{"id":2145,"date":"2019-11-20T11:19:04","date_gmt":"2019-11-20T10:19:04","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2145"},"modified":"2023-03-20T11:37:44","modified_gmt":"2023-03-20T10:37:44","slug":"quotients-colimits-of-dcpos-and-related-matters","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2145","title":{"rendered":"Quotients, colimits of dcpos, and related matters"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">How do we build colimits in the category <strong>Top<\/strong> of topological spaces?  This is easy: we take the quotient of a big disjoint sum.  How do we build colimits in the category <strong>Dcpo<\/strong> of dcpos?  This is a much more complicated question.  All colimits indeed exist in <strong>Dcpo<\/strong>, and this has been shown by various authors over time, but this is complex.  I will explain why.  Then I will explain what the definition of quotients should be in <strong>Dcpo<\/strong>, and how one can build them.  Funnily, this is related to other questions, such as the existence of d-completions, for example.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2102\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>How do we build colimits in the category Top of topological spaces? This is easy: we take the quotient of a big disjoint sum. How do we build colimits in the category Dcpo of dcpos? This is a much more &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=2145\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","footnotes":""},"categories":[1],"tags":[59,66],"class_list":["post-2145","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-dcpo","tag-quotient"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2145"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2145\/revisions"}],"predecessor-version":[{"id":2146,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/2145\/revisions\/2146"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2145"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2145"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}