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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.

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I'm not sure what axiom system you're using, but I always thought that in additive categories you don't necessarily have kernels for every morphism... in an abelian category you do, and that's used all the time in homological algebra. –  Dylan Wilson Oct 7 '10 at 2:38
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The usual term for an additive category which has kernels and cokernels, but need not obey the axiom you name, is preabelian. –  David Speyer Oct 7 '10 at 2:42
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T. Y. Lam has suggested that $\bar{u}$ be named the "makeover" of $u$, because it "usually looks better". He said the name was inspired by the Jenny Jones Show. –  S. Carnahan Oct 7 '10 at 4:16
    
Following David Speyer's comment, I'd suggest that there are finer distinctions to be made in between additive and abelian. In particular, there was some work done by J-P Schneiders on "quasi-abelian categories", which for me are exemplified by the category of Banach spaces and continuous linear maps. These have kernels and cokernels and some other nice relations between the two, but there are morphism which are mono and epi yet not iso. –  Yemon Choi Oct 7 '10 at 5:25
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Homological algebra can also be carried out (to an extent) in pre-abelian categories: springerlink.com/content/u2506k2608868457 and "Basic homological algebra" by M. Scott Osborne. –  Martin Brandenburg Oct 10 '10 at 23:09

3 Answers 3

up vote 11 down vote accepted

A good example of the situation you are thinking about is the category of filtered modules (over some ring). As Yemon Choi notes in a comment, Banach spaces give another example (and in general, filtrations behave very similarly to topologies, and are a closely related notion); but (at least for me) it is a bit easier to write down filtered modules concretely.

Suppose that you have two exact sequences in the category of filtered modules (exact in the strong sense that the filtrations on the outer two terms are induced by the filtration on the middle term, which is to say that these morphisms satisfy the condition that coimage = image) and a morphism between them; the snake lemma will then give a long sequence which is exact as a sequence of morphisms of modules, but which is no longer exact in the strong sense that its morphisms need no longer be strict: i.e. they need not satisfy the condition that coimage = image.

A trivial example is given by considering any morphism $$( 0 \to A \to B \to C \to 0) \longrightarrow (0 \to A' \to B'\to C' \to 0)$$ of exact sequences of non-trivial $R$-modules, and then declaring $A$, $B$, and $C$ to be filtered by having $F^0A = A$ and $F^1A = 0$, and similarly with $B$ and $C$, while equipping $A'$, $B'$, and $C'$ with filtrations such that $F^1A' = A'$ and similarly with $B'$ and $C'$.

The boundary map in the snake lemma from $ker(C \to C')$ to $coker(A\to A')$ now will not be strict (because $F^1$ on the kernel is $0$, while $F^1$ on the cokernel is everything).

One can nevertheless work out homological algebra in this context (e.g. one can form the "filtered derived category", which is the derived category of filtered modules) but one has to take care with the details, because of the phenomenon described above. See Laumon's paper "Sur la categorie derivees des $\mathcal D$-modules filtres" (Lecture Notes in Math 1016) for a careful development of the theory.

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In short or long exact sequences in abelian categories the following is used very often: If $0 \to A \to B \to 0$ is exact, then $A \to B$ is an isomorphism. This is the same as saying that every morphism, which is a mono- and an epimorphism, is an isomorphism, which is wrong in some preabelian categories such as the category of filtered modules, since there no reason why a pointwise inverse should respect the filtration.

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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.

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