Stratified noncommutative geometry

TL;DR

This paper develops a theory of stratifications for noncommutative stacks, providing a reconstruction framework compatible with monoidal structures, and applies it to equivariant stable homotopy theory and tensor-triangular geometry.

Contribution

It introduces a novel stratification and reconstruction theory for noncommutative stacks, extending to symmetric monoidal and operadic structures, with applications in homotopy theory.

Findings

Reconstruction theorem expresses noncommutative stacks via their strata and gluing data.

Provides a symmetric monoidal stratification of genuine G-spectra for compact Lie groups.

Establishes an adelic reconstruction theorem applicable to tensor-triangular geometry.

Abstract

We introduce a theory of stratifications of noncommutative stacks (i.e. presentable stable $\infty$-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 $E_n$-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 M\"obius 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.