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An abelian envelope without the quotient property
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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BibTeX
@misc{flake2026abelian,
title = {{An abelian envelope without the quotient property}},
author = {Johannes Flake and Jonathan Gruber and Thorsten Heidersdorf},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17467},
eprint = {2609.17467},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17467},
}
RIS
TY - UNPB
TI - An abelian envelope without the quotient property
AU - Johannes Flake
AU - Jonathan Gruber
AU - Thorsten Heidersdorf
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17467
UR - https://arxiv.org/abs/2609.17467
AB - 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.
ER -