A Category of Spectral Triples and Discrete Groups with Length Function

TL;DR

This paper introduces a new categorical framework for spectral triples, using discrete groups with length functions, and explores their properties and potential for future research.

Contribution

It defines a notion of morphism for spectral triples and demonstrates a covariant functor from discrete groups with length functions to spectral triples.

Findings

Spectral triples can be constructed functorially from discrete groups with length functions.

The paper proposes a categorical perspective on spectral triples and their morphisms.

Future research directions in the categorical properties of spectral triples are outlined.

Abstract

In the context of A. Connes' spectral triples, a suitable notion of morphism is introduced. Discrete groups with length function provide a natural example for our definitions. A. Connes' construction of spectral triples for group algebras is a covariant functor from the category of discrete groups with length functions to that of spectral triples. Several interesting lines for future study of the categorical properties of spectral triples and their variants are suggested.