Title: A Diagonal on the Associahedra
Authors: Samson Saneblidze and Ronald Umble
MSC-class: 57T30; 55U10; 55N20; 55N10
xxx.LANL.gov: math.AT/0011065
Author's Addresses:
A. Razmadze Mathematical Institute, M. Aleksidze St., 1, 380093 Tbilisi, Georgia
Department of Mathemaitcs, Millersville Univ. of PA, Millersville, PA 17551
Author's e-mail addresses:
sane@rmi.acnet.ge
ron.umble@millersville.edu
ABSTRACT:
An associahedral set is a combinatorial object generated by Stasheff
associahedra K_n and equipped with appropriate face and degeneracy operators.
Associahedral sets are similar in many ways to simplicial or cubical sets. In
this paper we give a formal definition of an associahedral set, discuss some
naturally occurring examples and construct an explicit geometric diagonal
\Delta :C_*(K_n) --> C_*(K_n) \otimes C_*(K_n) on the cellular chains C_*(K_n).
The diagonal \Delta, which is analogous to the Alexander-Whitney diagonal on
the simplices, gives rise to a diagonal on any associahedral set and leads
immediately to an explicit diagonal on the A_\infty operad. As an application
of this, we use the diagonal \Delta to define a tensor product in the A_\infty
category.