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

• URL: http://arxiv.org/abs/2603.22353v1
• Date: Sun, 22 Mar 2026 10:34:21 GMT
• Title: Contextuality as a Left Adjoint: A Categorical Generation of Orthomodular Structure
• Abstract summary: We show that non-distributive orthomodular structure arises canonically as a left adjoint from classical Boolean contexts.<n>We establish a precise correspondence between contextuality as a sheaf obstruction and non-distributivity as a colimit failure.
• Score: 2.1972716270608927
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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