The reductive Borel-Serre compactification as a model for unstable algebraic K-theory

Abstract

Let $A$ be an associative ring and $M$ a finitely generated projective $A$-module. We introduce a category $\operatorname{RBS}(M)$ and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories $\operatorname{RBS}(M)$ naturally arise as generalisations of the exit path $\infty$-category of the reductive Borel-Serre compactification of a locally symmetric space, and one of our main techniques is to find purely categorical analogues of some familiar structures in these compactifications.