# How much homological algebra can be done with topological abelian groups?

The category of topological abelian groups is known not to be an abelian category. Can we still do some homological algebra with it? Is there any structure of abelian groups that could make up for the Noether isomorphism theorem?

Note: I'm aware that we get an abelian category by only looking at groups that have a "nice" topology.

The category of topological abelian groups may not be abelian, but it is still homological; this implies that it satisfies the Noether isomorphisms theorems, the Five Lemma, the Nine Lemma, the Snake Lemma...

In fact the same is true for the models of any semi-abelian theory in the category of topological spaces; so you can replace abelian groups by groups, rings (without unit), Lie algebras over $\Bbb Z$...

All this is explained in the paper Topological semi-abelian algebras by Francis Borceux and Maria Manuel Clementino, where they aso consider some cases withe restrictions on the topology (Hausdorff, (locally) compact, connected, totally disconnected, profinite...); you can also look at these lecture notes of Dominique Bourn and Clementino from a 2007 workshop, where the case of (not necessarily abelian) groups is illustrated a bit more, and where you can also find more information on homological and semi-abelian categories.

For a different approach, the paper Almost abelian categories by Walter Rump develops a theory of additive categories satisfying some properties which are weaker than those of abelian categories, but are satisfied by the category of topological abelian group, as well as the category of topological vector spaces. These properties are apparently enough to do some homological algebra, but to be honest I don't know much about this approach.

N.B. : Despite the names, the notion of "semi-abelian categories" that appears in this last paper has nothing to do with the one in the other papers above.

• From what I can understand, Almost Abelian categories are almost examples of homological categories: being additive is equivalent to being pointed (!), protomodular (!) and all monos being normal. Left almost Abelian then is equivalent to every regular epi being stable under pullbacks. So such a category is almost regular too, except we are missing some finite limits (in particular pullbacks). – Stefan Perko Oct 17 '17 at 17:06
• @StefanPerko That's not correct. If the category is additive, then it has finite (bi)products, and if it is preabelian (co)equalizers can be computed as (co)kernels of the difference, so it does have finite (co)limits; so if cokernels are stable under pullback, it is also homological. In fact it is not true that all monos must be normal in an additive category : it is not true for the category of torsion-free abelian groups, for example, and for an almost exact category it is actually equivalent to abelianness. – Arnaud D. Oct 17 '17 at 18:24
• Oh right, it has finite limits and finite colimits. - Sorry, I mean "normal" in the sense of Bourceux-Bourn-Gran-... . A morphism which is a kernel is "proper mono" in this terminology. Every proper mono is normal but not vice versa. (c.f. "Categorical Foundations" p. 203, theorem 5.10) – Stefan Perko Oct 17 '17 at 18:30
• Ah my bad. With this in mind, what's really missing for an almost abelian category to be abelian would be that normal mono are kernels, which is equivalent to exactness. – Arnaud D. Oct 17 '17 at 18:45