Simplicial quantum contextuality

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