Definition 40.1 (Directed systems):

(i) A directed set is a set $\Lambda$ with a partial order $\leq$ such that for any pair $\alpha, \beta \in \Lambda$ there exists $\gamma \in \Lambda$ such that $\alpha \leq \gamma$ and $\beta \leq \gamma$.

(ii) A directed system of abelian groups is a family $\{G_{\alpha}\;/\;\alpha \in \Lambda\}$ of abelian groups indexed by a directed set $\Lambda$ together with a family of group homomorphisms $\{f_{\alpha\beta}:G_{\alpha}\longrightarrow G_{\beta}\;/\; \alpha \leq \beta\}$ satisfying the two conditions

(a)

$f_{\beta\gamma}\circ f_{\alpha\beta} = f_{\alpha\gamma}$ for any three $\alpha, \beta, \gamma \in \Lambda$ such that $\alpha\leq \beta\leq \gamma.$

(b)

$f_{\alpha\alpha} =$   id$_{G_{\alpha}}$ for each $\alpha \in \Lambda$.

(iii) By dropping the adjective abelian from (ii) we obtain a directed system of groups.

(iv) A directed system of topological spaces is a family $\{X_{\alpha}\;/\; \alpha \in \Lambda\}$ of topological spaces indexed by a directed set $\Lambda$ together with a family of continuous maps $\{f_{\alpha\beta}:X_{\alpha}\longrightarrow X_{\beta}\;/\; \alpha \leq \beta\}$ satisfying the two conditions (a) and (b) in (ii).

So we shall speak of a directed system in the categories Gr, AbGr or Top.