Abstract. We describe a unifying approach to a variety of homotopy decompositions of classifying spaces, mainly of finite groups. For a group G acting on a poset W and an isotropy presheaf d :|W| is a homotopy equivalence. Different choices of G-posets and isotropy presheaves on them lead to homotopy decompositions of classifying spaces. We analyze higher limits over the categories associated to isotropy presheaves W d ; in some important cases they vanish in dimensions greater than the length of W and can be explicitly calculated in low dimensions. We prove a cofinality theorem for functors F : C → O(G) into the category of G-orbits which guarantees that the associated map α F : hocolim C EG × G F (−) → BG is a mod-p-homology decomposition.

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The main results of the paper describe conditions which imply Morita equivalences of certain functor categories. It is shown that various well known results of homological algebra, which are related to Morita equivalences, can be obtained as a specialization of these results. A new application is obtained in the case of two categories associated to an operad.

Let C be a small category. We present some results which describe cohomology groups and homotopy colimits of functors defined over C using cohomology groups and homotopy colimits over certain categories associated to functors from C to posets.

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