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 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.