Convex Hull Realizations of the Multiplihedra

TL;DR

This paper provides a simple algorithm to realize the multiplihedra as convex polytopes, resolving an open question and connecting various mathematical approaches and applications involving these structures.

Contribution

It introduces a straightforward convex hull algorithm for multiplihedra, establishing their realization as convex polytopes and linking different mathematical frameworks.

Findings

Successfully realized multiplihedra as convex polytopes

Unified approaches to A_n-maps and higher homotopy theories

Suggested applications in category theory and related fields

Abstract

We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_infinity-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph associahedra.