In the early 1990s, a family of combinatorial CW-complexes named

permutoassociahedra was introduced by Kapranov, and it was realized

by Reiner and Ziegler as a family of convex polytopes. The polytopes

in this family are \hybrids" of permutohedra and associahedra. Since

permutohedra and associahedra are simple, it is natural to search for

a family of simple permutoassociahedra, which is still adequate for

a topological proof of Mac Lane's coherence. Such a family, intro-

duced in the paper A simple permutoassociahedron co-authored with

Zoran Petri c and -Dord e Barali c, will be presented in this talk. Beside

the combinatorial definition of the family, two different geometrical

realisations willalso be shown.