Framed Polytopes and Higher Categories

TL;DR

This work develops a convex-geometric framework for higher-categorical pasting diagrams via framed polytopes. It shows that a framed polytope yields a Steiner diagram if and only if it has no cellular loops, connects cellular strings to higher-dimensional obstructions, and uses this to disprove the Kapranov–Voevodsky conjecture in a strong sense. The paper then builds a rich tapestry linking orientals, cyclic cubes, higher Bruhat and Stasheff–Tamari structures, and Mnëv universality, culminating in a universal moduli picture for framed data and a correspondence with oriented matroids. Finally, it places framed polytopes within the theory of regular directed complexes and oriented graded posets, providing a unified convex-geometric bridge between polytope theory and higher-category pasting data with wide-ranging implications for models of 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.

Figures (14)

• Figure 1: The two frames $(v,w)$ and $(v',w)$ are $P$-equivalent for the regular hexagon $P$.
• Figure 2: The first row depicts $P$ and its projections $\pi_2(P)$ and $\pi_1(P)$. The faces in $s_0(P)$, $s_1(P)$ and $s_2(P)$, and their projections, are in red, while the faces in $t_0(P)$, $t_1(P)$ and $t_2(P)$, and their projections, are in blue. The second row shows the $0$- and $1$-sources and targets of the $2$-faces, projected onto the $\langle v_1, v_2\rangle$ plane. The $0$-sources and targets of the $1$-faces are computed similarly.
• Figure 3: A cellular $1$-string and a cellular $0$-string.
• Figure 4: The $3$-dimensional cross-polytope $P$ admits a basis which is loop-free, but not strongly loop-free.
• Figure 5: A cellular 1-loop in $P_5$ formed by 2-faces. It depicts the image of the vertices of $P_5$ and some of its edges under the projection $\pi_2 \colon \mathbb{R}^5 \to \mathbb{R}^2$.

Theorems & Definitions (56)

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