  We introduce a new framework for contextuality based on simplicial sets,
combinatorial models of topological spaces that play a prominent role in modern
homotopy theory. Our approach extends measurement scenarios to consist of
spaces (rather than sets) of measurements and outcomes, and thereby generalizes
nonsignaling distributions to simplicial distributions, which are distributions
on spaces modeled by simplicial sets. Using this formalism we present a
topologically inspired new proof of Fine's theorem for characterizing
noncontextuality in Bell scenarios. Strong contextuality is generalized
suitably for simplicial distributions, allowing us to define cohomological
witnesses that extend the earlier topological constructions restricted to
algebraic relations among quantum observables to the level of probability
distributions. Foundational theorems of quantum theory such as the Gleason's
theorem and Kochen-Specker theorem can be expressed naturally within this new
language.
