What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?

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groupoidwhose objects areXand whose arrows are XxG. In that way, you avoid the possibility of doing various non-sensical constructions (such as keeping track of different G-space structures on H - no offence). – André Henriques Sep 28 '10 at 16:56