Is it possible to formulate notion of continuous cohomology in terms of model categories?
If yes, then is there a reference for this?
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1$\begingroup$ I suppose what you are after is, or is closely related to, the cohomology of topological/Lie groups due to Segal "Cohomology of topological groups" and Brylinski "Differentiable Cohomology of Gauge Groups"? If so, then, yes, this is equivalent to the derived hom out of the nerve of the Lie group in the model category of simplicial presheaves over manifolds. This is theorem 4.4.36 in "Differential cohomology in a cohesive topos" arxiv.org/abs/1310.7930 $\endgroup$– Urs SchreiberCommented Feb 2, 2016 at 17:30
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$\begingroup$ @Urs: That's not the continuous cohomology from the reference. The continuous cohomology is the cohomology of the cochain complex of continuous functions and the standard group differential. This vanishes for a (simply connected) compact group (with connected coefficients). But the cohomology you are referring to is for compact groups and U(1)-coefficients the cohomology of the classifying space (with a degree shift). $\endgroup$– Christoph WockelCommented Feb 5, 2016 at 8:47
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$\begingroup$ Okay, if the question is about the naive continuous group cohomology then the answer is also "Yes." Use the global model structure on simplicial presheaves. (I.e. don't localize it at covers). $\endgroup$– Urs SchreiberCommented Feb 5, 2016 at 15:52
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