On General Closure Operators and Quasi Factorization Structures

TL;DR

This paper introduces and explores the concepts of general closure operators and quasi factorization structures in categories, establishing their properties, interrelations, and providing illustrative examples.

Contribution

It defines quasi weakly hereditary closure operators and quasi factorization structures, and demonstrates their interplay and conditions for their existence in categories.

Findings

Quasi factorization structures induce quasi right and left factorization structures.

Closure operators that are quasi weakly hereditary and quasi idempotent relate to specific factorization structures.

Examples illustrate the theoretical concepts and their applications.

Abstract

In this article the notions of (quasi weakly hereditary) general closure operator $\mb{C}$ on a category $\cx$ with respect to a class $\cm$ of morphisms, and quasi factorization structures in a category $\cx$ are introduced. It is shown that under certain conditions, if $(\ce, \cm)$ is a quasi factorization structure in $\cx$, then $\cx$ has quasi right $\cm$-factorization structure and quasi left $\ce$-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class $\cm$, every quasi factorization structure $(\ce, \cm)$ yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class $\cm$, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. Several illustrative examples are furnished.