Possibilistic collapse and extremality of simplicial distributions

Abstract

Consistent families of locally defined probability distributions that do not admit a joint global distribution are known as contextual, with primary examples arising in quantum theory. In this paper, we study such families of distributions using the theory of simplicial distributions, and further develop the theory for possibilistic distributions defined over the Boolean semiring. We characterize possibilistic collapses of simplicial distributions geometrically using bundle scenarios. Using this characterization together with a new connectivity condition on the total space of a bundle scenario, we provide a criterion for detecting extremal simplicial distributions. In parallel, we develop an analogous theory for presheaves on simplicial complexes, describe possibilistic collapses of empirical models on them using event scenarios together with a categorical extremality condition, and relate the two frameworks via a comparison result. We provide examples of contextual simplicial distributions that arise from our criteria on scenarios of interest in quantum foundations, such as Bell scenarios and boundaries of standard simplices, the latter connecting to Vorob'ev's classical theorem on acyclic complexes.