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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 → (G) we construct a natural G-map $hocoli{m}_{{}_d}G/d(-)\to \left|W\right|$ which is a (non-equivariant) homotopy equivalence, hence $hocoli{m}_{{}_d}EG{\times }_G{F}_d\to EG{\times }_G\left|W\right|$ 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...

We describe, for any compact connected Lie group G and any prime p, the monoid of self maps $B{G}_{{}^p}$ → $B{G}_{{}^p}$ which are rational equivalences. Here, $B{G}_{{}^p}$ denotes the p-adic completion of the classifying space of G. Among other things, we show that two such maps are homotopic if and only if they induce the same homomorphism in rational cohomology, if and only if their restrictions to the classifying space of the maximal torus of G are homotopic.