The categorical graph minor theorem

TL;DR

This paper introduces a categorical framework for graph minors, generalizes Gr"obner theory to categories with functors, and applies these to study homology of graph configuration spaces, extending finite generation results.

Contribution

It categorifies the Robertson--Seymour graph minor theorem and extends Gr"obner theory to new categorical settings, enabling advanced analysis of graph-related homology.

Findings

The category of contravariant representations of the graph minor category is locally Noetherian.

Generalization of Gr"obner theory to categories with functors to sets.

Improved finite generation results for homology groups of graph configuration spaces.

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.