Stratified noncommutative geometry

Abstract

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...

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

“…The identification (c) is direct from the definition of relative functor ∞-categories. Furthermore, there is a definitional identification of the right-lax coinvariants B(N op ⋉ M op ) ≃ (BM op ) / r.lax N over BN op (see Appendix A of [AMGR3]), which determines the identification (d). The identification (e) follows from the codification of the N op -action on Fun(BM op , X) in the statement of the proposition.…”

“…The identification (e) follows from the codification of the N op -action on Fun(BM op , X) in the statement of the proposition. The identification (g) is the definition of left-lax invariants (see Appendix A of [AMGR3]).…”

Separated and complete adelic models for one-dimensional Noetherian tensor-triangulated categories

“…Since we are restricting ourselves to the 1-dimensional case in this paper there is only one step to the assembly process, often described in terms of recollements. Discussions of this type can be found in [1,9,25], and we plan to return to the higher dimensional case elsewhere.…”