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...

We prove a reconstruction theorem for stratifications of noncommutative stacks (i.e. presentable stable ∞-categories). In particular, we obtain a reconstruction theorem for quasicoherent sheaves over an ordinary stratified scheme. We show that our reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as En-monoidal structures.Our main application is to equivariant stable homotopy theory: for any compact Lie group G, we give a symmetric monoidal stratification of genuine G-spectra, whose strata record their geometric fixedpoints (as non-genuinely equivariant spectra).We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory: the adelic stratification of the ∞-category of spectra. l.lax.B 2.5. The O-monoidal reconstruction theorem 2.6. Symmetric monoidal stratifications and tensor-triangular geometry Date: November 1, 2019.1 3. The geometric stratification of genuine G-spectra 63 3.1. The geometric stratification of genuine G-spectra 64 3.2. Gluing functors for genuine G-spectra: the proper Tate construction 70 3.3. Examples of reconstruction of genuine G-spectra 72 4. The metacosm reconstruction theorem 78 4.1. Stratifications of right-lax limits 79 4.2. The metacosm reconstruction theorem 87 Appendix A. Actions and limits, strict and lax 92 A.1. Strict and lax actions 93 A.2. Strict and lax limits 100 A.3. Subdivisions 104 A.4. Left-lax limits of right-lax modules and right-lax limits of left-lax modules 105 A.5. Left-lax equivariant functors between right-lax modules and right-lax equivariant functors between left-lax modules 108 A.6. An alternative description of right-lax limits of left-lax modules over posets 111 A.7. Connections with (∞, 2)-category theory 118 A.8. Lax actions and adjunctions 119 A.9. Diagrams in LMod r.lax l.lax.B 121 References 123 -its geometric localization functors are the geometric fixedpoints functors Sp gG Φ H − − → Sp hW(H) ; and its gluing functors are given by a version of the Tate construction.As explained in Remark 0.33, this provides a sense in which genuine G-spectra are the quasicoherent sheaves on a "nearly commutative" stack. 0.2. Detailed overview. In this subsection, we give a detailed overview of our work.Local Notation 0.2. Throughout this subsection, we fix a qcqs scheme X, a noncommutative stack X, and a poset P.This subsection is organized as follows. §0.2.1: We recall the notion of a recollement of X and the fact that a closed-open decomposition of X determines a recollement of QC(X). §0.2.2: We generalize closed-open decompositions of X to stratifications of X. §0.2.3: We define stratifications of X and state our main reconstruction theorem (Theorem A). We also explain how a stratification of X determines a stratification of QC(X); in retrospect, §0.2.1 describes the s...

We provide, among other things: (i) a Bousfield-Kan formula for colimits in ∞-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) ∞categorical generalizations of Barwick-Kan's Theorem Bn and Dwyer-Kan-Smith's Theorem Cn (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of diagram ∞-categories.

We functorially associate to each relative ∞-category (R, W) a simplicial space N R ∞ (R, W), called its Rezk nerve (a straightforward generalization of Rezk's "classification diagram" construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve N R ∞ (R, W) is precisely the one corresponding to the localization R W −1 ; and (ii) that the Rezk nerve functor defines an equivalence RelCat∞ W −1 BK ∼ − → Cat∞ from a localization of the ∞-category of relative ∞-categories to the ∞-category of ∞-categories.

Abstract. We formulate a theory of punctured affine formal schemes, suitable for certain problems within algebraic topology. As an application, we show that the Morava K-theoretic localizations of Morava E-theory corepresent a Lubin-Tate-type moduli problem in this framework.