Weak factorization systems for S-acts

Abstract

The concept of a weak factorization system has been studied extensively in homotopy theory and has recently found an application in one of the proofs of the celebrated flat cover conjecture, categorical versions of which have been presented by a number of authors including Rosický [15]. One of the main aims of this paper is to draw attention to this interesting concept and to initiate a study of these systems in relation to flatness of S−acts and related concepts.

Regular Injectivity and Exponentiability in the Slice Categories of Actions of Pomonoids on Posets

“…Also, we can show that (U, E S ) is a weak factorization system for the category Pos-S. Details of proof is similar to Theorem 3.1 in [2].…”

“…Consider the category Pos-S/B and let U be the class of all unitary S-poset monomorphisms (recall [2] that an S-poset monomorphism f : X → Y is unitary if y ∈ im(f ), whenever ys ∈ im(f ) for some s ∈ S) and E S be the class of all split S-poset epimorphisms (that is, an epimorphism that has a right inverse). One can prove that, each g ∈ E S is U-injective in this category.…”

Weak Factorization System for Actions of Po-monoids on Posets

“…So by the Proposition 2.4 and above theorem, we can say that (Emb, Emb ) is a weak factorization system for Pos-S. This implies that Emb is saturated (this means, every class in a category is closed under pushouts, transfinite compositions and retracts (see [2])).…”

“…Recently, Bailey and Renshaw in [2], provide a number of examples of weak factorization systems for S-acts such as the following theorem.…”

“…Recently, injective objects in slice categories (C/B) have been investigated in detail (see [1,6]), especially in relationship with weak factorization systems, a concept used in homotopy theory, in particular for model categories. More precisely, H-injective objects in C/B, for any B in C, form the right part of a weak factorization system that has morphisms of H as the left part (see [1,2]).…”