Quotients of the multiplihedron as categorified associahedra

Abstract

We describe a new sequence of polytopes which characterize A ∞ -maps from a topological monoid to an A ∞ -space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parametrize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence of composihedra, that is, the n th composihedron CK(n).

Convex hull realizations of the multiplihedra

“…The case for which multiplication in the range is strictly associative was found by Stasheff in [28] to yield the associahedra. It was long assumed that the case for which the domain was associative would likewise yield the associahedra, but we demonstrate in [7] that this is not so. In the limit as q → 0 the convex hulls instead approach a newly discovered sequence of polytopes.…”

“…The first is to describe important quotients of the multiplihedra, an effort brought to fruition in [7]. Another effort underway is the generalization of the multiplihedron and its quotients by analogy to the graph-associahedra introduced by Carr and Devadoss, in [5].…”

Marked tubes and the graph multiplihedron

“…Indeed, the graph multiplihedra are already beginning to appear in literature; for instance, in Devadoss, Heath and Vipismakul [6], they arise as realizations of certain bordered Riemann disks of Liu [11]. Similar to multiplihedra, the graph multiplihedra degenerates into two natural polytopes; these polytopes are akin to one measuring associativity in the domain of the morphism f and the other in the range (see Forcey [7]). …”

Extending the Tamari Lattice to Some Compositions of Species

“…Stasheff [19] described these maps combinatorially using cell complexes called multiplihedra, while Boardman and Vogt [4] used spaces of painted trees. Both the spaces of trees and the cell complexes are homeomorphic to convex polytope realizations of the multiplihedra as shown in [7].…”