Term rewriting on nestohedra

TL;DR

This work extends Huet’s coherence program to the realm of hypergraph polytopes, or nestohedra, by building terminating and confluent term rewriting systems on both the vertices and the faces of these polytopes. It shows that the vertex rewritings generalize the flip order on graph-associahedra and that face rewritings realize the facial weak order and the generalized Tamari order, with critical pairs corresponding to 2-faces. By introducing contextual families of nestohedra, the authors obtain uniform local-confluence diagrams that yield coherence theorems; in particular, associahedra and operahedra recover categorical coherence results for monoidal categories and categorified operads, respectively. The framework also suggests coherence phenomena for permutahedra and contextual graph-associahedra, linking polytope combinatorics with higher algebraic structures in a unified, polytopal setting.

Abstract

We define term rewriting systems on the vertices and faces of nestohedra, and show that the former are confluent and terminating. While the associated posets on vertices generalize Barnard--McConville's flip order for graph-associahedra, the preorders on faces generalize the facial weak order for permutahedra and the generalized Tamari order for associahedra. Moreover, we define and study contextual families of nestohedra, whose local confluence diagrams satisfy a certain uniformity condition. Among them are associahedra and operahedra, whose associated proofs of confluence for their rewriting systems reproduce proofs of categorical coherence theorems for monoidal categories and categorified operads.

Figures (1)

• Figure 1: A planar tree with $5$ vertices (left) and its line graph (right).

Theorems & Definitions (71)

• Remark 1.1
• Lemma 1.2
• proof
• Definition 1.3
• Remark 1.4
• Remark 1.5
• Definition 1.6
• Lemma 1.7
• proof
• Definition 1.8