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

Quantum Physics [Submitted on 22 Mar 2026] Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure Yukio-Pegio Gunji, Yoshihiko Ohzawa, Yuki Tokuyama, Yu Huang, Kyoko Nakamura 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 Submission history From: Yukio-Pegio Gunji [view email] [v1] Sun, 22 Mar 2026 10:34:21 UTC (182 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)