The diagonal of the operahedra

TL;DR

This paper develops a general theory for cellular approximations of the diagonal in polytopes, applies it to operahedra, and constructs a compatible topological operad structure to model homotopy operads.

Contribution

It introduces a unified framework for cellular diagonal approximations and applies it to operahedra, enabling explicit formulas for tensor products of homotopy operads.

Findings

Established a coherent cellular approximation method for polytopes.

Constructed a topological cellular operad structure on operahedra.

Provided a functorial formula for tensor products of homotopy operads.

Abstract

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 model for topological and algebraic homotopy operads and an explicit functorial formula for their tensor product.