The abelianization functor from the category of groups to that of abelian groups is right exact in the sense that it takes a short exact sequence $$1 \to K \to G \to H \to 1$$ to a shorter exact sequence $$K_{\mathrm{ab}} \to G_{\mathrm{ab}} \to H_{\mathrm{ab}} \to 0.$$ (See Show that the abelianization functor is right exact.) If the category of groups were an abelian category, then we would be able to define derived functors of abelianization. The category of groups isn't actually abelian, or even additive, but at least exactness still makes sense, so one might hope something would go through anyway.

1. What breaks down?

2. What can be salvaged?

3. Is there some morally allied modification to this naive hope that is actually studied?

  • $\begingroup$ 3. By Dold-Kan theorem (or by more general approaches) and the fact that simplicial homotopies are defined in terms of composition and equalities (without sums) you do not need an additive category. Not only for groups; e.g. André-Quillen homology is also constructed deriving the "abelianization" functor. $\endgroup$ – A.G Aug 23 '16 at 21:01
  • $\begingroup$ I understand (which is to say, my Google results show) André–Quillen homology to be about commutative rings, though ... do you mean there's enough structure in the category of groups that derived functors of abelianization still work? $\endgroup$ – jdc Aug 23 '16 at 21:20
  • $\begingroup$ Yes. This is done essentially (in much more generality but also particularizing to groups at some points) in Quillen, Homotopical Algebra, II.3 - II.5. For your particular case of groups, I think there should be more down to earth references, probably a lot, but I do not know any. $\endgroup$ – A.G Aug 24 '16 at 17:39
  • $\begingroup$ This is a book I've meant to read since undergrad; I guess now is a good time. Before I start, though, what facts are ultimately being applied to this construction? Is there a closed model category structure on Grp? Does one embed discrete groups as trivial simplicial groups and then apply the results to the category of simplicial groups? Do you remember, e.g., what the functor $\mathrm{ab}^1$ winds up being on objects? $\endgroup$ – jdc Aug 24 '16 at 21:06
  • $\begingroup$ For deriving additive functors on abelian categories with enough projectives, you take a projective resolution and apply the functor. You use that two projective resolutions are homotopic and then the functor, being additive, preserve homotopies. Applying the Dold-Kan functor $K$ from complexes to simplicial objects in the abelian category, projective resolutions go to "simplicial projective" resolutions and homotopies of complexes to simplicial homotopies. Now this functor commutes with homology, that is, ... $\endgroup$ – A.G Aug 24 '16 at 21:58

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