A naive approach to genuine $G$-spectra and cyclotomic spectra

Abstract

For any compact Lie group G, we give a description of genuine G-spectra in terms of the naive equivariant spectra underlying their geometric fixedpoints. We use this to give an analogous description of cyclotomic spectra in terms of naive T-spectra (where T denotes the circle group), generalizing Nikolaus-Scholze's recent work in the eventually-connective case. We also give an explicit formula for the homotopy invariants of the cyclotomic structure on a cyclotomic spectrum in these terms. Contents 0. Introduction 0.1. Overview 0.2. Genuine cyclotomic spectra 0.3. A naive approach to genuine equivariant spectra 0.4. A naive approach to cyclotomic spectra 0.5. The formula for TC 0.6. Miscellaneous remarks 0.7. Outline 0.8. Notation and conventions 0.9. Acknowledgments 1. Actions and limits, strict and lax 1.1. Strict and lax actions 1.2. Strict and lax limits 1.3. Left-lax limits of right-lax modules and right-lax limits of left-lax modules 1.4. Left-lax equivariant functors of right-lax modules and right-lax equivariant functors of left-lax modules 1.5. Lax equivariance and adjunctions 2. A naive approach to genuine G-spectra 2.1. Preliminaries on equivariant spectra 2.2. Examples of fracture decompositions of genuine G-spectra 2.3. Fractured stable ∞-categories 2.4. The canonical fracture of genuine G-spectra 2.5. A naive approach to cyclotomic spectra 3. The formula for TC 3.1. The proto Tate package 3.2. Cyclotomic spectra with Frobenius lifts Date: October 18, 2017. 1 3.3. Trivial cyclotomic spectra 75 3.4. Partial adjunctions 76 3.5. The formula for TC 77 References 82 the former paper, namely an unstable cyclotomic structure on spaces-enriched THH and an unstable cyclotomic trace map to the resulting unstable version of TC.0.2. Genuine cyclotomic spectra. In their influential paper [BHM93], Bökstedt-Hsiang-Madsen defined TC by observing that the THH spectrum (of an associative ring spectrum) carries certain additional structure maps -resulting from what is now called its cyclotomic structure -and taking a limit over those maps. Following work of Hesselholt-Madsen [HM97b], this construction was placed on firmer categorical footing by Blumberg-Mandell [BM15]: they defined a homotopy theory -precisely, a "spectrally-enriched model* category" -of (what we'll refer to as genuine) cyclotomic spectra, which we denote by Cyc g (Sp), and they showed that TC could be recovered as the (derived) hom-spectrumout of the sphere spectrum equipped with its trivial cyclotomic structure. In [BG], Barwick-Glasman refined this to a presentable stable ∞-category, which we continue to denote by Cyc g (Sp). These approaches to cyclotomic spectra are based in genuine equivariant homotopy theory. Recall that for a compact Lie group G, the ∞-category Sp gG of genuine G-spectra is an enhancement of the ∞-categoryof homotopy G-spectra. 2,3 Roughly speaking, this is obtained by "remembering the genuine Hfixedpoints" for all closed subgroups H ≤ G, instead of simply taking them to be the homotopy H-fixedpoints of the underlying homotopy G...

Recollements and stratification

“…Sections 2, 3, and 4 of this paper are lightly revised versions of the corresponding sections of [QS19], whereas section 5 on the application to stratified ∞-topoi is entirely new. Also, in the intervening time since we wrote [QS19], Ayala-Mazel-Gee-Rozenblyum released their work on stratified noncommutative geometry [AMGR21]; this is an expansion of [AMGR17] and bears greatly on many of the topics treated in this paper. As such, we have added a few remarks throughout (in particular, Remark 3.40 and the new §3.2.4) explaining how our work relates to [AMGR21].…”

“…[MNN17] or [QS21b]). In fact, we may further recast this problem using the stratification theory of Ayala-Mazel-Gee-Rozenblyum [AMGR17,AMGR21]. In their work, they construct a certain locally cocartesian fibration Sp G φ-locus P , where P is the poset of conjugacy classes of subgroups of G and the fiber over [H] is Fun(BW G H, Sp) for W G H = N G H/H the Weyl group, such that one has a canonical equivalence Fun cocart /P (sd(P ), Sp G φ-locus ) ≃ Sp G where sd(P ) is the barycentric subdivision 3 of P regarded as a locally cocartesian fibration over P via the functor that takes the maximum, and the lefthand side denotes the full subcategory spanned by those functors sd(P )…”

A universal characterization of noncommutative motives and secondary algebraic K-theory

“…While cyclotomic spectra were originally defined in terms of genuine T-spectra, we note that they can also be defined using the formalism of stratified categories [AMGR20,AMGR17c].…”

Natural symmetries of secondary Hochschild homology

“…Following the developments in [AMGR1], we expect Definition 0.18 of an unstable cyclotomic object to lend to a definition of a (stable) secondary cyclotomic object, and that Theorem Y.2 lends a secondary cyclotomic structure on secondary topological Hochschild homology. For secondary topological cyclotomic homology to be the invariants with respect to this structure,…”

“…When working over the sphere spectrum (which is to say V = (Spectra, ∧)) so that HH Spectra (A) = THH(A) is topological Hochschild homology , in [BHM] Bökstedt-Hsaing-Madsen extend this Taction as a cyclotomic structure on THH(A). In [AMGR1] it is demonstrated how this cyclotomic structure on THH(A) is derived from an action of the continuous monoid T ⋊ N × on the unstable version HH Spaces (A). Here, we prove Theorem Y.1, which constructs a canonical T 2 ⋊ Braid 3 -action on HH (2) (A), which is functorial in the 2-algebra A.…”