A Diagonal on the Associahedra

四类associahedra模型之间的等价性

Equivalence Between Four Models of Associahedra

本文研究了四类associahedra模型之间的等价性，揭示了Stasheff associahedra与Loday构造的同构关系，并探讨了multiplihedra和graph cubeahedra之间的联系。

Associahedra、Multiplihedra与$A_{\infty}$形式中的单位

Associahedra, Multiplihedra and units in $A_{\infty}$ form

本文通过构造$A_{m}$-结构并引入internal precategory理论，重新证明了$A_{\infty}$形式中两个单位定义在“loop-like”条件下的等价性。

Associahedra, cellular W-construction and products of $A_\infty$-algebras

Our aim is to construct a functorial tensor product of $A_\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of $A_\infty$-algebras.

The diagonal of the associahedra

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.

Diagonals on the Permutahedra, Multiplihedra and Associahedra

We construct an explicit diagonal Δ_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization through J. We introduce the notion of a permutahedral set Z and lift Δ_P to a diagonal on Z. We show that the double cobar construction Ω^2(C_*(X)) is a permutahedral set; consequently Δ_P lifts to a diagonal on Ω^2(C_*(X)). Finally, we apply the diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal generality.

The diagonal of the Stasheff polytope

We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze-Umble and Markl-Shnider based on the Boardman-Vogt cubical decomposition of the Stasheff polytope.