The stratified homotopy type of the reductive Borel-Serre compactification

TL;DR

This paper characterizes the homotopy type of the reductive Borel-Serre compactification using an exit path $ty$-category, linking it to rational parabolic subgroups and their radicals, and provides a combinatorial framework for sheaves.

Contribution

It identifies the exit path $ty$-category of the compactification with a nerve of a simple 1-category, offering new insights into its fundamental group and sheaf theory.

Findings

Identifies the exit path $ty$-category as a nerve of a 1-category.

Recovers the fundamental group of the compactification.

Provides a combinatorial model for constructible sheaves.

Abstract

We identify the exit path $\infty$-category of the reductive Borel-Serre compactification as the nerve of a $1$-category defined purely in terms of rational parabolic subgroups and their unipotent radicals. As an immediate consequence, we identify the fundamental group of the reductive Borel-Serre compactification, recovering a result of Ji-Murty-Saper-Scherk, and we obtain a combinatorial incarnation of constructible complexes of sheaves on the reductive Borel-Serre compactification as elements in a derived functor category.