colimit

Colimit is a fundamental construction in category theory that generalizes many familiar constructions such as coproducts, coequalizers, and pushouts. Given a small diagram F: J → C in a category C, a colimit consists of an object Colim F together with a cocone, i.e., a family of morphisms φ_j: F(j) → Colim F for each object j in J, satisfying that for every morphism f: j → k in J, φ_k ∘ F(f) = φ_j. This cocone is universal in the sense that for any other object X with a cocone ψ_j: F(j) → X, there exists a unique morphism u: Colim F → X such that ψ_j = u ∘ φ_j for all j.

Existence of colimits depends on the category. If C has all J-indexed colimits for a given shape J, the construction defines a functor colim_J: C^J → C, left adjoint to the diagonal functor Δ: C → C^J (which assigns to each object the corresponding constant diagram). Not all categories have all colimits, and some have only certain shapes of colimits.

Examples include: coproducts (colimits over discrete diagrams), coequalizers, and pushouts; in Set, the colimit of a diagram is the disjoint union of the involved sets modulo identifications induced by the diagram, reflecting the universal gluing of pieces.

Colimits are dual to limits. A limit of a diagram is a universal cone; a colimit is a universal cocone. This duality underpins many constructions in various areas of mathematics and computer science, where colimits capture the idea of freely gluing or identifying objects along a diagram.