Hopf substitutions in Species

TL;DR

This paper explores when species substitutions produce Hopf monoids, extending known results and analyzing properties of these constructions within the theory of species.

Contribution

It characterizes conditions under which species substitutions yield Hopf monoids and extends interpolation results to this context.

Findings

Identifies species for which substitution yields Hopf monoids.

Analyzes properties of Hopf monoids arising from species substitutions.

Extends interpolation results in species to new contexts.

Abstract

In the theory of species, the species $\mathbf{L}$ of linear orders and the substitution operation $\boldsymbol{\circ}$ combine for a compelling result: given any positive comonoid $\mathbf{p}$, $\mathbf{L}\boldsymbol{\circ}\mathbf{p}$ carries the structure of Hopf monoid, freely generated by $\mathbf{p}$. Leaving aside the universal property this implies, we ask, "for which $\mathbf{b}$ does $\mathbf{b}\boldsymbol{\circ}\mathbf{p}$ carry the structure of Hopf monoid?" After answering this question, we look at basic properties of our construction. We also extend a result of the present authors, on interpolation in species, to this new context.