Convex hull realizations of the multiplihedra

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 ∞ -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.

Marked tubes and the graph multiplihedron

“…Figure 1(b) shows the two-dimensional hexagon which is J 3 . Recently, Forcey [8] has provided a realization of the multiplihedron, establishing it as a convex polytope. Moreover, Mau and Woodward [13] have shown J n as the compactification of the moduli space of quilted disks.…”

Lifted generalized permutahedra and composition polynomials

“…The result is a connected "coat of paint" starting at the root of each tree in the forest. Figure 5 shows a painted B-forest for the building set of the graph in Figure 2 This notion is compatible with the notion of painted trees in [9]. When B(P n ) = {[i, j] : 1 ≤ i ≤ j ≤ n} is the nested set of the path graph P n , the painted B(P n )forests are in bijection with the painted trees of [9].…”

“…In his work on homotopy associativity for A ∞ spaces, Stasheff [24] defined the multiplihedron J n , a cell complex which has since been realized in different geometric contexts by Fukaya, Oh, Ohta, and Ono [10], by Mau and Woodward [13], and others. It was first realized as a polytope by Forcey [9].…”

A∞-actions and recognition of relative loop spaces

“…The correspondence between the multiplihedra and W (Ass b→w ) is not immediate and involves the so called level trees introduced in [BV73]. A nice description of such relations, can be found in [For08,Tsu12]. The multiplihedra can also be obtained as a compactification of a certain class of configuration spaces, as shown by Merkulov in [Mer11].…”