A categorical description of Bass-Serre theory

TL;DR

This paper generalizes Bass-Serre theory from monoids to categories, showing how Rees categories embed into their universal groups and connecting graph of groups to Rees categories.

Contribution

It extends the theory of Rees monoids to Rees categories, establishing embedding results and linking Bass-Serre theory to category embedding into groupoids.

Findings

Rees categories embed into their universal groups.

Construction of Rees categories from graphs of groups.

Bass-Serre theory as a special case of category embedding.

Abstract

Self-similar group actions may be encoded by a class of left cancellative monoids called left Rees monoids, a result obtained by combining pioneering work by Perrot with later work by the first author. Left Rees monoids that are also right cancellative are called Rees monoids. Irreducible Rees monoids have the striking property that they are embedded in their universal groups and these universal groups are HNN extensions by a single stable letter. In this paper, we generalize the above theory from monoids to categories. We study the structure of left Rees categories and prove that Rees categories embed into their universal groups. Furthermore, we show that from each graph of groups, we may construct a Rees category and prove that the fundamental group of the former is the universal group of the latter. In this way, Bass-Serre theory may be viewed as a special case of the general problem of embedding categories into groupoids.