Homotopy colimits and decompositions of function complexes

Abstract

Given a functor F : C→ GSp, the homotopy colimit hocolimCF is defined as the diagonal space of simplicial replacement of F. Let G be a finite group and F be a family of subgroups of G, the classifying space EFG can be taken as the homotopy colimit hocolimOF G(G/H) over the orbit category OFG. For G-spaces X and Y , let mapG(X, Y ) be the space formed by G-simplicial maps from X to Y . Given a functor F : C→ GSp and a G-space Y , there is an isomorphism mapG(hocolimCF , Y ) ∼= holimC(mapG(F, Y )) [1]. We give a proof for this isomorphism by writing explicit simplicial maps in both directions. As an application we show that the generalized homotopy fixed points set Y hF G := mapG(EFG, Y ) of a G-space Y can be calculated as the homotopy limit holimH∈OF GY H. Topological version of this is recently proved by D. A. Ramras in [2]. We also give some other applications of this isomorphism.