Some new examples of pretorsion theories

The notion of pretorsion theory is a natural extension of the classical notion of torsion theory in an abelian category. The idea is to associate, with any pair (T, F) of full replete subcategories of a category C, a corresponding notion of Z-trivial morphism, where Z = T ∩F is the induced subcategory of “trivial objects” in C. When C is pointed and Z = 0, with 0 the zero object of C, then the notion of pretorsion theory reduces to the usual notion of torsion theory. There are, however, several examples of pretorsion theories that are not torsion theories, for instance in the categories of preordered sets FF and of (small) categories? . In this talk we shall recall some basic properties of pretorsion theories FFG, the universal property BCG of the stable category (in the sense of A. Facchini and C. Finocchiaro FF), and some motivating examples. We shall then focus our attention on some new examples of pretorsion theories in the category PreOrd (Grp) of preordered groups GM and Cat of small categories BCGT. This work is in collaboration with Alberto Facchini, Carmelo Finocchiaro, Francis Borceux, Federico Campanini, Aline Michel and Walter Tholen. References BCG F. Borceux, F. Campanini and M. Gran, The stable category of preorders in a pretopos II: the universal property, Annali Mat. Pura Appl. 201 (2022) 2847–2869. BCGT F. Borceux, F. Campanini, M. Gran and W. Tholen, Groupoids and skeletal categories form a pretorsion theory in Cat, preprint (2022) arXiv: 2207. 08487. FF A. Facchini and C. Finocchiaro, Pretorsion theories, stable category and preordered sets, Annali Mat. Pura Appl. 199 (2020), 1073–1089. FFG A. Facchini, C. Finocchiaro and M. Gran, Pretorsion theories in general categories, J. Pure Appl. Algebra 225 (2021) 106503. GM M. Gran and A. Michel, Torsion theories and coverings of preordered groups, Algebra Universalis 82, 22 (2021). X J. Xarez, A pretorsion theory for the category of all categories, Cah. Top. Géom. Diff. Catég. 53 (2022), 25-34.