Trees, quivers, bigraphs: combinatorial bialgebras from monoidal M\"obius categories

Abstract

The Hopf algebras associated by G\'alvez-Carillo, Kock and Tonks to monoidal M\"obius categories are shown to be distinct from those associated by Cibils and Rosso to Hopf quivers, except in the trivial case of a group algebra. Combinatorial left-sided Hopf algebras in the sense of Loday and Ronco do, however, provide examples, including planar rooted trees. As a new example, Milner's bigraphs also define a bialgebra in this way.