Categorified Spectral Sheaves and Homotopical Invariants for Noncommuting Operators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912845907623936 |
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| author | Chang, Shih-Yu |
| author_facet | Chang, Shih-Yu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17597 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Categorified Spectral Sheaves and Homotopical Invariants for Noncommuting Operators Chang, Shih-Yu Category Theory Operator Algebras 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. |
| title | Categorified Spectral Sheaves and Homotopical Invariants for Noncommuting Operators |
| topic | Category Theory Operator Algebras |
| url | https://arxiv.org/abs/2601.17597 |