Framed Polytopes and Higher Categories

Abstract

In the early 1990s, Kapranov and Voevodsky proposed a geometric method for constructing higher-categorical pasting diagrams from generically framed convex polytopes. This work revisits their construction and identifies a convex-geometric condition that is both necessary and sufficient for the procedure to yield a well-defined pasting diagram. Our criterion, the absence of cellular loops, relates their construction to the theory of cellular strings, an active area of convex geometry originating in the Baues problem. This paper introduces higher-dimensional cellular strings and uses them to disprove the Kapranov-Voevodsky conjecture in the following strong sense. Not only do we exhibit framed polytopes admitting cellular loops, but we also construct examples for which every admissible frame produces one. As observed by these authors, Street's orientals arise from canonically framed cyclic simplices. We establish that this family is exceptional as any random $n$-simplex, canonically framed, almost surely exhibits cellular loops in the large $n$-limit.