I'm trying to prove that hom-sets in an abelian category have a canonical abelian group structure, working with this definition of an abelian category:

A category is abelian if

  1. It has a zero object
  2. It has all finite (co)products
  3. Every morphism has a (co)kernel
  4. Every monomorphism is a kernel and every epimorphism is a cokernel

My approach is to first construct pullbacks, and I think I can do this once I have equalizers, but I'm having trouble constructing those. Any help is appreciated!

  • 1
    $\begingroup$ Peter Freyd's book "Abelian Categories" is available on the web, and the result you want is Theorem 2.14 there. The road from the axioms to this theorem is somewhat longer than what I'd want to reproduce here. $\endgroup$ – Andreas Blass Mar 29 '16 at 23:18
  • $\begingroup$ It is not enough to assume that the category has all finite products and coproducts. You need the product and coproduct of any two objects to be isomorphic by the canonical map from the coproduct to the product (like the direct sum in $R$-modules). $\endgroup$ – Arun Kumar Mar 30 '16 at 6:08
  • 1
    $\begingroup$ In fact it is enough. The other axioms force the canonical comparison to be an isomorphism. $\endgroup$ – Zhen Lin Mar 30 '16 at 8:50

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