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arXiv · 2609.17467

An abelian envelope without the quotient property

Johannes Flake·Jonathan Gruber·Thorsten Heidersdorf
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Abstract

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 [op.cit.] 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.

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Johannes Flake, Jonathan Gruber, Thorsten Heidersdorf. 2026-09-15. An abelian envelope without the quotient property. https://arxiv.org/abs/2609.17467

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