An abelian envelope without the quotient property

RESUMO

We show that the universal rigid monoidal category on one object has an abelian envelope. Since it was proven by Coulembier-Etingof-Ostrik-Pauwels (2023) that this category cannot have an abelian envelope with the quotient property, this disproves the conjecture in [ this http URL .] saying that every abelian envelope has the quotient property. Monoidal Ringel duality (Flake-Gruber arXiv:2512.19558 ) yields a candidate envelope as a lower finite highest weight category. Our proof that this is an abelian envelope relies on Coulembier-Etingof's (2024) continuants to show the required universal property. AI was used to find these results.