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As I understand it, the category of groups (not just abelian groups) satisfies all of the definitions of an abelian category. It has all kernels/cokernels as well as products/coproducts. Further the coimage of a morhpism is isomorphic to the image. Does this not then mean that it is an abelian category?

Perhaps a more general question: what about the definition of abelian category captures the essential notion of commutativity?

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    $\begingroup$ You are missing a very important condition: in $\mathbf{Grp}$, coproducts and products do not coincide! $\endgroup$
    – Zhen Lin
    Dec 6, 2015 at 10:31
  • $\begingroup$ Another definition of abelian category ask $Hom(G,H)$ to be an abelian group which explains probably the terminology. $\endgroup$
    – user171326
    Dec 6, 2015 at 10:35
  • $\begingroup$ @N.H. I heard the terminology was inspired by the fact that the category of abelian groups was the archetype of such categories (see the full embedding theorem). You might be right, though. $\endgroup$
    – Arthur
    Dec 6, 2015 at 10:43
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    $\begingroup$ @ZhenLin is the coincidence of coproducts and products an alternative definition of abelian category? Or is it just a consequence of the definition? $\endgroup$
    – gdavtor
    Dec 6, 2015 at 10:48
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    $\begingroup$ @N.H. no need to be vague. An additive category has homs in Ab, and so does an abelian category, but both kinds of categories have more properties than just that: just having abelian groups for Homs is sometimes called being preadditive. $\endgroup$ Dec 8, 2015 at 4:06

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The answer depends on which of the equivalent definitions of "abelian category" you use. The definition in Freyd's book, "Abelian Categories" requires (among other things) that every monomorphism is the kernel of some morphism (Axiom A3 on page 35). In the category of groups, all inclusion maps of subgroups are monomorphisms, but they're not kernels unless the subgroup is normal.

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    $\begingroup$ For other equivalent definitions of "abelian category" Zhen Lin and N.H. have given answers in the comments. In Freyd's book, the facts that products coincide with coproducts and that Hom-sets are abelian groups are theorems rather than being part of the definition. $\endgroup$ Dec 6, 2015 at 10:41
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what about the definition of abelian category captures the essential notion of commutativity?

This question has a very precise answer: it is that in an abelian category finite products and coproducts coincide. This condition alone forces a category to automatically be enriched over commutative monoids in a unique way; see my old blog post A meditation on semiadditive categories for more on this.

So $\text{Grp}$ is disqualified before we even start talking about the regularity conditions on kernels and cokernels; finite coproducts are given by free products and these are much larger than finite products.

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