TITLE: abelian categories vs. additive categories
QUESTION [9 upvotes]: This must be common knowledge.
Where exactly in the development of homological algebra does one need the axiom that makes additivepre-abelian and abelian categories different? (I mean this statement: for every morphism $u: X \to Y$, the canonical morphism $\bar{u}: \mathrm{Coim}\ u \to \mathrm{Im}\ u$ is an isomorphism.)
My gut feeling is that it should be necessary for the Snake lemma, but I couldn't find a step in the proof that would use it.

REPLY [2 votes]: You might be interested in taking a quick look at the following paper:
A. V. Yakovlev - Homological algebra in pre-Abelian categories (1982)
(see http://www.springerlink.com/content/u2506k2608868457/)
Plenty of examples of pre-abelian categories that are not abelian come from categories of more "analytic" objects (such as the category of topological abelian groups), but it is also common for such analytic examples to fail to have the kernels and cokernels required to be even pre-abelian. Because of this, it may be more common when studying cohomology theories in these analytic cases to develop the relevant homological algebra in the context of (Quillen) exact categories rather than say pre-abelian categories (even though you can perhaps develop homological algebra for pre-abelian categories). But I am no expert.