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

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.