direct limit

English

Noun

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 $i\leq k,\ j\leq k$ in the indexing poset, then there exist $f_{ik}:A_{i}\rightarrow A_{k}$ and $f_{jk}:A_{j}\rightarrow A_{k}$. If $a_{i}\in A_{i},\ a_{j}\in A_{j}$ such that $f_{ik}(a_{i})=f_{jk}(a_{j})$ then $a_{i}\sim a_{j}$. 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. (category theory) a colimit

Synonyms

• inductive limit

Antonyms

• inverse limit

Related terms

• direct system

Anagrams

• credit limit