Categorified Fredholm Modules and Chern Characters

TL;DR

This paper advances noncommutative geometry by developing a categorified framework using small linear categories, defining Fredholm modules over them, and constructing a Chern character that links to cyclic cohomology.

Contribution

It introduces a categorified approach to noncommutative differential geometry, replacing rings with categories and defining associated Fredholm modules and Chern characters.

Findings

Categorification of noncommutative geometry via small linear categories.

Construction of Fredholm modules over these categories.

A well-behaved Chern character depending on homotopy classes.

Abstract

In this paper, we continue our program of systematic categorification of the Noncommutative Differential Geometry of Connes. We replace a ring with a small $\mathbb C$-linear category, seen as a ring with several objects in the sense of Mitchell. We introduced Fredholm modules over this category and construct a Chern character taking values in the cyclic cohomology of $\mathcal C$. We show that this categorified Chern character depends only on the homotopy class of the Fredholm module and is well-behaved with respect to the periodicity operator in cyclic cohomology.