Categorified Spectral Sheaves and Homotopical Invariants for Noncommuting Operators

TL;DR

This paper develops a higher-categorical, geometric framework for spectral theory of noncommuting operators, using spectral stacks and homotopical invariants to capture local spectra and global obstructions.

Contribution

It introduces a spectral stack approach and homotopical invariants to analyze noncommuting operators, bridging operator theory with higher geometry and homotopy theory.

Findings

Spectral stacks encode local spectra and automorphism data.

Homotopical invariants measure obstructions to global spectral assembly.

Framework unifies operator theory with higher geometric and homotopical methods.

Abstract

Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial solutions: joint spectra apply mainly to commuting families, noncommutative geometry emphasizes global invariants that obscure local structure, and topos-theoretic methods capture contextuality while losing higher coherence information. This paper proposes a geometric and higher-categorical reformulation of the spectral problem for noncommuting operators. Local classical spectra associated with commutative subalgebras are organized into a stack-valued object, called a spectral stack, which retains automorphism and unitary equivalence data between contexts. Noncommutativity is thereby interpreted as nontrivial descent data rather than a breakdown of spectral theory. Using homotopical and derived constructions, we define functorial invariants that measure obstructions to global spectral assembly and extend classical index-theoretic ideas. The resulting framework views noncommutative operator algebras as geometric objects equipped with a spectral atlas, providing a concise bridge between operator theory, homotopy theory, and higher geometry.