Abstract
This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.
Highlights
Abstract. — This paper introduces the first general method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes
On retrouve les cas classiques des simplexes et des cubes et on résout celui des associaèdres, appelés aussi polytopes de Stasheff
There are first the mathematicians who will apply it in their work: to compute the homology of fibered spaces in algebraic topology [Bro59, Pro11], to construct tensor products of string field theories [GZ97], or to consider the product of Fukaya A∞-categories in symplectic geometry [Sei08, Amo17]
Summary
(4) Point (2) shows that the facets of Kω correspond bijectively to two-vertex planar trees: the facet labeled by cp+1+r ◦p+1 cq, for p + q + r = n and 2 q n − 1, is the convex hull of the points M (t, ω) associated to planar binary trees of the form t = u ◦p+1 v, for u ∈ PBTp+1+r and v ∈ PBTq. Any face of Kω of codimension k, for 0 k n − 2, is defined as an intersection of k facets. Problem (1) Endow the collection of Loday realizations of the associahedra {Kn}n 1 with a nonsymmetric operad structure in the category Poly, whose induced set-theoretical nonsymmetric operad structure on the set of faces coincides with that of planar trees. In order to find an operadic cellular approximation to the diagonal of the associahedra, we introduce ideas coming from the theory of fiber polytopes [BS92] as follows
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