Noncommutative Geometry in the Framework of Differential Graded Categories

TL;DR

This survey explores a framework for noncommutative geometry using differential graded categories, introducing a category of noncommutative spaces, motives, and zeta functions to deepen understanding of noncommutative spaces.

Contribution

It presents a comprehensive framework connecting differential graded categories with noncommutative geometry, including the construction of noncommutative spaces and motivic measures.

Findings

Construction of the category of noncommutative spaces

Introduction of noncommutative motives and motivic measures

Definition of zeta functions for noncommutative spaces

Abstract

In this survey article we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces.