direct limit

direct limit (plural direct limits)

1. (algebra) 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 ${\displaystyle i\leq k,\ j\leq k}$ in the indexing poset, then there exist ${\displaystyle f_{ik}:A_{i}\rightarrow A_{k}}$ and ${\displaystyle f_{jk}:A_{j}\rightarrow A_{k}}$. If ${\displaystyle a_{i}\in A_{i},\ a_{j}\in A_{j}}$ such that ${\displaystyle f_{ik}(a_{i})=f_{jk}(a_{j})}$ then ${\displaystyle a_{i}\sim a_{j}}$. If k = j then ${\displaystyle f_{jj}(a_{j})=a_{j},\ f_{ij}(a_{i})=a_{j}}$.)

   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. (category theory) a colimit

• inductive limit

• inverse limit

• direct system

• credit limit