Pretorsion theories in prenormal categories
Sandra Mantovani, Mariano Messora
TL;DR
The paper extends pointed torsion theories to the non-pointed setting by employing a fixed class $\mathcal{Z}$ of trivial objects and relative kernels/cokernels within the framework of $\mathcal{Z}$-prenormal categories. It establishes how to recover characterizations of $\mathcal{Z}$-torsion and $\mathcal{Z}$-torsion-free subcategories and connects these notions to closure operators and to (stable) factorisation systems. It introduces the class $\operatorname{TKer}(\mathcal{C},\mathcal{Z})$ and analyzes the interplay between $\mathcal{Z}$-normal morphisms and pretorsion theory presentations, extending classical homological-torsion theory results to a broader, non-pointed context. The framework also permits specialization to hereditary torsion theories and yields new examples of pretorsion theories across various $\mathcal{Z}$-pre-normal categories, broadening applicability beyond pointed or regular settings.
Abstract
In this paper we extend several classical results on pointed torsion theories -- also known as torsion pairs -- to the setting of non-pointed torsion theories defined via kernels and cokernels relative to a fixed class of trivial objects (often referred to as pretorsion theories). Our results are developed in the recently introduced framework of (non-pointed) prenormal categories and other related contexts. Within these settings, we recover some characterisations of torsion and torsion-free subcategories, as well as the classical correspondences between torsion theories and closure operators. We also suitably extend a correspondence between torsion theories and (stable) factorisation systems on the ambient category, known in the homological case. Some of these results are then further specialised to an appropriate notion of hereditary torsion theory. Finally, we apply the developed theory to construct new examples of pretorsion theories.