Homological algebra using nonabelian groups

Can homological algebra be done with nonabelian groups? In particular, can homology or cohomology be defined on chain complexes of nonabelian groups? I know that Abelian categories are the choice settings for homological algebra, but the notions of kernel and cokernel (which seem to be all that is necessary to define homology) seem to make sense for nonabelian groups as well, if we define $\operatorname{coker}(f : G \to H)$ to be the quotient of $H$ by the normal subgroup generated by $\operatorname{im} f$.

For example, given a sequence of nonabelian groups $$ \dotsb \to C_3 \xrightarrow{\partial_3} C_2 \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0 \to 0 $$ with $\partial_{n} \circ \partial_{n+1} = 0$, is it useful to define the homology groups $H_n(C_\bullet) = \ker \partial_n/N$, where $N$ is the normal subgroup generated by $\operatorname{im} \partial_{n+1}$? By useful I mean whether it respects homological algebra, assembles into useful long exact sequences, all the usual stuff.

Solution 1:

A good setting for generalizing homological algebra is that of homological categories. There is a book on the subject due to Borceux and Bourn references at the linked article, as well as a very approachable shorter book by Bourn alone. The definition is noticeably more complex than that of an abelian category or a topos, but quite natural with sufficient explanation.

More to the point, the concept subsumes groups, as well as other nice algebraic categories which have a zero object that are "groupish" enough, such as rings without unit and Lie algebras, and more exotically, the opposite of the category of pointed sets. However, such categories as unital rings and monoids (even commutative monoids) are not homological, essentially because they lack a sufficiently strong notion of kernel. Most of the standard homological algebra results hold in a homological category, including the snake lemma and the resulting long exact sequence in homology.

Solution 2:

There's no reason to expect this to be useful, and as far as I know it's not. An abstract conceptual way to think about where homological algebra comes from is the Dold-Kan correspondence, which says that nonnegatively graded chain complexes of abelian groups (for simplicity) are equivalent to simplicial abelian groups, and this equivalence sends the homology groups of a chain complex to the simplicial homotopy groups of the corresponding simplicial abelian group. Why you would care about simplicial abelian groups is a long story, but the short version is that they have the same homotopy theory as topological abelian groups.

The Dold-Kan correspondence is valid more generally with abelian groups replaced by an abelian category, but is not at all valid for groups: simplicial groups (which hav the same homotopy theory as topological groups, which are very interesting!) are much more complicated than chain complexes of groups.