Pretorsion theories in general categories
Alberto Facchini, Carmelo Finocchiaro, and Marino Gran

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 (, ) of full replete subcategories in a category , the corresponding full subcategory of \emph{trivial objects} in . The morphisms which factor through are called -trivial, and these form an ideal of morphisms, with respect to which one can define -prekernels, -precokernels, and short -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 is reduced to the -object of . We study the basic properties of pretorsion theories, and examine some new examples in the category of…
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