Cofree compositions of coalgebras

TL;DR

This paper introduces a new composition operation for coalgebras, proves it preserves cofreeness, and explores conditions under which the composition forms a one-sided Hopf algebra, with applications to combinatorial structures.

Contribution

It defines the composition of coalgebras, proves cofreeness preservation, and establishes conditions for resulting structures to be one-sided Hopf algebras.

Findings

Composition of two cofree coalgebras is cofree.

Conditions for composition to be a one-sided Hopf algebra are identified.

Primitive elements are computed for coalgebras related to multiplihedra, composihedra, and hypercubes.

Abstract

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.