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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.2012-10-31
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    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 $G$ ?2012-11-01
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    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.2012-11-01

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