There are various notions of exact categories ([nlab](http://ncatlab.org/nlab/show/exact+category)). 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.



Exact categories which are not additive
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](https://doi.org/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](http://www.numdam.org/item/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).


