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Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure

arXiv Quantum Physics
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Researchers led by Yukio-Pegio Gunji propose a novel categorical framework demonstrating that quantum contextuality—long viewed as probabilistic—emerges structurally from classical Boolean logic via left adjoint functors. The team introduces a "gluing functor" that merges Boolean algebras by identifying only their minimal/maximal elements through pushouts, yielding orthomodular lattices that are inherently non-distributive, mirroring quantum logic’s core properties. This construction is proven to be left adjoint to a forgetful functor extracting Boolean subalgebras, providing a "free" generation of quantum structure from classical contexts without probabilistic assumptions. The work links non-distributivity to sheaf-theoretic obstructions, showing the pushout’s failure to stay Boolean corresponds to the absence of global sections in Abramsky-Brandenburger’s framework, unifying two key quantum phenomena. The findings offer a foundational, pre-probabilistic explanation for quantum logic’s necessity in information theory, clarifying its structural origins through category theory and lattice theory.
Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure

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Quantum Physics arXiv:2603.22353 (quant-ph) [Submitted on 22 Mar 2026] Title:Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure Authors:Yukio-Pegio Gunji, Yoshihiko Ohzawa, Yuki Tokuyama, Yu Huang, Kyoko Nakamura View a PDF of the paper titled Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure, by Yukio-Pegio Gunji and 4 other authors View PDF HTML (experimental) Abstract:Contextuality is widely regarded as a hallmark of quantum information, yet its structural origin is often obscured by probabilistic or operational formulations. In this work, we show that non-distributive orthomodular structure need not be postulated, but arises canonically as a left adjoint from classical Boolean contexts. We introduce a gluing functor that takes pairs of Boolean algebras and identifies only their minimal and maximal elements via a categorical pushout. The resulting lattice is orthomodular but generically non-distributive. We prove that this construction is left adjoint to a forgetful functor selecting Boolean subalgebras, thereby providing a free but constrained generation of quantum-logical structure from classical contexts. Furthermore, we demonstrate that the failure of this pushout to remain Boolean is equivalent to the absence of global sections in the sheaf-theoretic framework of Abramsky and Brandenburger. This establishes a precise correspondence between contextuality as a sheaf obstruction and non-distributivity as a colimit failure. Our results offer a categorical and lattice-theoretic reconstruction of contextuality that precedes probabilistic notions and clarifies the structural necessity of quantum logic in information-theoretic settings. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2603.22353 [quant-ph] (or arXiv:2603.22353v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.22353 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Yukio-Pegio Gunji [view email] [v1] Sun, 22 Mar 2026 10:34:21 UTC (182 KB) Full-text links: Access Paper: View a PDF of the paper titled Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure, by Yukio-Pegio Gunji and 4 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)

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Source: arXiv Quantum Physics