Co-universal $C^{\ast}$-algebras for product systems over finite aligned subcategories of groupoids

TL;DR

This paper introduces and studies co-universal $C^{ ext{*}}$-algebras for product systems over finite aligned subcategories of groupoids, establishing their existence under certain conditions.

Contribution

It defines compactly aligned product systems over finite aligned categories and proves the existence of co-universal algebras for their Nica covariant representations.

Findings

Existence of co-universal algebras for specified product systems.

Introduction of compactly aligned product systems over finite categories.

Development of Nica covariant representation theory in this context.

Abstract

The product systems over left cancellative small categories are introduced and studied in this paper. We also introduce the notion of compactly aligned product systems over finite aligned left cancellative small categories and its Nica covariant representations. The existence of co-universal algebras for injective, gauge-compatible, Nica covariant representations of compactly aligned product systems over finite aligned subcategories of groupoids is proved in this paper.