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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 $\text{coim}(f) \to \text{im}(f)$ 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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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? – Todd Trimble Oct 13 '10 at 12:55
Ok. There are kernels and cokernels, but c) is not satisfied. Perhaps the same problem with compact spaces. – Martin Brandenburg Oct 13 '10 at 13:23
The notion of exact category which I have seen ( 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. – Sasha Oct 13 '10 at 18:27
@Sasha: I know that and mentioned it in the first line of my question. – Martin Brandenburg Oct 13 '10 at 23:10
up vote 6 down vote accepted

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:, for non-abelian examples. And see the papers of Marco Grandis (e.g. this one: 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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Thanks ! – Martin Brandenburg Oct 13 '10 at 17:34

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