Quotients of the Multiplihedron as Categorified Associahedra

TL;DR

This paper introduces a new sequence of polytopes called composihedra, which are quotients of multiplihedra and are used to parameterize compositions in higher category theories, with a simple algorithm for their construction.

Contribution

It defines the composihedra as quotients of multiplihedra, demonstrating their non-equivalence to associahedra and their application in enriched bicategories and pseudomonoids.

Findings

Composihedra are not combinatorially equivalent to associahedra.

A simple algorithm for constructing the nth composihedron is provided.

Composihedra parameterize compositions in higher category theories.

Abstract

We describe a new sequence of polytopes which characterize A_infinity maps from a topological monoid to an A_infinity space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Later term(s) in our sequence of polytopes are demonstrated not to be combinatorially equivalent to the associahedron, as was previously assumed. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of composihedra, that is, the nth composihedron.