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arXiv:2606.16949

Categorified Spectral Duality: From Operator Systems to Spectral Stacks and Back

S. Chang

Published Jun 15, 2026• 0 citations
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Abstract

Classical Gelfand duality provides an equivalence between commutative C-star algebras and topological spaces, but fails to furnish a geometric object for noncommutative operator systems. To address this, we introduce a categorified notion of spectrum that captures the full operator-semantic structure. For an operator system A, we construct a spectral stack, denoted Spec(A), over the site of its commutative contexts. The construction proceeds in three stages. First, we encode the syntax of A via a colored operad called the synergy operad. Second, we aggregate local semantic data using a left Kan extension, providing an explicit coend formula. Third, we enforce descent via sheafification, yielding a stack that satisfies higher gluing conditions. We prove that Spec(A) satisfies a Yoneda-style universal property, making it the initial descent-complete semantic realization of A. This yields a contravariant functor from operator systems to spectral objects that admits a right adjoint, the global sections functor. Under semantic generation and descent completeness, the counit of this adjunction is an equivalence, establishing a reconstruction theorem: A is isomorphic to the endomorphisms of the structure sheaf on its spectrum. We further provide a recognition theorem characterizing the essential image of this functor and prove Morita invariance of the associated quasi-coherent sheaf categories. The construction naturally recovers classical Gelfand spectra and Bohrification as truncations of a Postnikov tower. Explicit computations for matrix algebras, Pauli systems, and the Mermin-Peres square yield a quantitative invariant of contextuality extracted from the inertia stack of Spec(A). Comparison theorems show that our framework subsumes Gelfand duality, Bohrification, and Tannaka reconstruction as special cases.

DOI
10.48550/arxiv.2606.16949

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Classical Gelfand duality provides an equivalence between commutative C-star algebras and topological spaces, but fails to furnish a geometric object for noncommutative operator systems. To address this, we introduce a categorified notion of spectrum that captures the full operator-semantic structure. For an operator system A, we construct a spectral stack, denoted Spec(A), over the site of its commutative contexts. The construction proceeds in three stages. First, we encode the syntax of A via a colored operad called the synergy operad. Second, we aggregate local semantic data using a left Kan extension, providing an explicit coend formula. Third, we enforce descent via sheafification, yielding a stack that satisfies higher gluing conditions. We prove that Spec(A) satisfies a Yoneda-style universal property, making it the initial descent-complete semantic realization of A. This yields a contravariant functor from operator systems to spectral objects that admits a right adjoint, the global sections functor. Under semantic generation and descent completeness, the counit of this adjunction is an equivalence, establishing a reconstruction theorem: A is isomorphic to the endomorphisms of the structure sheaf on its spectrum. We further provide a recognition theorem characterizing the essential image of this functor and prove Morita invariance of the associated quasi-coherent sheaf categories. The construction naturally recovers classical Gelfand spectra and Bohrification as truncations of a Postnikov tower. Explicit computations for matrix algebras, Pauli systems, and the Mermin-Peres square yield a quantitative invariant of contextuality extracted from the inertia stack of Spec(A). Comparison theorems show that our framework subsumes Gelfand duality, Bohrification, and Tannaka reconstruction as special cases.
This paper develops the obstruction theory and derived geometric classification of categorified spectral objects, building upon the intrinsic geometric framework established in the companion paper. We prove that contextuality is not merely detected by geometry but is fundamentally a geometric phenomenon: it is the universal obstruction to classical geometric descent. We establish the Contextual Obstruction Theorem, which constructs a canonical obstruction class that vanishes if and only if the system admits a global classical semantic realization. We prove that this obstruction is universal among all contextual obstructions, and that its vanishing is equivalent to non-contextuality. We then establish the Contextuality-Singularity Correspondence, showing that contextuality is precisely the failure of smoothness of the categorified spectrum. We further prove that the spectral curvature tensor is naturally identified with the Atiyah-Kodaira-Spencer obstruction class, yielding the Curvature Criterion for Contextuality. We define the complete geometric spectral invariant as the tuple comprising all canonical geometric data, and prove the Completeness Theorem, establishing that geometry completely characterizes semantics. We develop a geometric stratification of the categorified spectrum by local geometric signatures, proving the Stratification Theorem and showing that contextuality is detected by the presence of strata with positive obstruction rank. Explicit computations for the Pauli system and the Mermin-Peres square demonstrate the full geometric encoding of contextual phenomena. Finally, we introduce the derived moduli functor of operator-semantic systems and formulate a Representability Conjecture asserting that this functor is represented by a derived stack. Conditional on this conjecture, we establish the Derived Spectral Moduli Principle, which unifies Hochschild cohomology, the tangent complex of the moduli stack, and the derived global sections of the tangent complex of the categorified spectrum. Thus this paper proves that contextuality is the universal obstruction to classical geometric descent, and that the derived geometry of the categorified spectrum provides a complete, functorial, and Morita-invariant classification of operator-semantic systems.
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Domain D0 established the internal spectral geometry of a single Localization Act and proved the Recursive Localization Theorem, demonstrating that the non-vanishing of the radical ZZZ forces the generation of an infinite inductive hierarchy of local acts {An,τn}\{A_n, \tau_n\}{An​,τn​}. The present document establishes the global compatibility of this hierarchy under standard assumptions of quantum probability. We formally construct the category of Localization Acts $\Act$ and define a compatibility functor $\mathcal{C}: \Act \to \Prob$ mapping each act to its spectral measure space. By applying the Quantum Central Limit Theorem (Accardi--Fagnola--Quaegebeur) to the tensor hierarchy induced by the localization endofunctor, and invoking the method of \textit{radial reduction} via symmetric group representations, we prove that the local spectral flows {τn}\{\tau_n\}{τn​} admit a unique global affine parameter ttt. We rigorously demonstrate that macroscopic time t∈Rt \in \Rt∈R is the canonical coordinate on the Gelfand spectrum of the colimit of the induced direct system of commutative C∗C^*C∗-algebras, satisfying the universal property of the colimit in $\CAlg$. The derivation of the macroscopic spatial manifold M4M_4M4​ is strictly deferred to Domain D2.
This paper is the third part of a program aimed at building a unified operadic and multicategorical foundation for operator theory and quantum processes. Building on the multicategory HilbMult and the previously introduced Synergy Operad, we investigate the intrinsic algebraic structure underlying compositional quantum dynamics. Three main themes are developed. First, we introduce n-adjoints, a hierarchy of generalized adjoint operations that extend classical adjoints to multicategorical settings and capture higher-order operator interactions. We prove a multicategorical version of Stinespring's theorem formulated in terms of n-adjoints. Second, we study the algebraic semantics of quantum processes and show that completely positive trace-preserving maps admit a canonical monadic description. In particular, we prove a monadicity theorem identifying quantum processes, up to operadic operational equivalence, with algebras of a synergy monad induced by a symmetric quantum interaction operad, yielding a faithful representation that uniformly accounts for sequential and parallel composition. Third, we present an algebraic formulation of quantum no-go principles by encoding constraints such as the no-cloning theorem as operadic ideals. Together, these results provide a coherent algebraic framework that unifies operator theory, category theory, and quantum information.
Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A){\mathcal{C}(A)}C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A){\mathcal{C}(A)}C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” A‾{\underline A}A​ of A, which is a commutative Rickart C*-algebra in the topos of functors from CA{\mathcal{C}A}CA to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A‾{\underline A}A​ for A = B(H).
In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutative C*-algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.