Cofree Compositions of Coalgebras

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 that these conditions are satisfied when one coalgebra is a graded Hopf operad ${\mathcal{D}}$ and the other is a connected graded coalgebra with coalgebra map to ${\mathcal{D}}$ . We conclude by computing the primitive elements for compositions of coalgebras built on the vertices of multiplihedra, composihedra, and hypercubes.