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There are various notions of exact categories (nlab). In a lecture I've seen the following definition of an exact category, which is basically (exact) = (abelian) − (additive):

A category $C$ is called exact if a) it contains a zero object, b) every morphism has a kernel and a cokernel, c) the canonical morphism $\operatorname{coim}(f) \to \operatorname{im}(f)$ is an isomorphism for every morphism $f$.

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.

Questions: 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.

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    $\begingroup$ I am not familiar with this notion of exact category (and my gut feeling is that it isn't a very useful notion, although I would welcome being shown otherwise). But is the category of pointed compactly generated Hausdorff spaces an example? What if you take $f$ to be the identity function from the real line with the discrete topology to the real line with its standard topology? $\endgroup$
    – Todd Trimble
    Commented Oct 13, 2010 at 12:55
  • $\begingroup$ Ok. There are kernels and cokernels, but c) is not satisfied. Perhaps the same problem with compact spaces. $\endgroup$
    – Martin Brandenburg
    Commented Oct 13, 2010 at 13:23
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    $\begingroup$ The notion of exact category which I have seen (en.wikipedia.org/wiki/Exact_category) is different --- the typical examples of exact categories are the category of vector bundles, or the category of filtered vector spaces. Both don't satisfy your definition. $\endgroup$
    – Sasha
    Commented Oct 13, 2010 at 18:27
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    $\begingroup$ @Sasha: I know that and mentioned it in the first line of my question. $\endgroup$
    – Martin Brandenburg
    Commented Oct 13, 2010 at 23:10

1 Answer 1

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These categories are called Puppe-exact or p-exact categories. See paragraph 1.1, of Jordan-Hölder, modularity and distributivity in non-commutative algebra, by Francis Borceux and Marco Grandis (JPAA 208 (2007), 665-689 doi:10.1016/j.jpaa.2006.03.004), for non-abelian examples. And see the papers of Marco Grandis (e.g. On the categorical foundations of homological and homotopical algebra (Numdam)) 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).

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  • $\begingroup$ Dear Mathieu, I think what Martin proposes is not necessarily Puppe-exact. For example, the category of pointed sets is not Puppe-exact. The epimorphisms are not cokernels in general. $\endgroup$
    – user12580
    Commented Oct 31, 2021 at 16:19
  • $\begingroup$ Since the answer was accepted, I assume the OP was happy with it. $\endgroup$
    – David White
    Commented Sep 6, 2022 at 12:53

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