Let \(\mathcal C\) be a category, \(I\) a small category, and \(F\colon I\to\mathcal C\) a functor. A cocone on \(F\) is an object \(Y\) of \(\mathcal C\) with morphisms \(\psi_i\colon F(i)\to Y\) for \(i\in I\) such that \(\psi_j\circ F(u) = \psi_i\) for every morphism \(u\colon i\to j\) in \(I\). A colimit of \(F\) is a cocone \((X, (\iota_i)_{i\in I})\) such that for every cocone \((Y, (\psi_i)_{i\in I})\) there exists a unique morphism \(\varphi\colon X\to Y\) with \(\varphi\circ\iota_i = \psi_i\) for all \(i\in I\).
A colimit is unique up to unique isomorphism compatible with the maps \(\iota_i\), and is written \(\colim_I F\). Reversing all arrows gives the dual notion, a limit.
The following constructions are colimits.
A coproduct \(\Disjoint_{i\in I} X_i\) is the colimit of a functor on a category \(I\) with no non-identity morphisms.
The pushout of \(X \leftarrow Z \to Y\) is the colimit of that diagram.
The direct limit of a directed system \((A_\alpha, f_{\alpha\beta})\) is its colimit.
In abelian groups, \(\ZZ[1/p] \cong \colim\qty{\ZZ \mapsvia{p} \ZZ \mapsvia{p} \ZZ \mapsvia{p} \cdots}\), with the \(k\)th copy of \(\ZZ\), for \(k\geq 0\), mapped to \(\ZZ[1/p]\) by \(1\mapsto p^{-k}\).
[@Wei94].
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