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This paper offers a novel homotopical characterization of strongly contextualsimplicial distributions with binary outcomes, specifically those defined onthe cone of a 1-dimensional space. In the sheaf-theoretic framework, suchdistributions correspond to non-signaling distributions on measurementscenarios where each context contains 2 measurements with binary outcomes. Toestablish our results, we employ a homotopical approach that includescollapsing measurement spaces and introduce categories associated withsimplicial distributions that can detect strong contextuality.

Homotopical characterization of strongly contextual simplicial distributions on cone spaces