exact categories which are not additive - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:59:10Z http://mathoverflow.net/feeds/question/41998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41998/exact-categories-which-are-not-additive exact categories which are not additive Martin Brandenburg 2010-10-13T09:21:03Z 2010-10-13T17:32:45Z <p>There are various notions of exact categories (<a href="http://ncatlab.org/nlab/show/exact+category" rel="nofollow">nlab</a>). In a lecture I've seen the following definition of an exact category, which is basically (exact) = (abelian) - (additive):</p> <p>A category $C$ is called <strong>exact</strong> if a) it contains a zero object, b) every morphism has a kernel and a cokernel, c) the canonical morphism $\text{coim}(f) \to \text{im}(f)$ is an isomorphism for every morphism $f$.</p> <p>So for example, the category of pointed sets is an exact category in this sense. I think also the category of pointed compactly generated hausdorff spaces is an example.</p> <p><strong>Questions</strong>: 1) Which theorems and constructions of homological algebra carry over from abelian categories to exact categories in the above sense? 2) Where can I find literature about these categories? I can only find some about the other definitions.</p> http://mathoverflow.net/questions/41998/exact-categories-which-are-not-additive/42034#42034 Answer by Mathieu Breckes for exact categories which are not additive Mathieu Breckes 2010-10-13T17:26:45Z 2010-10-13T17:32:45Z <p>These categories are called Puppe-exact or p-exact categories. See “Jordan-Hölder, modularity and distributivity in non-commutative algebra”, paragraph 1.1, by Francis Borceux and Marco Grandis (JPAA 208 (2007), 665-689 ; available here: <a href="http://www.dima.unige.it/~grandis/BGwe.Abs.html" rel="nofollow">http://www.dima.unige.it/~grandis/BGwe.Abs.html</a>), for non-abelian examples. And see the papers of Marco Grandis (e.g. this one: <a href="http://www.numdam.org:80/numdam-bin/item?id=CTGDC_1992__33_2_135_0" rel="nofollow">http://www.numdam.org:80/numdam-bin/item?id=CTGDC_1992__33_2_135_0</a>) and Mitchell's book “Theory of categories” for homological results in this context (as a general rule, all homological lemmas non involving direct products hold).</p>