Hopf substitutions in Species

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.