{"id":1182,"date":"2011-01-25T01:02:50","date_gmt":"2011-01-25T01:02:50","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1182"},"modified":"2011-01-25T01:02:50","modified_gmt":"2011-01-25T01:02:50","slug":"embedding-abelian-categories","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1182","title":{"rendered":"Embedding abelian categories"},"content":{"rendered":"<p>Let <em>A<\/em> be an abelian category. In the stacks project this means that <em>A<\/em> has a set of objects, and that<\/p>\n<ul>\n<li><em>A<\/em> is a pre-additive category with a zero object and direct sums, i.e., an additive category,<\/li>\n<li><em>A<\/em> has all kernels and cokernels (and hence all finite limits and all finite colimits), and<\/li>\n<li>Coim(f) = Im(f) for all morphisms f in <em>A<\/em><\/li>\n<\/ul>\n<p>Martin Olsson pointed out that there is a simple direct argument which proves that in such a category any epimorphism (called a surjection in the following) is a universal epimorphism, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05PK\">Lemma Tag 05PK<\/a>. Using this fact we obtain a site <em>C<\/em> whose underlying category is simply <em>A<\/em> and where a covering is the same thing as a single surjective morphism. Then the Yoneda functor gives a fully faithful, exact functor<\/p>\n<p><em>A<\/em> &#8212;> Ab(<em>C<\/em>),  X &#8212;> h_X<\/p>\n<p>into the category of abelian sheaves, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05PN\">Lemma Tag 05PN<\/a>. Combining this with results on abelian sheaves one obtains a proof of Mitchell&#8217;s embedding theorem for abelian categories, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05PR\">Remark Tag 05PR<\/a>.<\/p>\n<p>I like the argument phrased in this way because I already know about sites, sheaves, etc. It in some sense explains to me (and hopefully an additional handful of readers here) why the embedding theorem should be true. Moreover, I want to make the point that for all applications I can imagine the embedding into the category of abelian sheaves on a site is sufficient.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let A be an abelian category. In the stacks project this means that A has a set of objects, and that A is a pre-additive category with a zero object and direct sums, i.e., an additive category, A has all &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1182\">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-1182","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\/1182","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=1182"}],"version-history":[{"count":12,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1182\/revisions"}],"predecessor-version":[{"id":1194,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1182\/revisions\/1194"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1182"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1182"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}