5,654 research outputs found
Professor Grzegorz Malinowski in Honorem
Zadanie „ Wdrożenie platformy Open Journal System dla czasopisma „ Bulletin of the Section of Logic” finansowane w ramach umowy 948/P-DUN/2016 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę
Two isomorphism criteria for directed colimits
Using the general notions of finitely presentable and finitely generated
object introduced by Gabriel and Ulmer in 1971, we prove that, in any (locally
small) category, two sequences of finitely presentable objects and morphisms
(or two sequences of finitely generated objects and monomorphisms) have
isomorphic colimits (=direct limits) if, and only if, they are confluent. The
latter means that the two given sequences can be connected by a back-and-forth
chain of morphisms that is cofinal on each side, and commutes with the
sequences at each finite stage. In several concrete situations, analogous
isomorphism criteria are typically obtained by ad hoc arguments. The abstract
results given here can play the useful r\^ole of discerning the general from
the specific in situations of actual interest. We illustrate by applying them
to varieties of algebras, on the one hand, and to dimension groups---the
ordered of approximately finite-dimensional C*-algebras---on the other.
The first application encompasses such classical examples as Kurosh's
isomorphism criterion for countable torsion-free Abelian groups of finite rank.
The second application yields the Bratteli-Elliott Isomorphism Criterion for
dimension groups. Finally, we discuss Bratteli's original isomorphism criterion
for approximately finite-dimensional C*-algebras, and show that his result does
not follow from ours.Comment: 10 page
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