Let \((I,\leq)\) be a directed set. A directed system of abelian groups indexed by \(I\) consists of abelian groups \(A_\alpha\) for \(\alpha\in I\) and homomorphisms \(f_{\alpha\beta}\colon A_\alpha\to A_\beta\) for \(\alpha\leq\beta\), with \(f_{\alpha\alpha}=\id_{A_\alpha}\) and \(f_{\beta\gamma}\circ f_{\alpha\beta}=f_{\alpha\gamma}\) for \(\alpha\leq\beta\leq\gamma\). Its direct limit is the set \[ \varinjlim_\alpha A_\alpha\coloneqq\qty{\Disjoint_{\alpha\in I}A_\alpha}/\sim, \] where \(\sim\) is the equivalence relation generated by \(a\sim f_{\alpha\beta}(a)\) for \(a\in A_\alpha\) and \(\alpha\leq\beta\), with the group operation \([a]+[b]\coloneqq[f_{\alpha\gamma}(a)+f_{\beta\gamma}(b)]\) for \(a\in A_\alpha\), \(b\in A_\beta\), and any \(\gamma\geq\alpha,\beta\) [@Hat02]. For each \(\alpha\), the map \(\iota_\alpha\colon A_\alpha\to\varinjlim A_\alpha\), \(a\mapsto[a]\), is a homomorphism with \(\iota_\beta\circ f_{\alpha\beta}=\iota_\alpha\).
Because \(I\) is directed, \(a\in A_\alpha\) and \(b\in A_\beta\) satisfy \(a\sim b\) if and only if \(f_{\alpha\gamma}(a)=f_{\beta\gamma}(b)\) for some \(\gamma\geq\alpha,\beta\); this makes the group operation well defined. With the maps \(\iota_\alpha\), the direct limit is the colimit of the system in the category of abelian groups.
Direct limits over a directed set are exact: if \(A_\alpha\mapsvia{u_\alpha}B_\alpha\mapsvia{v_\alpha}C_\alpha\) is an exact sequence for each \(\alpha\in I\), and the \(u_\alpha\) and \(v_\alpha\) commute with the maps of directed systems \((A_\alpha)\), \((B_\alpha)\), \((C_\alpha)\), then the induced sequence \(\varinjlim A_\alpha\mapsvia{u}\varinjlim B_\alpha\mapsvia{v}\varinjlim C_\alpha\) is exact.
If a CW complex \(X\) is the union of a family of subcomplexes \(X_\alpha\) directed by inclusion, then the inclusions induce an isomorphism \(\varinjlim_\alpha H_n(X_\alpha;G)\cong H_n(X;G)\) for every \(n\) and every abelian group \(G\) [@Hat02].