After translating this category-theoretic statement into a purely convex-geometric one, we were led to the study of globular structures and higher cellular strings on polytopes. Specifically, the absence of cellular loops is a necessary condition for the claim. We strongly disprove it by constructing polytopes for which every frame leads to a cellular loop.
An important infinite family of framed polytopes without cellular loops is defined by the canonically framed cyclic simplices. These happen to be exceptional since we show that, as the dimension of a canonically framed random simplex grows, the probability that it has a cellular loop tends to 1.
We conclude this work relating globular structures on simplices to
oriented flag matroids, and use this connection to prove a
universality theorem showing how complicated the moduli space of
frames can be.
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