Benedetta Andina: The category of n-excisive functors from finite G-CW-complexes to spectra

Abstract: 

In this work we extend some already known results, in the setting of model categories, from a non-equivariant to a \(G\)-equivariant context. We focus on the functor category from finite pointed \(G\)-\(\operatorname{CW}\)-complexes to a stable model category, like the category of Spectra.

We give this category a new \(n\)-excisive model structure using a right Bousfield localization of its projective model structure and we prove multiple Quillen equivalences of this model category with other model categories with the projective model structure. First of all, we have that the restriction map induces a Quillen equivalence \[n\text{-}[G\text{-}\operatorname{CW}^f_*,\mathcal{D}]\xrightarrow{\simeq}[G\text{-}\mathcal{F}^{\le n}_*,\mathcal{D}],\] for \(G\) a compact Lie group and \(\mathcal{D}\) a suitable stable model category, and \(G\text{-}\mathcal{F}^{\le n}_*\) the category \(G\)-spaces with at most \(n\) orbits plus a disjoint base point.

In the case of \(G\) a finite group, we get an additional Pirashvili-type Quillen equivalence to the functor category from pointed \(G\)-spaces with at most \(n\) orbits with only active epimorphisms, \(G\text{-}\mathcal{E}^{\le n}_*\), to \(\mathcal{D}\): \[[G\text{-}\mathcal{F}^{\le n}_*,\mathcal{D}]\simeq[G\text{-}\mathcal{E}^{\le n}_*, \mathcal{D}].\]

These equivalences can be useful to get a good approximation to excisive functors given by the filtration \[\dots\hookrightarrow (n-1)\text{-}[G\text{-}\operatorname{CW}^f_*,\mathcal{D}]\hookrightarrow n\text{-}[G\text{-}\operatorname{CW}^f_*,\mathcal{D}]\hookrightarrow\dots\hookrightarrow [G\text{-}\operatorname{CW}^f_*,\mathcal{D}].\] We also study in detail the layers of this filtration, and get a \(G\)-equivariant extension of previous known formulas in the case of \(G=\{1\}\), after extending the definitions of some already well known objects in the literature.

Once again in the case of \(G\) a finite group, with \(\mathcal{D}=\operatorname{Sp}\), we also show that this \(n\)-excisive model category can be seen as a spectral presheaf category, using the Schwede-Shipley theorem. We indeed construct a set of compact generators \(\mathcal{H}\subset n\text{-}[G\text{-}\operatorname{CW}^f_*,\operatorname{Sp}]\), that allows us to prove that \[n\text{-}[G\text{-}\operatorname{CW}^f_*,\operatorname{Sp}]\simeq[\mathcal{H}^{\text{op}},\operatorname{Sp}].\]