{"id":415,"date":"2013-12-02T17:41:16","date_gmt":"2013-12-02T15:41:16","guid":{"rendered":"http:\/\/logica.dmi.unisa.it\/lucaspada\/?p=415"},"modified":"2017-10-05T09:45:48","modified_gmt":"2017-10-05T08:45:48","slug":"two-isomorphism-criteria-for-directed-colimits","status":"publish","type":"post","link":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/415-two-isomorphism-criteria-for-directed-colimits\/","title":{"rendered":"Two isomorphism criteria for directed colimits"},"content":{"rendered":"<p>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 <em>confluent<\/em>. The latter means \u00a0that the two given sequences \u00a0can 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, \u00a0analogous isomorphism criteria are typically obtained by <em>ad hoc<\/em> arguments. \u00a0The abstract results given here can \u00a0play the useful \u00a0r\u00f4le of discerning \u00a0the general from the specific in situations of actual interest. We illustrate by applying them to varieties \u00a0of algebras, on the one hand, and to <em>dimension groups<\/em>&#8212;the ordered $K_{0}$ of approximately finite-dimensional \u00a0C*-algebras&#8212;on the other. The first application encompasses such classical examples as Kurosh&#8217;s isomorphism criterion for countable torsion-free Abelian groups of finite rank. The second application yields the Bratteli-Elliott \u00a0Isomorphism Criterion for dimension groups. Finally, we \u00a0discuss \u00a0Bratteli&#8217;s original isomorphism criterion for approximately finite-dimensional C*-algebras, and show that his result does not follow from ours.<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-content\/uploads\/IsomorphismCriterion.pdf\">Two isomorphism criteria for directed colimits<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>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. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,17],"tags":[25,26,24,20,19,18,22,21,23,27],"class_list":["post-415","post","type-post","status-publish","format-standard","hentry","category-news","category-preprint","tag-af-c-algebra","tag-bratteli-elliott-isomorphism-criterion","tag-dimension-group","tag-direct-limit","tag-directed-colimit","tag-filtered-colimit","tag-finitely-generated-object","tag-finitely-presentable-object","tag-partially-ordered-abelian-group","tag-variety-of-algebras"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/415","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/comments?post=415"}],"version-history":[{"count":4,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/415\/revisions"}],"predecessor-version":[{"id":1151,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/415\/revisions\/1151"}],"wp:attachment":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/media?parent=415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/categories?post=415"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/tags?post=415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}