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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 2

### Quotients of the multiplihedron as categorified associahedra

Pages: 227 – 256

DOI: http://dx.doi.org/10.4310/HHA.2008.v10.n2.a12

#### Author

#### Abstract

We describe a new sequence of polytopes which characterize $A_{\infty}$-maps from a topological monoid to an $A_{\infty}$-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 $CK(n)$.

#### Keywords

enriched categories; $n$-categories; monoidal categories; polytopes

#### 2010 Mathematics Subject Classification

18D20, 18D50, 52B12, 55P43