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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.

flag	asked Oct 13 2010 at 9:21 Martin Brandenburg 29.1k●2●45●129

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 2010 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 2010 at 13:23

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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. – Sasha Oct 13 2010 at 18:27

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@Sasha: I know that and mentioned it in the first line of my question. – Martin Brandenburg Oct 13 2010 at 23:10

Mathieu Breckes · Accepted Answer · 2010-10-13 17:32:45Z UTC

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: http://www.dima.unige.it/~grandis/BGwe.Abs.html), for non-abelian examples. And see the papers of Marco Grandis (e.g. this one: http://www.numdam.org:80/numdam-bin/item?id=CTGDC_1992__33_2_135_0) 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).

exact categories which are not additive

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