The Infinite Seminar

There is a paper by Loday on the arXiv since about a month ago, entitled The diagonal of the Stasheff polytope. The basic idea of the paper is to introduce a new operad ${\mathrm{AA}}_{\mathrm{\infty }}$, built on the simplicial chains of Lodays triangulation of the associahedron, and using the relative simplicity of forming diagonals on simplicial complexes to generate a reasonably natural diagonal on this new operad.

With quasiisomorphisms from our familiar $A_{\mathrm{\infty }}$-operad to ${\mathrm{AA}}_{\mathrm{\infty }}$ and back again, he then constructs a diagonal on $A_{\mathrm{\infty }}$, formed by going to ${\mathrm{AA}}_{\mathrm{\infty }}$ and computing a simplicial diagonal, which finally gets deformed into a diagonal on $A_{\mathrm{\infty }}$.

Thus, this construction is based in a slightly different approach to diagonal computation than both the Saneblidze-Umble construction and the Markl-Schnider paper; but Loday conjectures equality between the two constructions based on a comparison of the results for the diagonal 5-ary operation.

If anyone out there has read more of the paper than I have (or if Loday himself is reading this sporadic blog), I would appreciate some nudges on how to internalize the deformation enough to fix it in computer code.