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 $${\text {RBS}}(M)$$ RBS ( M ) and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories $${\text {RBS}}(M)$$ 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.