Colimits

Colimits are a central concept in category theory that generalizes how a diagram of objects and morphisms can be glued together into a single object. Given a small category J and a functor F: J → C, a colimit of F consists of an object colim F in C and a cocone consisting of morphisms ι_j: F(j) → colim F for each object j in J, such that the following universal property holds: for any object N in C and any cocone η_j: F(j) → N, there exists a unique morphism u: colim F → N with η_j = u ∘ ι_j for all j in J. Equivalently, the colimit represents the functor sending N to the set of cocones from F to ΔN, where ΔN denotes the constant diagram at N.

Colimits generalize several familiar constructions. If J is a discrete category with as many objects as a family {A_i}, the colimit is the coproduct (a disjoint union in Set). If J has two objects and two parallel arrows, the colimit is a coequalizer. A pushout arises when J forms a span A → B ← C. More generally, colimits can be taken over any shape J and, in many categories, they can be built by composing simpler colimits.

Existence and preservation: Not every category has all small colimits, but many common categories do (for example Set, Ab, Cat, Top). Colimits are stable under change of diagram shape in the sense that they intertwine with functors. A fundamental fact is that left adjoint functors preserve colimits, while right adjoints preserve limits.

In practice, colimits provide a unifying framework for constructions such as quotients, unions, and gluing along common subobjects across various mathematical contexts.