TORSION THEORIES BESIDE THE POINT -- HOMOTOPICAL AND TWO-DIMENSIONAL APPROACHES

This thesis investigates the notion of torsion theory beyond the classical pointed setting, that is, in categories which need not have a zero object. Introduced in the context of abelian categories, with torsion and torsion-free abelian groups being a motivating example, torsion theories are by now well established in various more general pointed contexts, supported by a wide range of results and examples. More recently, several approaches have been proposed in the literature to extend this notion to non-pointed categories, in order to capture situations that resemble the classical picture but lie outside its scope. The aim of the present work is to contribute to this programme by both extending and unifying existing approaches and by introducing entirely new ones. The first part of this thesis focuses on torsion theories defined relative to a chosen class of trivial objects, sometimes called pretorsion theories. Several classical results on torsion theories – such as certain characterisations of torsion(-free) subcategories, and the well-known interplays of torsion theories with closure operators and factorisation systems – rely in the pointed case on additional structural assumptions and are not known to hold for pretorsion theories. To provide a framework where such results can be in part recovered, a new notion is introduced: prenormal categories. This notion, which in the pointed case generalises normal categories, is investigated in depth through its properties and many pointed and non-pointed examples, and it is shown to allow several of the aforementioned results to be suitably extended to pretorsion theories. In the second part, we introduce the new notion of homotopy torsion theories, developed in the known setting of categories with nullhomotopies. The general framework of homotopy torsion theories unifies pretorsion theories with other approaches from the literature, and in particular shows that factorisation systems arise precisely as homotopy torsion theories on arrow categories. Moreover, we demonstrate through some paradigmatic examples that this notion can incorporate into torsion theories a degree of two-dimensionality. Finally, the third part introduces bitorsion theories in bicategories with a bizero object, lifting the classical definition of torsion theory to a genuinely two-dimensional level. The basic theory is developed and illustrated with a range of examples, including two-dimensional analogues of classical torsion theories, as well a class of examples where connected and discrete objects are shown to form a bitorsion theory.