From Six Functors Formalisms to Derived Motivic Measures

TL;DR

This paper develops a method to derive motivic measures from six functors formalisms, enabling new liftings of existing measures like the Gillet-Soué measure within a unified categorical framework.

Contribution

It introduces a general procedure to produce derived motivic measures from abstract six functors formalisms, extending the scope of motivic measure theory.

Findings

Defined a lifting of the Gillet-Soué motivic measure

Established a framework connecting six functors formalisms to motivic measures

Provided a systematic method for deriving new motivic measures

Abstract

In this paper, we generally describe a method of taking an abstract six functors formalism in the sense of Khan or Cisinski-D\'{e}glise, and outputting a derived motivic measure in the sense of Campbell-Wolfson-Zakharevich. In particular, we use this framework to define a lifting of the Gillet-Sou\'e motivic measure.