{"id":6818,"date":"2023-05-21T16:49:33","date_gmt":"2023-05-21T14:49:33","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6818"},"modified":"2023-05-21T16:52:18","modified_gmt":"2023-05-21T14:52:18","slug":"exponentiable-locales-i-every-exponentiable-locale-is-continuous","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6818","title":{"rendered":"Exponentiable locales I: every exponentiable locale is continuous"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The exponentiable objects of <strong>Top<\/strong> are exactly the core-compact spaces.  Through Stone duality, the core-compact spaces are related to the continuous frames.  So here is a wild guess: would the exponentiable locales be exactly the continuous frames?  That is indeed true, as was proved by Martin Hyland in 1979 (published in 1981).  I will concentrate on one half of the this result for this time, and I will explain why every exponentiable locale must be a continuous dcpo.  The proof is very close to the similar result in <strong>Top<\/strong>, but, as usual, locales are so much more abstract that similar arguments tend to be harder to understand in <strong>Loc<\/strong>; I will do my best.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6714\" data-type=\"page\" data-id=\"6714\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The exponentiable objects of Top are exactly the core-compact spaces. Through Stone duality, the core-compact spaces are related to the continuous frames. So here is a wild guess: would the exponentiable locales be exactly the continuous frames? That is indeed &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=6818\">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":[24,71,21,72,42],"class_list":["post-6818","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-cartesian-closeness","tag-category-theory","tag-continuous-lattice","tag-exponentiability","tag-locale"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6818","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=6818"}],"version-history":[{"count":2,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6818\/revisions"}],"predecessor-version":[{"id":6821,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/6818\/revisions\/6821"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6818"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6818"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6818"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}