Stratified noncommutative geometry / David Ayala, Aaron Mazel-Gee, Nick Rozenblyum.

Format:

Book

Author/Creator:

Ayala, David, 1982- author.

Mazel-Gee, Aaron, author.

Rozenblyum, Nick, 1984- author.

Series:

Memoirs of the American Mathematical Society ; v. 1485.

Memoirs of the American Mathematical Society, 0065-9266 ; v. 1485

Language:

English

Subjects (All):

Geometry, Algebraic.

Algebraic topology.

Categories (Mathematics).

Physical Description:

v, 260 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2024.

Summary:

We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable ∞-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as En-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. 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. In the case that G is finite, this expresses genuine G-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). 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.

Contents:

Chapter 1. Introduction

Chapter 2. Detailed overview and fundamental examples

Chapter 3. Stratified noncommutative geometry

Chapter 4. Fundamental operations

Chapter 5. The O-monoidal reconstruction theorem

Chapter 6. The geometric stratification of genuine G-spectra

Chapter 7. The metacosm reconstruction theorem

Chapter 8. Variations on the metacosm reconstruction theorem

Appendix A. Actions and limits, strict and lax

Appendix B. Some(∞, 2)-category theory.

Notes:

Includes bibliographical references (pages 257-260).

ISBN:

1470469626

9781470469627

OCLC:

1437991837