# Skew products of finitely aligned left cancellative small categories and Cuntz-Krieger algebras

## TL;DR

This paper explores the structure of Cuntz-Krieger algebras associated with finitely aligned small categories under group cocycles, revealing their crossed product nature and applications to group actions and coactions.

## Contribution

It introduces a framework for analyzing skew product categories and their Cuntz-Krieger algebras via crossed products and coactions, extending the understanding of group actions on these algebras.

## Key findings

- Cuntz-Krieger algebras can be described as crossed products by group coactions.
- Group actions on small categories induce coactions on associated algebras.
- Universal groups of small categories relate to skew product connectedness.

## Abstract

Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSC's, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.