首页/论文/1902.08059

arXiv 1902.080592019-02-21math.ATmath.CO

The diagonal of the associahedra

Naruki Masuda, Hugh Thomas, Andy Tonks, Bruno Vallette

备注：: 19 pages, 7 figures. Comments are welcome

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Quotients of the Multiplihedron as Categorified Associahedra

We describe a new sequence of polytopes which characterize A_infinity maps from a topological monoid to an A_infinity space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Later term(s) in our sequence of polytopes are demonstrated not to be combinatorially equivalent to the associahedron, as was previously assumed. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We present a sim…

2008-05-08math.ATmath.COmath.CT

Root systems and graph associahedra

It is known that a connected simple graph $G$ associates a simple polytope $P_G$ called a graph associahedron in Euclidean space. In this paper we show that the set of facet vectors of $P_G$ forms a root system if and only if $G$ is a cycle graph and that the root system is of type A.

2016-02-12math.ATmath.CO

A Diagonal on the Associahedra

Let C_*(K) denote the cellular chains on the Stasheff associahedra. We construct an explicit combinatorial diagonal Δ: C_*(K) --> C_*(K) \otimes C_*(K); consequently, we obtain an explicit diagonal on the A_\infty-operad. We apply the diagonal Δto define the tensor product of A_\infty-(co)algebras in maximal generality.

2004-06-07math.AT

Convex Polytopes from Nested Posets

Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes. However, these poset associahedra fall in a different category altogether than generalized permutohedra.

2013-06-18math.ATmath.COmath.QA

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.

2004-08-21math.AT

Pseudograph associahedra

Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.

2010-05-14math.ATmath.COmath.QA

更多相关论文收起更多论文

Associahedra 上对角映射的比较

Comparing Diagonals on the Associahedra

本文证明了作者2004年给出的 associahedron 对角逼近公式与 Markl–Shnider、Loday 及 Masuda–Thomas–Tonks–Vallette 分别独立得到的“神奇公式”完全一致。

2024-02-23math.ATmath.CO

Upper and lower bound theorems for graph-associahedra

From the paper of the first author it follows that upper and lower bounds for $γ$-vector of a simple polytope imply the bounds for its $g$-,$h$- and $f$-vectors. In the paper of the second author it was obtained unimprovable upper and lower bounds for $γ$-vectors of flag nestohedra, particularly Gal's conjecture was proved for this case. In the present paper we obtain unimprovable upper and lower bounds for $γ$-vectors (consequently, for $g$-,$h$- and $f$-vectors) of graph-associahedra and some its important subclasses. We use the constructions that for an $(n-1)$-dimensional graph-associahedr…

2010-05-17math.AGmath.ATmath.CO

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.

2003-12-14math.ATmath.QA

The diagonal of the multiplihedra and the tensor product of A-infinity morphisms

We define a cellular approximation for the diagonal of the Forcey--Loday realizations of the multiplihedra, and endow them with a compatible topological cellular operadic bimodule structure over the Loday realizations of the associahedra. This provides us with a model for topological and algebraic A-infinity morphisms, as well as a universal and explicit formula for their tensor product. We study the monoidal properties of this newly defined tensor product and conclude by outlining several applications, notably in algebraic and symplectic topology.

2023-02-17math.ATmath.CO

The diagonal of the operahedra

The primary goal of this article is to set up a general theory of coherent cellular approximations of the diagonal for families of polytopes by developing the method introduced by N. Masuda, A. Tonks, H. Thomas and B. Vallette. We apply this theory to the study of the operahedra, a family of polytopes ranging from the associahedra to the permutahedra, and which encodes homotopy operads. After defining Loday realizations of the operahedra, we make a coherent choice of cellular approximations of the diagonal, which leads to a compatible topological cellular operad structure on them. This gives a…

2022-05-19math.ATmath.CO

Convex Hull Realizations of the Multiplihedra

We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_infinity-categories, deformation theory, and moduli spaces. We also include …

2008-06-10math.ATmath.COmath.CT

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.

2007-10-23math.ATmath.QA