Year: 2026
Type: preprint
Abstract: Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.
Source: arXiv (Cornell University)
Institution: University of Turin
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", European Commission, Fonds Wetenschappelijk Onderzoek, Vrije Universiteit Brussel, Junta de Andalucía, Ministero dell'Università e della Ricerca, Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni, Agencia Estatal de Investigación
OpenAlex ID: W7212520390