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$H^1$ has an interpretation as torsors. But what about the higher $H^i$ (in the setting of algebraic geometry, and étale or flat cohomology)? For example $H^2(X, \mathbf{G}_m)$ is (often) isomorphic to the Brauer group, i.e. equivalence classes of Azumaya algebras.

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What sort of coefficients are you thinking of? –  Tim Porter Feb 16 '11 at 21:48
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Depending a bit on your context, try $n$-gerbes. That should handle all the usual cases. Look at papers by Larry Breen and more recently Aldrovandi and Noohi in Advances (Butterflies is the keyword to look for!).

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I would say have a look at Duskin's paper '$K(\pi,n)$-torsors and the interpretation of "triple" cohomology' (pdf) and his student Glenn's paper 'Realization of cohomology classes in arbitrary exact categories' J. Pure Appl. Algebra, vol. 25, (1982) pp. 33-105. Also Duskin's 'Higher dimensional torsors and the cohomology of topoi : The abelian theory' Lecture Notes in Mathematics, Volume 753 (1979), pp255-279.

I'm afraid you have to move beyond just working with schemes to simplicial schemes and what-not.

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