Syntactic aspects of hypergraph polytopes

This paper introduces an inductive tree notation for all the faces of polytopes arising from a simplex by truncations, which allows viewing face inclusion as the process of contracting tree edges. These polytopes, known as hypergraph polytopes or nestohedra, fit in the interval from simplices to permutohedra (in any finite dimension). This interval was further stretched by Petri? to allow truncations of faces that are themselves obtained by truncations. Our notation applies to all these polytopes. As an illustration, we detail the case of Petri?’s permutohedron-based associahedra. As an application, we present a criterion for determining whether edges of polytopes associated with the coherences of categorified operads correspond to sequential, or to parallel associativity.