What is the meaning of Direct_limit?

A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If $i \leq k, j \leq k$ in the indexing poset, then there exist $f_{i k} : A_{i} \to A_{k}$ and $f_{j k} : A_{j} \to A_{k}$ . If $a_{i} \in A_{i}, a_{j} \in A_{j}$ such that $f_{i k} (a_{i}) = f_{j k} (a_{j})$ then $a_{i} \sim a_{j}$ . If k = j then $f_{j j} (a_{j}) = a_{j}, f_{i j} (a_{i}) = a_{j}$ .)

a colimit

Source: wiktionary.org

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