Does every abelian envelope have the quotient property?
An abelian envelope of a monoidal category is a universal abelian tensor category receiving it. Coulembier, Etingof, Ostrik and Pauwels conjectured in 2023 that every abelian envelope has the quotient property, and proved that the universal rigid monoidal category on one object cannot have an abelian envelope with that property. Does every abelian envelope have the quotient property?
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Tensor categories
- Posed by
- Kevin Coulembier, Pavel Etingof, Victor Ostrik and Bregje Pauwels (2023)
- Year posed
- 2023
- Years open
- 3y
- Solved
- 2026-09-15
- Model
- GPT-6 Astra; Claude Opus 5
- Vendor
- OpenAI; Anthropic
- Collaborators
- Johannes Flake, Jonathan Gruber, Thorsten Heidersdorf
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
No. The universal rigid monoidal category on one object has an abelian envelope, which by the 2023 no-go result cannot have the quotient property.
What the AI did
From the paper's use-of-AI section: the first version of the proof was found by GPT-6 Astra on 5 September 2026; the authors then simplified and checked it and wrote it up by hand, before using GPT-6 Astra and Claude Opus to check the document. The abstract says outright that AI was used to find these results.
Verification
Checked here on 22 September 2026 against arXiv:2609.17467: the abstract states that the universal rigid monoidal category on one object does have an abelian envelope, which disproves the conjecture because Coulembier, Etingof, Ostrik and Pauwels had shown it cannot have one with the quotient property. The candidate envelope comes from monoidal Ringel duality and the universal property is established with Coulembier-Etingof continuants. The mathematics was not checked here; seven days old, no referee.