A generalization of higher rank graphs

TL;DR

This paper introduces generalized higher rank $k$-graphs, extending existing graph categories with a new notion of size, and demonstrates their construction via Zappa-Szép products of groupoids and higher rank graphs.

Contribution

It defines generalized higher rank $k$-graphs, broadening the framework of higher rank graphs and Levi categories, and provides a method to construct them using Zappa-Szép products.

Findings

Generalized higher rank $k$-graphs extend traditional higher rank graphs.

Construction of examples via Zappa-Szép products of groupoids and higher rank graphs.

Unification of graph categories with a new size notion.

Abstract

We introduce what we call `generalized higher rank $k$-graphs' as a class of categories equipped with a notion of size. They extend not only the higher rank $k$-graphs, but also the Levi categories introduced by the first author as a categorical setting for graphs of groups. We prove that examples of generalized higher rank $k$-graphs can be constructed using Zappa-Sz\'ep products of groupoids and higher rank graphs.