How to pronounce “direct limit”

Meaning

• [Noun]
• (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
• ≤
• k
• ,
•  
• j
• ≤
• k
• {\displaystyle i\leq k,\ j\leq k}
• in the indexing poset, then there exist
• f
• i
• k
• :
• A
• i
• →
• A
• k
• {\displaystyle f_{ik}:A_{i}\rightarrow A_{k}}
• and
• f
• j
• k
• :
• A
• j
• →
• A
• k
• {\displaystyle f_{jk}:A_{j}\rightarrow A_{k}}
• . If
• a
• i
• ∈
• A
• i
• ,
•  
• a
• j
• ∈
• A
• j
• {\displaystyle a_{i}\in A_{i},\ a_{j}\in A_{j}}
• such that
• f
• i
• k
• (
• a
• i
• )
• =
• f
• j
• k
• (
• a
• j
• )
• {\displaystyle f_{ik}(a_{i})=f_{jk}(a_{j})}
• then
• a
• i
• ∼
• a
• j
• {\displaystyle a_{i}\sim a_{j}}
• . If k = j then
• f
• j
• j
• (
• a
• j
• )
• =
• a
• j
• ,
•  
• f
• i
• j
• (
• a
• i
• )
• =
• a
• j
• {\displaystyle f_{jj}(a_{j})=a_{j},\ f_{ij}(a_{i})=a_{j}}
• .)
• (category theory) a colimit

Practice Nearby Words

inverse limit, direct system, inverse system, bilimit, direct sum, limit, colimit, dedekind cut, function, categorical product, unlink, free category, subduction, inverse image, equalizer, inclusion, adjunction, directed set, exact sequence, digraphon, left inverse, cointersection, field of quotients, diagram, kernel, inverse, hom-set, right inverse, full functor, ring, disjoint union, profinite group, coproduct, connex, injection, diagonal morphism, semidirect product, conjugacy class, strictification, quotient space, inclusion function, kleene algebra, split monomorphism, limit point, partial ordering relation, identity, equivalence class, downset, quotient set, boundary homomorphism, ultrafilter, rank, difference, boundary group, endomorphism, isogeny, cohomology, hausdorff gap, difference set, disjoint set, faithful functor, matroid, morphism set, ideal, disjunction, correspondence, bijection, distributive lattice, homology, implicant, nilsystem, limit cardinal, dynkin diagram, isomorphism, injectivity, split epimorphism, block system, cycle group, monotone function, identification map, category, monoid, homology group, section, discrete category, adjoint, factor space, dipath, continuous function, relation, projective module, lower set, direct product, isoclass, strict ordering, heyting algebra, inclusion map, canonical disjunctive normal form, cofinality, setoid

Similar Sounds

debt limit, directorate, direct hit, direct loan, direct line, delimit, drilling mud, dry cleaned, doctorate, bractlet