Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets

Abstract

Let \(S\) be a pomonoid. In this paper, 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 Pos-\(S.\) We show that if \(S\) is a pogroup, or the identity element of \(S\) is the bottom (or top) element, then \((\mathcal{DU}, SplitEpi)\) is a weak factorization system in Pos-\(S,\) where \(\mathcal{DU}\) and \(SplitEpi\) are the class of du-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 Pos-\(S/B\) under a particular case that \(B\) has trivial action. We show that every regular injective object in Pos-\(S/B\) is topological functor. Finally, we characterize them under a special case, where \(S\) is a pogroup.