\(G\)-functors, \(G\)-posets and homotopy decompositions of \(G\)-spaces (Q2759158)

Let \(G\) be a discrete group and let \(C\) be a small category equipped with a \(G\)-action. The authors introduce the notion of a \(G\)-structure on a functor from \(C\) to the category of spaces. The key property of a \(G\)-functor \(X:C\to Sp\) (where \(Sp\) is a category of simplicial sets or topological spaces) is that there is a natural action of \(G\) on the homotopy colimit \(\text{hocolim}_CX\). Of course such an action exists whenever \(X\) takes values in the category of \(G\)-spaces. They 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 isotopy presheaf \(d:W\to S(G)\) they construct a natural \(G\)-map \(\text{hocolim}_{W_d} G/(-)\to|W|\) which is a (non-equivariant) homotopy equivalence, hence \(\text{hocolim}_{W_d} EG\times_G F_d\to EG\times_G |W|\) is a homotopy equivalence. Different choices of \(G\)-posets and isotopy presheaves on them lead to homotopy decompositions of classifying spaces. They 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. The authors prove a cofinality theorem for functors \(F:C\to O(G)\) into the category of \(G\)-orbits which guarantees that the associated map \(\alpha_F :\text{hocolim}_C EG\times_GF(-)\to BG\) is a mod-\(p\)-homology decomposition.