directlimit (pluraldirect limits)

1. (algebra) A set ofequivalence classes which partition thedisjoint union of the members of adirect 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${\displaystyle i\le k,j\le k}$  in the indexing poset, then there exist${\displaystyle f_{ik}:A_i\to A_k}$  and${\displaystyle f_{jk}:A_j\to A_k}$ . If${\displaystyle a_i\in A_i,a_j\in A_j}$  such that${\displaystyle f_{ik}(a_i)=f_{jk}(a_j)}$  then${\displaystyle a_i\sim a_j}$ . Ifk =j then${\displaystyle f_{jj}(a_j)=a_j,f_{ij}(a_i)=a_j}$ .)

   Adirect 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. (category theory) acolimit

• inductive limit

• inverse limit

• direct system

• credit limit