  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.
