Weak Factorization System for Actions of Po-monoids on Posets

TL;DR

This paper explores weak factorization systems in the category of $S$-posets, establishing conditions under which certain classes of maps form such systems, and characterizes regular injective objects in specific cases.

Contribution

It introduces and proves the existence of weak factorization systems in {f Pos}-$S$ under certain conditions and characterizes regular injective objects in these categories.

Findings

$( ext{C}_D, ext{E}_S)$ forms a weak factorization system when the identity is bottom in $S$.

Characterization of regular injective objects in {f Pos}-$S/B$ under specific conditions.

$(Emb, Top)$ is a weak factorization system when $S$ is a pogroup.

Abstract

Let $S$ be a pomonoid. In this paper, {\bf Pos}-$S$, the category of $S$-posets and $S$-poset maps, is considered. One of the main aims of this paper is to draw attention to the notion of weak factorization systems in {\bf Pos}-$S.$ We show that if the identity element of $S$ is the bottom element, then $(\mathcal{C_D}, \mathcal{E_S})$ is a weak factorization system in {\bf Pos}-$S,$ where $\mathcal{C_D}$ and $\mathcal{E_S}$ are the class of down-closed embedding $S$-poset maps and the class of all split $S$-poset epimorphisms, respectively. Among other things, we use a fibrewise notion of complete posets in the category {\bf Pos}-$S/B$ under a particular case where $B$ has trivial action. We get a necessary condition for regular injective objects in {\bf Pos}-$S/B$. Finally, we characterize them under a spacial case, where $S$ is, a pogroup and conclude $(Emb, Top)$ is a weak factorization system in {\bf Pos}-$S$.