Quotients of the multiplihedron as categorified associahedra. (English) Zbl 1159.18002

Forcey, Stefan

Homology Homotopy Appl. 10, No. 2, 227-256 (2008).

Stasheff’s famous associahedra are polytopes which model coherence relations between associativity homotopies. The sequence of multiplihedra, defined by Iwase and Mimura, model maps preserving such homotopy associative structures up to homotopy.
Formally, the associahedra form a topological operad and homotopy associative structures are represented by spaces equipped with a full action of this operad. In the literature, an \(A_\infty\)-space refers to such a space, equipped with an action of Stasheff’s operad of associahedra, an \(A_\infty\)-map refers to a morphism between \(A_\infty\)-spaces preserving structures up to homotopy.
The author studies quotients of multiplihedra, called composihedra, which model \(A_\infty\)-maps from strict monoids to \(A_\infty\)-spaces. He gives a recursive combinatorial description of these new polytopes and an algorithm to realize them as convex hulls of points in \(\mathbb R^n\).

Reviewer: Benoît Fresse (Villeneuve d’Ascq)

MSC:

18D50	Operads (MSC2010)
52B11	\(n\)-dimensional polytopes
55P48	Loop space machines and operads in algebraic topology
18D20	Enriched categories (over closed or monoidal categories)

Keywords:

A-infinity structures; polytopes; associahedron; multiplihedron; composihedron

Software:

polymake