Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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
1  
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 '10 at 18:27
1  
@Sasha: I know that and mentioned it in the first line of my question. –  Martin Brandenburg Oct 13 '10 at 23:10
add comment

1 Answer 1

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: 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).

share|improve this answer
    
Thanks ! –  Martin Brandenburg Oct 13 '10 at 17:34
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.