{"id":139,"date":"2010-03-08T02:48:23","date_gmt":"2010-03-08T02:48:23","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=139"},"modified":"2010-03-08T02:48:23","modified_gmt":"2010-03-08T02:48:23","slug":"cocontinuous-functors","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=139","title":{"rendered":"Cocontinuous functors"},"content":{"rendered":"<p>In the stacks project a site is defined as in Artin&#8217;s notes on Grothendieck topologies, and not as in SGA4. Hence also our notion of a <em>cocontinuous functor<\/em> u : C &#8212;&gt; D between sites C and D is a bit different (than Verdier&#8217;s original one). Namely, it means that, given any object U of C, and any covering {V_j &#8212;&gt; u(U)}_j in D there should exist a covering {U_i &#8212;&gt; U} in C such that the family of morphisms {u(U_i) &#8212;&gt; u(U)}_i refines the given family {V_j &#8212;&gt; u(U)}_j.<\/p>\n<p>The reason this definition is convenient is twofold. On the one hand, it is easy to check that a functor is cocontinuous, and on the other hand, it is true that a cocontinuous functor u : C &#8212;&gt; D gives rise to a morphism of topoi g : Sh(C) &#8211;&gt; Sh(D). For example, for a sheaf G on D the sheaf g^{-1}(G) is the sheaf associated to the presheaf U |&#8212;&gt; G(u(U)).<\/p>\n<p>Here are two examples<\/p>\n<ul>\n<li>Let f : X &#8212;&gt; Y be an open continuous map of topological spaces. Then the functor u(U) = f(U) is a cocontinuous functor between the site of opens of X and the site of opens of Y. The induced morphism of topoi Sh(X) &#8212;&gt; Sh(Y) is the usual one.<\/li>\n<li>Let f : X &#8212;&gt; Y be a morphism of schemes. The &#8220;forgetful&#8221; functor u : (Sch\/X)_{fppf} &#8212;&gt; (Sch\/Y)_{fppf} is cocontinuous and the associated morphism of topoi is the usual morphism of big topoi f_{big} : Sh((Sch\/X)_{fppf}) &#8212;&gt; Sh((Sch\/Y)_{fppf}).<\/li>\n<\/ul>\n<p>A little less standard are the following examples, which are related to the discussion in <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=126\">the previous post<\/a>. Suppose that i : X_0 &#8212;&gt; X is a closed immersion of schemes defined by a sheaf of ideals of square zero. Consider the functor of sites u : X_{lisse-etale} &#8212;&gt; (X_0)_{lisse-etale}, or u : (Sch\/X)_{syntomic} &#8212;&gt; (Sch\/X_0)_{syntomic} given by the rule V |&#8212;&gt; V_0 = X_0 \\times_X V. Then you can check that u is cocontinuous (in both cases). Hence we obtain a morphisms of topoi<\/p>\n<ul>\n<li>g_{lisse-etale} : Sh(X_{lisse-etale}) &#8212;&gt; Sh((X_0)_{lisse-etale})<\/li>\n<li>g_{syntomic} : Sh((Sch\/X)_{syntomic}) &#8212;&gt; Sh((Sch\/X_0)_{syntomic})<\/li>\n<\/ul>\n<p>These maps are somehow contracting the topos associated to X onto the topos associated to X_0. Now in the second case the functor u also gives rise to a morphism of topoi in the opposite direction, namely i_{big} (for the syntomic topology), but I think neither i_{big} nor g_{syntomic} is an equivalence of topoi. In the first case, even though u is continuous, it does not define a morphism of topoi in the other direction.<\/p>\n<p>In any case, cocontinuous functors are very useful and often easier to deal with than the better known continuous ones. For more information see the chapter on <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/sites.pdf\">Sites and Sheaves<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the stacks project a site is defined as in Artin&#8217;s notes on Grothendieck topologies, and not as in SGA4. Hence also our notion of a cocontinuous functor u : C &#8212;&gt; D between sites C and D is a &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=139\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-139","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/139","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=139"}],"version-history":[{"count":9,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/139\/revisions"}],"predecessor-version":[{"id":148,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/139\/revisions\/148"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=139"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=139"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=139"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}