Definition 40.2 (Inductive limit):

Given a directed system $\{M_{\alpha}\;/\; \alpha \in \Lambda\}$ in one of the categories Gr, AbGr or Top and a family of morphisms $\{f_{\alpha\beta}:M_{\alpha}\longrightarrow M_{\beta}\;/\; \alpha \leq \beta\}$ in the same category satisfying the conditions in definition (40.1), an inductive limit is an object $M$ together with a family of morphisms $\{f_{\alpha}:M_{\alpha}\longrightarrow M\}$ such that the following two conditions hold:

(1)

For every pair $\alpha, \beta \in \Lambda$ with $\alpha \leq \beta$, $f_{\beta}\circ f_{\alpha\beta} = f_{\alpha}$, summarized as a commutative diagram:

$$\xymatrix{ M_{\alpha} \ar[rr]^{f_{\alpha\beta}}\ar[rd]_{f_{\alpha}} & & M_{\beta} \ar[ld]^{f_{\beta}}\\ & M }$$

(2)

Universal property: Given an object $L$ and a family of morphisms $g_{\alpha}:M_{\alpha}\longrightarrow L$ satisfying

$$g_{\beta}\circ f_{\alpha\beta} = g_{\alpha},\quad \alpha, \beta \in \Lambda, \alpha \leq \beta,$$

there exists a unique morphism $\psi : M\longrightarrow L$ such that

$$\psi\circ f_{\alpha} = g_{\alpha}.$$