{"id":6681,"date":"2023-04-20T21:40:52","date_gmt":"2023-04-20T19:40:52","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6681"},"modified":"2023-05-21T16:52:07","modified_gmt":"2023-05-21T14:52:07","slug":"topological-functors-ii-the-cartesian-closed-category-of-c-maps","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6681","title":{"rendered":"Topological Functors II: the Cartesian-closed category of C-maps"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Some time ago, I gave an introduction to <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4133\" data-type=\"page\" data-id=\"4133\">topological functors<\/a>.  They form a pretty brilliant categorical generalization of topological spaces.  The point of today&#8217;s post is to give one particular example of the fact that you can somehow generalize some results on topological spaces to topological functors.  I will concentrate on showing that a (pretty amazing) construction of certain Cartesian-closed full subcategories of <strong>Top<\/strong>, due to Mart\u00edn Escard\u00f3, Jimmie Lawson, and Alex Simpson, generalizes pretty smoothly to a pretty large class of topological functors\u2014the so-called well-fibered topological constructs.  More precisely, I will concentrate on the first part of this construction, which builds a Cartesian-closed category <strong>Map<\/strong><span style=\"text-decoration: underline;\"><sub>C<\/sub><\/span> out of a so-called strongly productive class <span style=\"text-decoration: underline;\">C<\/span> of objects of a category <strong>C<\/strong> that forms the domain of a topological construct with discrete terminal objects.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6613\" data-type=\"page\" data-id=\"6613\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Some time ago, I gave an introduction to topological functors. They form a pretty brilliant categorical generalization of topological spaces. The point of today&#8217;s post is to give one particular example of the fact that you can somehow generalize some &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6681\">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":{"footnotes":""},"categories":[1],"tags":[24,71,72,39],"class_list":["post-6681","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-cartesian-closeness","tag-category-theory","tag-exponentiability","tag-topological-functor"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6681","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=6681"}],"version-history":[{"count":2,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6681\/revisions"}],"predecessor-version":[{"id":6683,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6681\/revisions\/6683"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6681"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6681"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6681"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}