# Abelian categories: examples and nonexamples

I've been talking a little about abelian categories these days. That's because I've been going over Weibel's An Introduction to Homological Algebra. It's a book I read before, and I still feel pretty confident about the material. This time, though, I think I'm going to explore a few different paths that I haven't really given much thought to before, such as diagram proofs in abelian categories, group cohomology (more in-depth), and Hochschild homology.

Back to abelian categories. An abelian category is a category with the following properties:
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# Image factorisation in abelian categories

Let $R$ be a ring and $f:B\to C$ be a morphism of $R$-modules. The image of $f$ is of course
$${\rm im}(f) = \{ f(x) : x\in B \}.$$The image of $f$ is a submodule of $C$. It is pretty much self-evident that $f$ factors as
$$B\xrightarrow{e} {\rm im}(f)\xrightarrow{m} C$$where $e$ is a surjective homomorphism and $m$ is an injective homomorphism. In fact, there is nothing special about working in the category of $R$-modules at all. The same thing holds in the category of sets and a proof for the category of sets works perfectly well for the category of $R$-modules. This set-theoretic reasoning is very natural.

However, we can't always work with categories whose objects are sets with additional structure and whose morphisms are set functions that respect the additional structure (concrete categories). Sometimes we have to work with abelian categories. What's an abelian category? Briefly, it is a category $\Acl$ such that:
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# Wild Spectral Sequences Ep. 2: Five, Isomorphism!

Last time on Wild Spectral Sequences, we conquered the snake lemma using a spectral sequence argument. This time, we meet a new beast: the five lemma. The objective is the usual: prove the five lemma using spectral sequences.

Recall that the five lemma states that given a diagram

in an abelian category, if the rows are exact and $a,b,d,e$ are isomorphisms, then so is $c$. Actually, the hypotheses are too strong. It suffices to have $b,d$ isomorphisms, $a$ an epimorphism and $e$ a monomorphism. One can deduce this via J. Leicht's "strong four lemma" (which we might try and prove via a spectral sequence too) or just by using the regular diagram-chasing proof of the five lemma.
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