ j = v. This universality makes the pushout the most general way to amalgamate B and C along A.

A pushout can be viewed as the colimit of the diagram A -> B and A -> C. It is dual to the pullback, which is a limit of the dual diagram B -> A <- C.

Examples: In the category of sets, the pushout of f: A -> B and g: A -> C is the disjoint union B ⊔ C with f(a) identified with g(a) for all a in A. If A is empty, the pushout is simply the disjoint union. In groups, the pushout is the amalgamated free product B *_A C, where A embeds into B and C via f and g. In topological spaces, pushouts glue spaces along a common subspace.

Existence: pushouts exist in any category with finite colimits, such as Set, Ab, Grp, and Top, and more generally in categories that have binary coproducts and coequalizers.