# The tight groupoid of the inverse semigroups of left cancellative small categories

## TL;DR

This paper constructs and analyzes the tight groupoid of inverse semigroups from left cancellative small categories, providing insights into their $C^*$-algebras' structure, simplicity, and amenability.

## Contribution

It introduces a path model for the filter space, computes the tight groupoid, and characterizes simplicity and amenability of the associated $C^*$-algebras.

## Key findings

- Representation of $C^*$-algebras as groupoid algebras
- Criteria for simplicity of the $C^*$-algebras
- Conditions for amenability of the tight groupoid

## Abstract

We fix a path model for the space of filters of the inverse semigroup $\mathcal{S}_\Lambda$ associated to a left cancellative small category $\Lambda$. Then, we compute its tight groupoid, thus giving a representation of its $C^*$-algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.