Zappa-Sz\'ep products for partial actions of groupoids on Left Cancellative Small Categories

TL;DR

This paper explores the structure of groupoid actions on left cancellative small categories, focusing on their Zappa-Szép product constructions, properties of the associated tight groupoids, and conditions for amenability.

Contribution

It introduces a framework connecting left cancellative small categories with Zappa-Szép products and characterizes properties of the resulting tight groupoids, including amenability.

Findings

Certain categories can be represented as Zappa-Szép products.

Characterization of when the tight groupoid is Hausdorff, effective, and minimal.

Conditions under which the tight groupoid is amenable.

Abstract

We study groupoid actions on left cancellative small categories and their associated Zappa-Sz\'ep products. We show that certain left cancellative small categories with nice length functions can be seen as Zappa-Sz\'ep products. We compute the associated tight groupoids, characterizing important properties of them, like being Hausdorff, effective and minimal. Finally, we determine amenability of the tight groupoid under mild, reasonable hypotheses.