TITLE: exact categories which are not additive
QUESTION [11 upvotes]: 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.

REPLY [11 votes]: 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).