Homotopy torsion theories

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 $\mathrm{Arr}\left(A\right)$ of arrows, we give a new proof of the correspondence between orthogonal factorization systems in $A$ and homotopy torsion theories in $\mathrm{Arr}\left(A\right)$, 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.