direct limit

English

direct limit (plural direct limits)

(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.
(category theory) a colimit

单词	direct limit
释义	direct limit English Noun direct limit (plural direct limits) (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. (category theory) a colimit Synonyms inductive limit Antonyms inverse limit Related terms direct system Anagrams credit limit
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