Connected objects in categories of $S$-acts

TL;DR

This paper investigates the property of compactness in categories of $S$-acts, exploring how it relates to structural properties and introducing a unifying framework for these categories.

Contribution

It introduces the notion of a concrete category with unique decomposition of objects to unify the study of $S$-acts and analyzes the impact of compactness within this context.

Findings

Characterization of compact $S$-acts

Introduction of concrete categories with unique decomposition

Unified approach to categories of $S$-acts

Abstract

In this paper, the categorial property of compactness of an object, i. e. commuting of the corresponding $\Hom$ functor with coproducts, is studied in categories of $S$-acts and the corresponding structural properties of compact $S$-acts are shown. In order to establish a general context and to unify the approach to both of the most important categories of $S$-acts, the notion of a concrete category with unique decomposition of objects is introduced and studied.