process

64 A. Bartels, S. Echterhoff, and W. Lückto A Ì m G. The proof for the continuity of H ‹. I Ktop A;r/ follows from Theorem 4.1in [10].Notice that we have proved all promised results of the introduction as soon as wehave completed the proof of Theorem 6.1 which we have used as a black box so far.7 From spectra over groupoids to equivariant homology theoriesIn this section we explain how one can construct equivariant homology theories fromspectra over groupoids.A spectrum E Df.E.n/; .n// j n 2 Zg is a sequence of pointed spaces fE.n/ jn 2 Zg together with pointed maps called structure maps .n/W E.n/^S 1 ! E.nC1/.A(strong) map of spectra (sometimes also called function in the literature) f W E ! E 0is a sequence of maps f .n/W E.n/ ! E 0 .n/ which are compatible with the structuremaps .n/, i.e., we have f.nC 1/ ı .n/ D 0 .n/ ı .f .n/ ^ id S 1/ for all n 2 Z. Thisshould not be confused with the notion of a map of spectra in the stable category (see[1, III.2.]). Recall that the homotopy groups of a spectrum are defined by i .E/ WD colim k!1 iCk .E.k//;where the system iCk .E.k// is given by the composition iCk .E.k// S ! iCkC1 .E.k/ ^ S 1 / .k/ ! iCkC1 .E.k C 1//of the suspension homomorphism and the homomorphism induced by the structuremap. We denote by Spectra the category of spectra.A weak equivalence of spectra is a map f W E ! F of spectra inducing an isomorphismon all homotopy groups.Given a small groupoid G , denote by Groupoids # G the category of small groupoidsover G , i.e., an object is a functor F 0 W G 0 ! G with a small groupoid as sourceand a morphism from F 0 W G 0 ! G to F 1 W G 1 ! G is a functor F W G 0 ! G 1 satisfyingF 1 ı F D F 0 . We will consider a group as a groupoid with one object and as set ofmorphisms. An equivalence F W G 0 ! G 1 of groupoids is a functor of groupoids F forwhich there exists a functor of groupoids F 0 W G 1 ! G 0 such that F 0 ı F and F ı F 0 arenaturally equivalent to the identity functor. A functor F W G 0 ! G 1 of small groupoids isan equivalence of groupoids if and only if it induces a bijection between the isomorphismclasses of objects and for any object x 2 G 0 the map aut G0 .x/ ! aut G1 .F .x// inducedby F is an isomorphism of groups.Lemma 7.1. Let be a group. Consider a covariant functorEW Groupoids # ! Spectrawhich sends equivalences of groupoids to weak equivalences of spectra.