# Functorial invariants of trees and their cones

## TL;DR

This paper investigates functorial invariants of trees and their cones using category theory, establishing finite generation results that relate to homology, intersection homology, and Kazhdan-Lusztig coefficients.

## Contribution

It introduces a categorical framework for trees and cones, proving Noetherian properties and finite generation of modules related to homology and intersection homology.

## Key findings

- Modules over tree categories are Noetherian.
- Finite generation of homology modules leads to growth bounds.
- Kazhdan-Lusztig coefficients are connected to these invariants.

## Abstract

We study the category whose objects are trees (with or without roots) and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian, and we study two natural families of modules over these categories. The first takes a tree to a graded piece of the homology of its unordered configuration space, or to the homology of the unordered configuration space of its cone. The second takes a tree to a graded piece of the intersection homology of the reciprocal plane of its cone, which is a vector space whose dimension is given by a Kazhdan-Lusztig coefficient. We prove finite generation results for each of these modules, which allow us to obtain results about the growth of Betti numbers of configuration spaces and of Kazhdan-Lusztig coefficients of graphical matroids.