Are there homology groups for cosimplicial groups?

Question

Hi,

Assume you have a cosimplicial group $G$, so that for each $n \ge 0$ there is a group $G_n$, and you have the usual cofaces and codegeneracies.

Is there a known way to associate to this a collection of homology/homotopy groups in a sensible way?

"Sensible" means at least that it should provide a generalisation of the following two particular cases:

(1) When each $G_n$ is abelian, one can get a cochain complex (I believe this is called the Dold-Kan construction), and one can consider its cohomology groups.

(2) (I apologize for the vagueness of this one.) In low degrees, one can sometimes use a couple of tricks. I have, for example, come across the following situation: the map $x\mapsto d^0(x) d^2(x)$ was a homomorphism $G_2 \to G_3$, as was the map $x\mapsto d^3(x)d^1(x)$; so I could consider their equalizer. Moreover the map $x\mapsto d^0(x)d^1(x)^{-1}d^2(x)$ was a homomorphism $G_1 \to G_2$, whose image was normal in the preceding equalizer. Taking the quotient gave a generalisation of $H^2$ in this lucky situation. The "general" definition which I'm asking for, should it exist, would hopefully coincide with this equalizer trick whenever it makes sense.

Let me also point out that, in the example above, I had originally started with some bigger cosimplicial group $(\Gamma_n)$ and then decided to restrict to the smaller $(G_n)$ precisely so that I could use this little trick. I do believe that the details of this example are completely irrelevant to the general discussion; I mention it because someone might know how to go from a cosimplicial group to a "nicer" one somehow.

Thank you for reading this.

Pierre

Answer 1

If we put some (to me) pretty reasonable-looking axioms, then the answer is no.

For a cosimplicial group $G$ let $h_0G$ be the equalizer of the maps $d_0,d_1:G_0\to G_1$.

Suppose there is another functor $h_1$ from cosimplicial groups to groups such that for a short exact sequence $$1\to G\to H\to K\to 1$$ of cosimplicial groups the exact sequence $$1\to h_0G\to h_0H\to h_0K$$ extends to an exact sequence $$ 1\to h_0G\to h_0H\to h_0K\to h_1G\to h_1H\to h_1K $$ with the middle map natural in the obvious sense and the other new maps given by the functoriality of $h_1$. Suppose that $h_1$ coincides with the usual thing in the case of cosimplicial abelian groups, and suppose also that $h_1G$ is trivial when both $G_0$ and $G_1$ are trivial.

Make a cosimplicial group $H$ where $H_0$ has order $2$ and $H_1$ is nonabelian of order $6$ and the two face maps are not equal. Let $G$ be the kernel of abelianization $H\to H^{ab}$. Then $h_0H$ is trivial, $h_0H^{ab}$ has order $2$, so $h_1G$ must have a subgroup of order $2$. On the other hand $G_0$ is trivial and $G_1$ has order $3$, which implies that $h_1G$ injects into $h_1G^{ab}$, which has order $3$. Contradiction.

Answer 2

You will probably find interesting the paper by Mariam Pirashvili called "Second cohomotopy and nonabelian cohomology". Cohomotopy groups $\pi^nG^\bullet$ of cosimplicial groups $G^\bullet$ have been considered in very low dimensions $n=0,1$ since long time ago, e.g. in the context of non-abelian cohomology. Actually, the only group is $\pi^0G^\bullet$, which is the equalizer of the cofaces

$$G^0\mathop{\rightrightarrows}\limits^{d^0}_{d^1}G^1.$$ Then $\pi^1G^\bullet$ is just a pointed set and there seems to be no way of going higher in general (unless of course $G^\bullet$ is abelian, which is a very strong restriction). Nevertheless, M. Pirashvili has found a nice condition, satisfied by many examples of interest, under which the previous $\pi^0G^\bullet$ and $\pi^1G^\bullet$ are an abelian group and a group, respectively, and one can define a pointed set $\pi^2G^\bullet$. There are also nice exact sequences associated to cosimplicial subgroups, etc.