|
We develop the notion of the composition of two coalgebras, which arises naturally in
higher category theory and in the theory of species.
We prove that the composition of two cofree coalgebras is again cofree, and we give
sufficient conditions that ensure the composition is a one-sided Hopf algebra.
We show these conditions are satisfied when
one coalgebra is a graded Hopf operad D and the other is a connected graded
coalgebra with coalgebra map to D. We conclude by computing the primitive elements
for compositions of coalgebras built on the vertices of multiplihedra, composihedra, and
hypercubes.
|
