Refinement of Higher-Rank Graph Reduction

TL;DR

This paper extends the concept of graph reduction to higher-rank graphs, enabling a geometric classification of their $C^*$-algebras and uncovering new Morita equivalence classes.

Contribution

It formalizes higher-rank graph moves and extends reduction techniques, bridging geometric and categorical approaches for classifying higher-rank graph $C^*$-algebras.

Findings

Extension of reduction to higher-rank graphs

Identification of new Morita classes of $k$-graphs

Formalization of higher-rank graph moves

Abstract

Given a row-finite, source-free, graph of rank k, we extend the definition of reduction introduced by Eckhardt et al. This constitutes a large step forward in the extension of the geometric classification of finite directed graph $C^*$-algebras presented by Eilers et al. to higher-rank graph $C^*$-algebras. This new move acts as an inverse to delay, directly extends the previous version, and provides previously undocumented Morita classes of k-graphs. In pursuit of this extension, we formalize what constitutes a higher-rank graph move. Specifically, we use this formalization as a bridge between the new geometric reasoning and the classical category theoretic construction.