Models of \(G\)-spectra as presheaves of spectra (Q6592996)

Let \(G\) be a finite group. The main result of the paper under review is the construction of a certain category \(G\mathcal A\) for which there is a chain of Quillen equivalences between the category of \(G\)-spectra and the category of spectrally enriched contravariant functors from \(G\mathcal A\) to spectra. While the description of \(G\)-spectra as spectrally enriched functors is a consequence of a theorem of \textit{S. Schwede} and \textit{B. Shipley} [Topology 42, No. 1, 103--153 (2003; Zbl 1013.55005)], the point of the present paper is that the category \(G\mathcal A\) is constructed directly and not defined in terms of \(G\)-spectra. In fact, \(G\mathcal A\) is built by applying a suitable infinite loop space machine to a certain category of spans of finite \(G\)-sets, using amongst others techniques developed by the authors in [Algebr. Geom. Topol. 17, No. 6, 3259--3339 (2017; Zbl 1394.55008); New York J. Math. 26, 37--91 (2020; Zbl 1435.55009)]. The result can be viewed as one implementation of the slogan that \(G\)-spectra are spectral Mackey functors.\N\NAs the authors also note in the paper, there was a lot of progress in equivariant homotopy theory in the 13 years that elapsed between first appearance of this manuscript as a preprint and its publication. There are many works that refer to the paper under review, and several generalizations of and alternative approaches to its main result appeared. These include \textit{C. Barwick}'s work on spectral Mackey functors [Adv. Math. 304, 646--727 (2017; Zbl 1348.18020)] and the streamlined equivalence between \(G\)-spectra and spectral Mackey functors in Appendix A of \textit{D. Clausen} et al. [J. Eur. Math. Soc. (JEMS) 22, No. 4, 1149--1200 (2020; Zbl 1453.18011)]. The use of \(\infty\)-categorical methods in these newer approaches avoids many of the technical difficulties dealt with in the present paper.