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

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.

Figures (3)

• Figure 1: From coproduct to quantum logic via pushout. (Top left) The coproduct $B_1 + B_2$ of Boolean algebras, characterized by the universal property without identifications. (Bottom left) The pushout $B_1 \amalg_{\mathbf{2}} B_2$, obtained by imposing the identification of bottom and top elements through morphisms from the two-element Boolean algebra $\mathbf{2}$. (Top right) Hasse diagrams of a $2^3$-Boolean algebra and a $2^2$-Boolean algebra representing two classical contexts. (Bottom right) The resulting non-distributive orthomodular lattice obtained by gluing the two Boolean algebras along their extremal elements. The gluing functor $\mathcal{G}$ produces a quantum-logical structure while preserving each Boolean block as a context.
• Figure 2: Gluing Boolean contexts of different sizes. (Top) Hasse diagrams of a $2^3$-Boolean algebra and a $2^2$-Boolean algebra. (Bottom) The orthomodular lattice obtained by gluing these Boolean algebras along their shared extremal elements. The failure of distributivity persists despite the asymmetry in context size, demonstrating that orthomodularity is the minimal consistent extension independent of the relative complexity of the Boolean blocks.
• Figure 3: Contrasting views on the origin of quantum logic. (A) The traditional picture, in which an orthomodular lattice is taken as the ambient structure, and Boolean algebras appear as embedded classical substructures. (B) The constructive picture proposed in this work, where multiple Boolean algebras are glued via the operation $+_{\mathbf{2}}$, generating quantum logic as a non-distributive orthomodular lattice. In this perspective, quantum logic is not assumed but arises as the minimal structure compatible with multiple classical contexts.

Theorems & Definitions (19)

• Definition 1
• Definition 2: Forgetful functor
• Theorem 1: Adjunction
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof