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

TL;DR

This paper presents a categorical framework showing that quantum contextuality and orthomodular structures naturally arise from classical Boolean contexts via a left adjoint construction, linking sheaf-theoretic obstructions to lattice non-distributivity.

Contribution

It introduces a categorical gluing functor that generates orthomodular lattices from Boolean algebras, providing a structural and conceptual foundation for quantum contextuality.

Findings

Orthomodular structures arise as a left adjoint from Boolean contexts.

Failure of Boolean pushouts corresponds to contextuality as a sheaf obstruction.

The construction clarifies the structural necessity of quantum logic in quantum information.

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.