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I wonder if there is any way to characterise Abelian groups (in Grp) in the language of category theory. This is so basic that I can't imagine that this is not possible, but I cannot think of a way how.

One idea was to say a group is Abelian iff it is equal to its Abelianisation, but this does not work, because in order to define the Abelianisation we already need to know what Abelian groups are.

Then I thought maybe requiring that all subgroups are normal would be enough, but this is not strong enough since there are non-Abelian groups with only normal subgroups.

There's probably something I'm not thinking about. Would appreciate some input.

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2 Answers 2

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The Abelian groups are precisely the group objects in the category $\mathbf{Grp}$ (and $\mathbf{Mon}$). See https://en.wikipedia.org/wiki/Group_object

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  • $\begingroup$ Thanks that's nice, exactly the kind of thing I was looking for. $\endgroup$
    – user2520938
    Commented Nov 10, 2016 at 10:29
  • $\begingroup$ @user2520938 You're very welcome. $\endgroup$
    – user259242
    Commented Nov 10, 2016 at 10:30
  • $\begingroup$ I don't care about points, and if you would have taken a look at my profile you would have known that I usually accept answers. Sometimes I either wait for more answers before accepting, or I am busy with other stuff for a while and will work out my accepts after a while when I have time to do so. I would advise you to not ask for accepts, since it annoys most people. $\endgroup$
    – user2520938
    Commented Dec 15, 2016 at 18:26
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Another idea is that the category of groups is Bourn-unital and the commutative objects (a notion that makes sense in this context) are precisely the Abelian groups.

See "Mal’cev, protomodular, homological and semi-abelian categories" by Borceux and Bourn.

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