{"id":4602,"date":"2021-12-20T19:07:05","date_gmt":"2021-12-20T18:07:05","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=4602"},"modified":"2023-02-20T19:28:44","modified_gmt":"2023-02-20T18:28:44","slug":"sheaves-and-streams-ii-sheafification-and-stratified-etale-maps","status":"publish","type":"post","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=4602","title":{"rendered":"Sheaves and streams II: sheafification, and stratified \u00e9tale maps"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In part I, I explained how one can build the <em>\u00e9tale space<\/em> of a presheaf <em>F<\/em> over a topological space <em>X<\/em>.  I will show how one can retrieve a sheaf from an \u00e9tale map, leading to a nice adjunction and its associated monad, <em>sheafification<\/em>. This is all well-known, but then I would like to apply all that to the presheaf of locally monotone functions of a prestream, which we had already started to examine last time.  We will obtain a funny structure that I will call <em>stratified \u00e9tale maps<\/em>.  Read the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4466\" data-type=\"page\" data-id=\"4466\">full post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In part I, I explained how one can build the \u00e9tale space of a presheaf F over a topological space X. I will show how one can retrieve a sheaf from an \u00e9tale map, leading to a nice adjunction and &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?p=4602\">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":[36,34,35],"class_list":["post-4602","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-etale-map","tag-sheaf","tag-stream"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/4602","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=4602"}],"version-history":[{"count":1,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/4602\/revisions"}],"predecessor-version":[{"id":4603,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/posts\/4602\/revisions\/4603"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4602"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4602"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}