colimitation

Colimitation, in category theory, is the notion associated with colimits—a universal construction that amalgamates a diagram of objects into a single object via a cocone. The term is occasionally used to denote the act of forming a colimit, or, in older literature, to refer to the colimit itself; in modern usage "colimit" is standard.

Setup: Let D: J → C be a diagram in a category C. A colimit of D consists of an object L and a cocone i_j: D(j) → L such that for every other cocone (N, j_j: D(j) → N) there is a unique morphism u: L → N with j_j = u ∘ i_j for all j. The pair (L, i_j) is universal among cocones on D.

Existence and examples: Colimits exist in many categories (a category is cocomplete if it has all small colimits). Examples include Set (all small colimits), Ab, and Top. Typical colimits include coproducts (disjoint unions), pushouts, coequalizers, and filtered colimits (direct limits).

Construction and properties: In many cases a colimit can be built as a quotient of a coproduct by identifying certain subobjects, i.e., as a coequalizer of a pair of maps between coproducts. Colimits are preserved by left adjoint functors, while limits are preserved by right adjoints.

Notes: The term "colimitation" is less common today; most texts use "colimit" for the universal object and "to take the colimit" for the construction. The concept is dual to that of a limit.