# Non-abelian simplicial cohomology

Is there a theory of simplicial cohomology with coefficients in a non-abelian group ? I've found next to nothing on Google so far...

I'm interested in particular in the cohomology of graphs with coefficients in the symmetric group $S_3$...

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If you care only about graphs, then I'd imagine you may only care about $H^1$? In this case, there is no problem to define it: for any group $G$ we have a classifying space $BG=K(G,1)$, and this represents $H^1(-;G)$. However, to obtain $BBG=K(G,2)$ we need $G$ to be abelian. On the other hand, I'm pretty sure there's a well-developed theory of non-abelian cohomology; probably Ronnie Brown will eventually show up to comment further on this. –  Aaron Mazel-Gee Oct 31 '12 at 16:10
I think I don't have the level to understand the answer below... From a practical point of view, I know how to compute simplicial homology, for example, say, of the skeleton of a triangle, with coefficients in $\mathbb{Z}$; could you show me how to do it with coefficients in any non-abelian group $G$ ? –  MarcSimon Nov 1 '12 at 9:45
I think you should probably look up "Cech cohomology" -- this agrees with singular cohomology for nice spaces (certainly including graphs), and there it's quite clear how to generalize to not-necessarily-abelian groups. –  Aaron Mazel-Gee Nov 1 '12 at 12:56

The idea of nonabelian cohomology in dimensions higher than 1 goes back to Dedecker, and you can search his papers on MathSciNet. But even for dimension 1 there are variant ways of expressing it. These are discussed in the paper Fibrations of Groupoids; the point is that 1-dimensional cohomology involves derivations $d: G \to A$ but these are conveniently seen as sections (which are morphisms!) of the projection $G \rtimes A \to G$. (I learned this idea from Philip Higgins.)

Dedecker's insight for dimension 2 is that $H^2(X,A)$ for $A$ nonabelian is not really functorial in $A$ since the definition involves automorphisms of $A$. So he generalises it to coefficients in a crossed module $\alpha: A \to P$.

The further development of this idea is if that your coefficients are going to involve crossed modules then it is helpful to define a construction $\Pi$ from groups $G$ or simplicial sets $X$ to crossed complexes, which involve crossed modules, so that a cocycle with coefficents in a crossed module $\mathcal M$ is essentially a morphism $\Pi X \to \mathcal M$; and the cohomology set is essentially just homotopy classes of morphisms. This puts lots of seeming complications into the construction of the functor $\Pi$ and allows one to use the techniques of model categories for homotopy theories.

The full details of this are available in a book published in 2011 by the EMS and with a pdf available here.

One interesting reason why all this development can be done is by working directly with filtered spaces rather than just spaces. But many useful spaces, and certainly simplicial sets, come equipped with a filtration, so this is no great hardship, except to make the conceptual leap of looking more at the structure of the space.

As suggested by Aaron, you may need only the case of $H^1$. But note that graphs have lots of vertices so you may really need groupoids rather than groups, and I have a paper called Groupoids as coefficients which I suspect has hardly been cited!

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This is a great answer! I've never really known where this whole crossed business came from. Doesn't the correspondence between modules and infinitesimal extensions go back to Quillen's "Co/homology of rings"? (That's where I've seen it, at least.) Or is what you're describing a precursor? –  Aaron Mazel-Gee Nov 1 '12 at 12:51
Also, your last comment is a little mysterious to me. Why would small complexes admit group-valued cohomology whereas large complexes only admit groupoid-valued cohomology? –  Aaron Mazel-Gee Nov 1 '12 at 12:56
Aaron: I am just saying that it could be useful if the coefficients –  Ronnie Brown Nov 1 '12 at 14:45
Aaron: I am just saying that it could be useful if the coefficients admit structure modelling that of the object of the cohomology. Hence my guess about groupoid as coefficients for cohomology of graphs, and indeed for anything with lots of vertices. Actually I gave up a version of nonabelian cohomology in my old book because I found the van Kampen theorem was better treated using just fundamental groupoids on a set of base points. The origins of the crossed notion for groups are well treated in the Deutsche-MV review of our new EMS book, linked from the link given. –  Ronnie Brown Nov 1 '12 at 14:53