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I have just started reading MacLane's chapter Abelian Categories in Categories for the Working Mathematician. I am stuck in the second page, where he discusses the equivalence relation on a set of arrows with a fixed codomain in a category with a zero object. He defines a preorder <= with f<=g if and only if f factors through g. This is probably very silly of me but, can we not always use the zero arrow between the domains of the two arrows (in any direction) to provide such a factorization?

Thanks very much for any help, Mark

(Also, would MacLane be your suggestion for beginning to study Abelian Categories from a category theorist's prespective?)

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Ittay has already answered your question and suggested the very good book of Freyd. I would also suggest the book of Mitchell which I find to be more in-depth and precise than Freyd (mostly just because it is longer and so takes more time). –  Alex Youcis Feb 17 '13 at 7:23
    
Your idea gives a triangle, but you need a commutative triangle. –  Hurkyl Feb 19 '13 at 1:43
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2 Answers

Suppose $f:x\to y$ and $g:x\to z$. A factorization of $f$ through $g$ is a morphism $h:z\to y$ such that $f=h\circ g$. If $h$ is the unique zero map then $f$ is also the zero map (anything factoring through a zero map is a zero map), so a unless $f$ is a zero map then a factorization of $f$ through $g$ is certainly not automatic.

As for your second question, that chapter is certainly a nice first introduction, but it doesn't not get very far. I really like Freyd's classical little book "abelian categories", particularly for its numerous fun and deep exercises and remarks.

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As an alternative to the classics (Freyd, Grothendieck, Mac Lane) I recommend the notes by Daniel Murfet. They can be found on his website. They are mostly elaborations of textbooks and usually quite detailed.

The notes on abelian categories can be found here.

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