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I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$.

So i started to think that probably he used this terminology to create an analogy with the cohomology of a space with coefficient in a sheaf. So i started to think that maybe there is a canonical way to turn a sheaf in to a module over the fundamental group, in such a way that fixed points turns out to be global section, so i got to the following question:

1)Is there a natural way to associate to each sheaf(possibly with additional structure) over a (reasonable) space a module over his fundamental group(possibly with additional structure), and possibly the other way around? That is: is there an equivalence between sheaves(possibly with more structure) and modules over the fundamental group(possibly with more structure), inducing isomorphisms on the cohomology(respectively Cech, and group cohomology)?

2)(this is a very wild guess question, but i would be amused if the answer is yes) Wondering about 1) i started to think that in many cases the obstruction to define a global section comes from the fact that after a non trivial loop(i have the complex logarithm in mind) a local object gets a different value in the starting point; so this seemed to me a possible link between the two concepts: if 1) is true, then the module in 1) should take account of all this local data, and the loops acts exactly via the change of value of the "multi-function" after that the loop is performed, in a way that well defined global sections are exactly those that keep fixed by this procedure. Does something like this exist?

I know that 2) is not a precise question, but i hope that the link between the 2 concepts that i am asking suggest a reference to a precise statement of the form that i am hoping, from some of you.

Many thanks and apologize for the ignorance(in usual life i think about arithmetic)

Mimmo
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    I sadly don't have time to write a nice answer about this, but this question and its answers are relevant: http://mathoverflow.net/q/17786/33088 – Takumi Murayama Mar 23 '15 at 02:05
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    Sheaf cohomology and group cohomology are both special cases of topos cohomology. – Zhen Lin Mar 23 '15 at 08:30
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    @Zhen Lin: ok thanks for the comment. But my question was not "can i see these 2 things as special cases of a more general thing", but "is there an equivalence between sheaves over X and modules over $\pi(X)$ that gives isomorphisms on the cohomology"? So if your answer, answers 1) and 2) of my question, i'd like to ask you to expand a little bit. – Mimmo Mar 23 '15 at 08:45
  • @TakumiMurayama: many thanks for the link! – Mimmo Mar 23 '15 at 08:46
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    There are too many sheaves on $X$. Only some sheaves on $X$ come from modules over $\pi_1 (X)$ – in the case where $X$ is an Eilenberg–MacLane space, these are the locally constant sheaves. On the other hand, there is a topos such that the sheaves of abelian groups are literally $G$-modules. – Zhen Lin Mar 23 '15 at 11:25

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