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**1 - 1**of**1**### CONTINUOUS FUNCTORS AS A MODEL FOR THE EQUIVARIANT STABLE HOMOTOPY CATEGORY

, 2005

"... Abstract. In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category WG of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of [10], we show that there is a “stable model structure ” on this c ..."

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Abstract. In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category WG of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of [10], we show that there is a “stable model structure ” on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. We construct a second “absolute stable model structure ” which is Quillen equivalent to the “stable model structure”. Our main result is a concrete identification of the fibrant objects in the absolute stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors Z such that for A ∈ WG the collection {Z(A ∧ S W)} form an Ω-G-prespectrum as W varies over the universe U. We show that a functor is fibrant if and only if it takes G-homotopy pushouts to G-homotopy pullbacks and is suitably compatible with equivariant Atiyah duality for orbit spaces G/H+ which embed in U. Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces. The description of fibrant objects in the absolute stable model structure makes it clear that in the equivariant setting we cannot hope for a comparison between the category of equivariant continuous functors and equivariant Γ-spaces, except when G is finite. We provide an explicit analysis of the failure of