The categorical graph minor theorem
Abstract
We define the graph minor category and prove that the category of contravariant representations of the graph minor category over a Noetherian ring is locally Noetherian. This can be regarded as a categorification of the Robertson--Seymour graph minor theorem. In addition, we generalize Sam and Snowden's Gr\"obner theory of categories to the setting of pairs consisting of a category along with a functor to sets, and we apply this theory to the edge functor on the graph minor category. As an application, we study homology groups of unordered configuration spaces of graphs, improving upon various finite generation results in this subject.
Keywords
Cite
@article{arxiv.2004.05544,
title = {The categorical graph minor theorem},
author = {Dane Miyata and Nicholas Proudfoot and Eric Ramos},
journal= {arXiv preprint arXiv:2004.05544},
year = {2022}
}
Comments
A gap has been discovered in the proof the main theorem, and specifically in the justification for Lemma 3.5. For now, the Categorical graph minor theorem will remain a conjecture. One can view the applications of this work to graph configuration spaces as being consequences of this conjecture, as their proofs are unaffected by the gap