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The Sheaf-Theoretic Contextuality (STC) theory developed by Abramsky andcolleagues is a very general account of whether multiply overlapping subsets ofa set, each of which is endowed with certain "local'" structure, can be viewedas inheriting this structure from a global structure imposed on the entire set.A fundamental requirement of STC is that any intersection of subsets inheritone and the same structure from all intersecting subsets. I show that when STCis applied to systems of random variables, it can be recast in the language ofthe Contextuality-by-Default (CbD) theory, and this allows one to extend STC toarbitrary systems, in which the requirement in question (called "consistentconnectedness'" in CbD) is not necessarily satisfied. When applied todeterministic systems, such as systems of logical statements with fixed truthvalues, a problem arises of distinguishing lack of consistent connectednessfrom contextuality. I show that it can be resolved by considering systems withmultiple possible deterministic realizations as quasi-probabilistic systemswith Bayesian priors assigned to the realizations. Although STC and CbD havedistinct native languages and distinct aims and means, the conceptualmodifications presented in this paper seem to make them essentiallycoextensive.

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