{"id":6694,"date":"2023-05-15T07:39:13","date_gmt":"2023-05-15T05:39:13","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6694"},"modified":"2023-05-15T07:39:13","modified_gmt":"2023-05-15T05:39:13","slug":"topological-functors-and-cartesian-closed-categories","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6694","title":{"rendered":"Topological functors and Cartesian-closed categories"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">This is a talk I gave at the seminar on categorical topology at <a href=\"https:\/\/www.buap.mx\/\">Benem\u00e9rita Universidad Aut\u00f3noma de Puebla<\/a>, Puebla, M\u00e9xico, on May 24th, 2023 (or rather, for which I recorded a series of videos to be played there), on invitation from <a href=\"https:\/\/sites.google.com\/site\/fredmynardswebpage\/\">Fr\u00e9d\u00e9ric Mynard<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Abstract<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The category <strong>Top<\/strong> of topological spaces and continuous maps is not Cartesian-closed.  Mart\u00edn Escard\u00f3, Jimmie Lawson and Alex Simpson gave a wonderful construction of many Cartesian-closed, coreflective full subcategories of <strong>Top<\/strong>.  I will show that this construction generalizes beyond <strong>Top<\/strong> to topological functors.  In a sense, this whole talk is an excuse to give an introduction to the wonderful notion of topological functors.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Reference<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">This is based on the following paper:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Jean Goubault-Larrecq. <a href=\"https:\/\/hal.inria.fr\/hal-03189894\/\">Exponentiable streams and prestreams<\/a>. <a href=\"https:\/\/link.springer.com\/journal\/10485\"><em>Applied Categorical Structures<\/em><\/a>\u00a022,\u00a0pages\u00a0515\u2013549, 2014. The published version, available from <a href=\"https:\/\/link.springer.com\/article\/10.1007\/s10485-013-9315-x\">Springer Link<\/a>, contains two mistakes, which are repaired in the <a href=\"https:\/\/hal.inria.fr\/hal-03189894\/\">HAL report<\/a>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The construction is explained in more detail in <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6613\" data-type=\"page\" data-id=\"6613\">this post<\/a> (April 20th, 2023).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Slides<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Puebla-may-24-2023\/els_puebla_step_by_step_compressed.pdf\">full slides<\/a>, with all animation steps; the <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Puebla-may-24-2023\/els_puebla_compressed.pdf\">shorter<\/a> presentation, without them.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Videos<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Puebla-may-24-2023\/els_puebla_full.mov\">full-length video<\/a>; or the same one, in three parts:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Puebla-may-24-2023\/els_puebla_part1.mp4\">Cartesian-closed categories, and the Escard\u00f3-Lawson-Simpson construction<\/a> (23:30)<\/li>\n\n\n\n<li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Puebla-may-24-2023\/els_puebla_part2.mp4\">Topological functors<\/a> (11:29)<\/li>\n\n\n\n<li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Puebla-may-24-2023\/els_puebla_part3.mp4\">Building Cartesian-closed categories with topological functors<\/a> (16:57)<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" class=\"wp-image-993\" width=\"60\" height=\"83\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\">Jean Goubault-Larrecq<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a talk I gave at the seminar on categorical topology at Benem\u00e9rita Universidad Aut\u00f3noma de Puebla, Puebla, M\u00e9xico, on May 24th, 2023 (or rather, for which I recorded a series of videos to be played there), on invitation &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6694\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-6694","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/6694","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"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=6694"}],"version-history":[{"count":17,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/6694\/revisions"}],"predecessor-version":[{"id":6730,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/6694\/revisions\/6730"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}