  This paper offers a novel homotopical characterization of strongly contextual
simplicial distributions with binary outcomes, specifically those defined on
the cone of a 1-dimensional space. In the sheaf-theoretic framework, such
distributions correspond to non-signaling distributions on measurement
scenarios where each context contains 2 measurements with binary outcomes. To
establish our results, we employ a homotopical approach that includes
collapsing measurement spaces and introduce categories associated with
simplicial distributions that can detect strong contextuality.
