  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.
