Constructing Cofree Compositional Coalgebras
Frank Sottile

   Based on ideas from higher category theory, Stefan
Forcey defined nine families of polytopes, together with
cellular maps that fit into a commutative grid.  Their 
faces are indexed by certain types of trees.  Building
on a previous study of a Hopf object built from the
vertices of the multiplihedron (the middle family of 
polytopes in the gris), Forcey, Lauve and I show how 
to put Hopf structures on all these objects in a 
uniform manner, through a general functorial construction 
that we call a composition of two coalgebras.

   This construction preserves cofreeness, and we give
conditions under which the resulting object is a one-sided
Hopf algebra.   The objects and maps corresponding to our
polytopes come from composing the sequence of Hopf algebra
maps with itself:

 Permutations -> Binary plane trees -> divided powers