  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.
