Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

TL;DR

This paper develops a new framework for equivariant coarse homology theories using left-exact $$-categories and applies it to the algebraic K-theory related to the Novikov conjecture.

Contribution

It introduces a novel construction of equivariant $X$-controlled objects in left-exact $$-categories, enabling new applications to the Novikov conjecture.

Findings

Established new equivariant coarse homology theories from algebraic K-theory.

Applied injectivity results for assembly maps to algebraic K-theory.

Provided a framework connecting controlled objects with the Novikov conjecture.

Abstract

We associate to every $G$-bornological coarse space $X$ and every left-exact $\infty$-category with $G$-action a left-exact infinity-category of equivariant $X$-controlled objects. Postcomposing with algebraic K-theory leads to {new} equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact $\infty$-categories.