The Contextuality-by-Default View of the Sheaf-Theoretic Approach to Contextuality

Abstract: The Sheaf-Theoretic Contextuality (STC) theory developed by Abramsky and colleagues is a very general account of whether multiply overlapping subsets of a set, each of which is endowed with certain "local'" structure, can be viewed as inheriting this structure from a global structure imposed on the entire set. A fundamental requirement of STC is that any intersection of subsets inherit one and the same structure from all intersecting subsets. I show that when STC is applied to systems of random variables, it can be recast in the language of the Contextuality-by-Default (CbD) theory, and this allows one to extend STC to arbitrary systems, in which the requirement in question (called "consistent connectedness'" in CbD) is not necessarily satisfied. When applied to deterministic systems, such as systems of logical statements with fixed truth values, a problem arises of distinguishing lack of consistent connectedness from contextuality. I show that it can be resolved by considering systems with multiple possible deterministic realizations as quasi-probabilistic systems with Bayesian priors assigned to the realizations. Although STC and CbD have distinct native languages and distinct aims and means, the conceptual modifications presented in this paper seem to make them essentially coextensive.