Homology, Homotopy and Applications

Quotients of the multiplihedron as categorified associahedra

Stefan Forcey

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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 nth polytope in the sequence of composihedra, that is, the nth composihedron $CK(n)$.

Article information

Homology Homotopy Appl., Volume 10, Number 2 (2008), 227-256.

First available in Project Euclid: 1 September 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D20: Enriched categories (over closed or monoidal categories) 18D50: Operads [See also 55P48] 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 52B12: Special polytopes (linear programming, centrally symmetric, etc.)

Enriched categories n-categories monoidal categories polytopes


Forcey, Stefan. Quotients of the multiplihedron as categorified associahedra. Homology Homotopy Appl. 10 (2008), no. 2, 227--256. https://projecteuclid.org/euclid.hha/1251811075

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