On the Q construction for exact quasicategories

TL;DR

This paper develops a higher categorical Q construction for exact quasicategories, enabling computation of their K-theory and establishing their homotopy-theoretic properties, including infinite loop space structures and a proto-devissage criterion.

Contribution

It introduces a novel higher categorical Q construction for exact quasicategories, linking K-theory to infinite loop spaces and providing a new criterion for K-theory equivalences.

Findings

The K-theory of an exact quasicategory can be computed via a higher categorical Q construction.

The resulting homotopy type forms an infinite loop space compatible with canonical structures.

A proto-devissage criterion characterizes when a nilimmersion induces a K-theory equivalence.

Abstract

We prove that the K-theory of an exact quasicategory can be computed via a higher categorical variant of the Q construction. This construction yields a quasicategory whose weak homotopy type is a delooping of the K-theory space. We show that the direct sum endows this homotopy type with the structure of a infinite loop space, which agrees with the canonical one. Finally, we prove a proto-devissage result, which gives a necessary and sufficient condition for a "nilimmersion" of stable quasicategories to be a K-theory equivalence. In particular, we prove that a well-known conjecture of Ausoni and Rognes is equivalent to the weak contractibility of a particular quasicategory.