Bipermutahedron and biassociahedron

Abstract

Abstract. We give a simple description of the face poset of a version of the biassociahedra that generalizes, in a straightforward manner, the description of the faces of the Stasheff's associahedra via planar trees. We believe that our description will substantially simplify the notation of [8] making it, as well as the related papers, more accessible. References 25 Contents History and pitfallsIn this introductory section we recall the history and indicate the pitfalls of the 'quest for the biassociahedron,' hoping to elucidate the rôle of the present paper in this struggle.History. Let us start by reviewing the precursor of the biassociahedron. J. Stasheff in his seminal paper [9] introduced A ∞ -spaces (resp. A ∞ -algebras, called also strongly homotopy or sh associative algebras) as spaces (resp. algebras) with a multiplication associative up to a coherent system of homotopies. The central object of his approach was a cellular operad K = {K m } m≥2 whose mth piece K m was a convex (m − 2)-dimensional polytope called the Stasheff associahedron. A ∞ -space was then defined as a topological space on which the operad K acted, while A ∞ -algebras were algebras over the operad C * (K) of cellular chains on K. Let us briefly recall the basic features of the construction of [9], emphasizing the algebraic side. More details can be found for instance in [7, II.1.6] or in the original source [9].