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 (DU, SplitEpi) is a weak factorization system in Pos-S, where 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.