colimit
A colimit is the dual of a limit: instead of an object that maps optimally into a diagram, it is the object that a diagram maps optimally into — the universal way to glue a configuration of objects together along their specified relations. Where a limit takes an intersection or inverse limit, a colimit takes a union, a quotient, a cokernel, a pushout, or a direct limit. Colimits are how category theory builds big things by assembling small ones.
Precisely, for a diagram D : J -> C, a cocone under D to an object X is a family of morphisms D(j) -> X commuting with all the arrows of the diagram. A colimit of D is a universal cocone: an object colim D under D such that every other cocone factors through it by a unique morphism. This is exactly a limit computed in the opposite category C^op, so every theorem about limits dualizes to one about colimits.
By varying the shape J one recovers all the standard gluing constructions: a coproduct is the colimit over a discrete diagram, a coequalizer over two parallel arrows, a pushout over a span, and a direct (filtered) limit over a directed system. A category with all small colimits is cocomplete. Dually to the limit fact, left adjoint functors preserve all colimits, which explains why tensoring and free constructions commute with direct sums and quotients.
The group Q/Z is the direct limit (colimit over the directed system of inclusions) of the cyclic groups (1/n)Z / Z ordered by divisibility; equivalently it is the union of all p-power roots of unity. The Prüfer group Z(p^∞) is the analogous colimit using only powers of a single prime p.
A direct limit is a colimit over a directed system of inclusions.