Homotopy torsion theories

TL;DR

This paper introduces homotopy torsion theories within categories with nullhomotopies, generalizing existing torsion theories and establishing new correspondences with factorization systems without requiring pullbacks or pushouts.

Contribution

It defines homotopy torsion theories in a broad categorical context and links them to orthogonal and weakly orthogonal factorization systems, providing new proofs and extensions.

Findings

Established a correspondence between homotopy torsion theories and orthogonal factorization systems.

Extended the correspondence to weakly orthogonal factorization systems and weak homotopy torsion theories.

Provided a new proof avoiding the need for pullbacks and pushouts in the category.

Abstract

In the context of categories equipped with a structure of nullhomotopies, we introduce the notion of homotopy torsion theory. As special cases, we recover pretorsion theories as well as torsion theories in multi-pointed categories and in pre-pointed categories. Using the structure of nullhomotopies induced by the canonical string of adjunctions between a category A and the category Arr(A) of arrows, we give a new proof of the correspondence between orthogonal factorization systems in A and homotopy torsion theories in Arr(A), avoiding the request on the existence of pullbacks and pushouts in A. Moreover, such a correspondence is extended to weakly orthogonal factorization systems and weak homotopy torsion theories.