Monoidal Gröbner systems and categories of affine Brauer type
Abstract
We introduce an analogue of Gröbner bases, or equivalently, Bergman's diamond lemma, for linear (super)monoidal categories. It gives a systematic way to study monoidal ideals and to prove basis theorems. The central notion is that of a monoidal Gröbner system and is based on higher linear rewriting theory. We then apply the theory to categories of affine Brauer type: linear (super)monoidal categories that have the same hom-basis as the affine Brauer category, but possibly distinct composition and tensor product. Our main result is a criterion for a category to be of affine Brauer type, reducing the question to an explicit list of local computations, which we partially implement in the computer algebra system FORM. This gives new combinatorial proofs of the basis theorems for the affine Brauer category, the nil-Brauer category and the affine VW supercategory, and yields new examples. In particular, we construct the odd nil-Brauer category, a conjectural supercategorification of the split imathquantum group of rank one, and the quantized affine VW supercategory, whose non-affine part is the quantized periplectic Brauer category and which we expect to satisfy a higher quantum Schur--Weyl duality. Finally, we classify the categories of affine Brauer type admitting a sufficiently simple presentation.
Related
- Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the integral type AIII case
- Affine and cyclotomic Brauer categorification via imath-Kac--Moody $2$-categories: the half-integral type AIII case
- On closed symmetric monoidal enriched ∞-categories
- The symmetric skew monoidal structure of Gray-categories
- On pushouts of strict symmetric monoidal categories
BibTeX
@misc{barbier2026monoidal,
title = {{Monoidal Gröbner systems and categories of affine Brauer type}},
author = {Sigiswald Barbier and Léo Schelstraete},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25159},
eprint = {2609.25159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25159},
}
RIS
TY - UNPB
TI - Monoidal Gröbner systems and categories of affine Brauer type
AU - Sigiswald Barbier
AU - Léo Schelstraete
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25159
UR - https://arxiv.org/abs/2609.25159
AB - We introduce an analogue of Gröbner bases, or equivalently, Bergman's diamond lemma, for linear (super)monoidal categories. It gives a systematic way to study monoidal ideals and to prove basis theorems. The central notion is that of a monoidal Gröbner system and is based on higher linear rewriting theory. We then apply the theory to categories of affine Brauer type: linear (super)monoidal categories that have the same hom-basis as the affine Brauer category, but possibly distinct composition and tensor product. Our main result is a criterion for a category to be of affine Brauer type, reducing the question to an explicit list of local computations, which we partially implement in the computer algebra system FORM. This gives new combinatorial proofs of the basis theorems for the affine Brauer category, the nil-Brauer category and the affine VW supercategory, and yields new examples. In particular, we construct the odd nil-Brauer category, a conjectural supercategorification of the split imathquantum group of rank one, and the quantized affine VW supercategory, whose non-affine part is the quantized periplectic Brauer category and which we expect to satisfy a higher quantum Schur--Weyl duality. Finally, we classify the categories of affine Brauer type admitting a sufficiently simple presentation.
ER -