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.

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 $\operatorname{coim}(f) \to \operatorname{im}(f)$ 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.