Categorical Properties of Sequentially Dense Monomorphisms of Semigroup Acts

Abstract

Let $\mathcal M$ be a class of (mono)morphisms in a category $\mathcal A$. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair $(\mathcal{A},\mathcal{M})$. In this paper we take $\mathcal A$ to be the category Act-S of acts over a semigroup $S$, and ${\mathcal M}_d$ to be the class of sequentially dense monomorphisms (of interest to computer scientists, too) and study the categorical properties, such as limits and colimits, of the pair $(\mathcal{A},\mathcal{M})$. Injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, and the authors who used it to obtain information about injectivity relative to monomorphisms.