Homotopi eşlimitler ve fonksiyonlar komplekslerinin ayrışımları

Given a functor F:C→GSp, the homotopy colimit hocolim_CF 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 E_FG can be taken as the homotopy colimit hocolim(O_FG)(G/H) over the orbit category O_FG. For G-spaces X and Y, let map_G(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 map_G(hocolim_CF,Y)≈ holim_C(map_G(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^(h_FG):=map_G(E_FG,Y)$ of a G-space Y can be calculated as the homotopy limit holim_(O_FG)(Y^H). Topological version of this is recently proved by D. A. Ramras in [2]. We also give some other applications of this isomorphism.