M Theory Lesson 99
$Q_n \rightarrow A_n \rightarrow P_n$
also have left and right adjoints, giving a triple (cyclic) double adjunction involving these polytopes. This amazing categorical relation is a generalisation of an ordinary adjunction between just two categories. Since the associahedra were given as a 1-operad it is natural to try to view this triple as a 3 dimensional structure. Can we extend the use of vertices and edges to faces in a 2-category replacement for posets?
Aside: Regarding the Hopf algebras here, P. Cartier provides the clearest exposition. MZVs appear on page 65.