---
title: "Frobenius functors and one-sided Hopf algebras in braided monoidal categories"
authors:
  - "Lucrezia Bottegoni"
  - "Davide Ferri"
  - "Paolo Saracco"
year: 2026
venue: "arXiv:math.QA"
arxiv: "2609.13028"
doi: null
source: "https://arxiv.org/abs/2609.13028"
topics:
  - "curiosity"
  - "maths-foundations"
  - "category-theory"
  - "quantum-algebra"
seed_rank: 1828
seed_batch: "curiosity-2026-09-15"
reviewed: "2026-09-15"
pool: "maths-foundations"
relevance_score: 10
lineage: braided-hopf-frobenius
cites:
  - title: "Frobenius functors and one-sided Hopf algebras in braided monoidal categories"
    url: "https://arxiv.org/abs/2609.13028"
    year: 2026
    arxiv: "2609.13028"
    doi: null
see:
  - "1585-quantum-hopf-rigidity-in-symmetric-monoidal-categories"
  - "1780-crossed-module-crossed-braided-categories"
  - "1781-on-pushouts-of-strict-symmetric-monoidal-categories"
---

# Frobenius functors and one-sided Hopf algebras in braided monoidal categories

## One-sentence takeaway

In braided monoidal categories, a bimonoid is one-sided Hopf with invertible antipode exactly when its free Hopf-module functor is Frobenius.

## Why it's lovely

Why you might love this: the classical Hopf↔Frobenius handshake (antipode exists iff free Hopf modules are Frobenius) lifts cleanly into any braided monoidal world with mild assumptions — one-sided Hopf monoids included. It is the kind of dictionary that makes quantum algebra feel like category theory rather than calculation.

## Key ideas

- Works in any braided monoidal category with mild exactness/dualizability hypotheses.
- Links existence of antipode-like maps on bimonoid-like objects to the Frobenius property of the free Hopf-module functor.
- Characterizes one-sided Hopf monoids with invertible antipode via that Frobenius condition.
- Extends a classical Hopf-algebra fact to the braided setting Broadside already tracks (1585, braided cards).

## Caveats

Depends on braided monoidal background and technical dualizability hypotheses; not a new knot or TQFT invariant by itself. Read beside quantum Hopf rigidity (1585) and braided shelf.

## Links

- arXiv abs: https://arxiv.org/abs/2609.13028
- PDF: https://arxiv.org/pdf/2609.13028.pdf