{"id":174,"date":"2010-03-24T18:09:52","date_gmt":"2010-03-24T18:09:52","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=174"},"modified":"2010-03-24T18:09:52","modified_gmt":"2010-03-24T18:09:52","slug":"push-and-pull","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=174","title":{"rendered":"Push and pull"},"content":{"rendered":"<p>Here are some examples of morphisms f of topoi such that f^{-1}f_*F &#8212;&gt; F is always surjective for any sheaf of <em>sets<\/em>:<\/p>\n<ul>\n<li>If f : X &#8212;&gt; Y is a continuous map of topological spaces which induces a homeomorphism of X with a subset of Y.<\/li>\n<li>If f : Sets &#8212;&gt; G-Sets is the morphism of topoi coming from mapping the point to the &#8220;classifying space&#8221; of the group G.<\/li>\n<li>If C is a site and f : Sh(C) &#8212;&gt; PSh(C) is the morphism of topoi with f^{-1} equal to sheafification and f_* the forgetful functor.<\/li>\n<\/ul>\n<p>In the first case the map f^{-1}f_*F &#8212;> F is actually always an isomorphism but in the second case it isn&#8217;t if G is nontrivial. Bhargav pointed out the last one and he also pointed out that you can similarly produce lots of examples for exampl X_{etale} &#8212;&gt; X_{Zar} by comparing topologies.<\/p>\n<p>By the way I think it is true that if f : X &#8212;&gt; Y is a continuous map of Kolmogorov topological spaces with the property that f^{-1}f_*F &#8212;&gt; F is always surjective for any sheaf of <em>sets<\/em>, then f induces a homeomorphism of X with a subset of Y. (I haven&#8217;t written out all the details however.)<\/p>\n<p>Here is an example: f : X &#8212;&gt; Y and X = {p, q} with discrete topology and Y= {*}. Then for any sheaf of <em>abelian<\/em> groups F the map f^{-1}f_*F &#8212;&gt; F is surjective, but this does not hold for every sheaf of <em>sets<\/em>. Namely, a sheaf of sets (resp abelian groups) on X corresponds to a pair of sets (resp abelian groups) F_p, F_q (namely the stalks of F at p and at q). Then f_*F corresponds to F_p \\times F_q. Thus we see that f^{-1}f_*F &#8212;&gt; F is surjective if and only if the projections F_p \\times F_q &#8212;> F_p and F_p \\times F_q &#8212;> F_q are surjective. This is the case if and only if either both F_p and F_q are nonempty or both are empty. But for sheaves of sets F_p not empty and F_q empty can occur!<\/p>\n<p>Of course this is somehow incredibly trivial. But since I&#8217;m used to thinking mostly about sheaves of abelian groups it is also very confusing. Namely, any sheaf of abelian groups on {p, q} is globally generated but as seen above it is not the case that every sheaf of sets on {p, q} is &#8220;globally generated&#8221; (i.e., the target of an epimorphism from a constant sheaf).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here are some examples of morphisms f of topoi such that f^{-1}f_*F &#8212;&gt; F is always surjective for any sheaf of sets: If f : X &#8212;&gt; Y is a continuous map of topological spaces which induces a homeomorphism of &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=174\">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-174","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\/174","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=174"}],"version-history":[{"count":14,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/174\/revisions"}],"predecessor-version":[{"id":188,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/174\/revisions\/188"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}