Pretorsion theories in general categories

TL;DR

This paper introduces pretorsion theories in general categories, extending classical torsion theories, and explores their properties and examples in categories like endomappings of finite sets and preordered sets.

Contribution

It develops the concept of pretorsion theories in arbitrary categories, generalizing classical torsion theories, and provides new examples and foundational properties.

Findings

Defines pretorsion theories in general categories.

Establishes properties and structure of pretorsion theories.

Provides new examples in specific categories such as endomappings and preordered sets.

Abstract

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of \emph{trivial objects} in $\mathcal C$. The morphisms which factor through $\mathcal Z$ are called $\mathcal Z$-trivial, and these form an ideal of morphisms, with respect to which one can define $\mathcal Z$-prekernels, $\mathcal Z$-precokernels, and short $\mathcal Z$-preexact sequences. This naturally leads to the notion of pretorsion theory, which is the object of study of this article, and includes the classical one in the abelian context when $\mathcal Z$ is reduced to the $0$-object of $\mathcal C$. We study the basic properties of pretorsion theories, and examine some new examples in the category of all endomappings of finite sets and in the category of preordered sets.