Generalised morphisms of k-graphs: k-morphs

TL;DR

This paper introduces k-morphs as a systematic framework to generalize and unify constructions of higher-rank graphs, linking them to C*-correspondences and extending the functorial relationship to C*-algebras.

Contribution

The paper defines k-morphs, establishing a categorical framework that unifies various graph constructions and relates them to C*-correspondences.

Findings

Introduces k-morphs as a unifying concept for higher-rank graph constructions.

Establishes a functor from the category of k-graphs with k-morphs to C*-algebras.

Provides a systematic approach connecting graph theory and operator algebras.

Abstract

In a number of recent papers, (k+l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C*-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k-morphs, which provide a systematic unifying framework for these various constructions. We think of k-morphs as the analogue, at the level of k-graphs, of C*-correspondences between C*-algebras. To make this analogy explicit, we introduce a category whose objects are k-graphs and whose morphisms are isomorphism classes of k-morphs. We show how to extend the assignment \Lambda \mapsto C*(\Lambda) to a functor from this category to the category whose objects are C*-algebras and whose morphisms are isomorphism classes of C*-correspondences.