direct limit

1. 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<k,j<k in the indexing poset, then there exist f_ik:A_i→A_k and f_jk:A_j→A_k. If a_i∈A_i,a_j∈A_j such that f_ik(a_i)=f_jk(a_j) then a_i∼a_j. If k = j then f_jj(a_j)=a_j,f_ij(a_i)=a_j.)

   A direct limit has “canonical functions” which map each element of the disjoint union to its equivalence class.

   Direct limits in the algebraic sense are models of category-theoretic colimits.

2. A colimit

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