t-structures are normal torsion theories

We characterize $t$-structures in stable $\infty$-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure $\mathfrak{t}$ on a stable $\infty$-category $\mathbf{C}$ is equivalent to a normal torsion theory $\mathbb{F}$ on $\mathbf{C}$, i.e. to a factorization system $\mathbb{F}=(\mathcal{E},\mathcal{M})$ where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.Comment: Minor typographical corrections from v1; 25 pages; to appear in "Applied Categorical Structures

Monotone-light factorisation systems and torsion theories

Given a torsion theory (Y,X) in an abelian category C, the reflector I from C to the torsion-free subcategory X induces a reflective factorisation system (E, M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e that (E, M) induces a monotone-light factorisation system (E',M*) by simultaneously stabilising E and localising M, whenever the torsion theory is hereditary and any object in C is a quotient of an object in X. We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules.Comment: 12 page

Spinning probes and helices in AdS3_3

We study extremal curves associated with a functional which is linear in the curve's torsion. The functional in question is known to capture the properties of entanglement entropy for two-dimensional conformal field theories with chiral anomalies and has potential applications in elucidating the equilibrium shape of elastic linear structures. We derive the equations that determine the shape of its extremal curves in general ambient spaces in terms of geometric quantities. We show that the solutions to these shape equations correspond to a three-dimensional version of Mathisson's helical motions for the centers of mass of spinning probes. Thereafter, we focus on the case of maximally symmetric spaces, where solutions correspond to cylindrical helices and find that the Lancret ratio of these equals the relative speed between the Mathisson-Pirani and the Tulczyjew-Dixon observers. Finally, we construct all possible helical motions in three-dimensional manifolds with constant negative curvature. In particular, we discover a rich space of helices in AdS$_3$ which we explore in detail.Comment: 28 pages, 5 figure

Semi-localizations of semi-abelian categories

A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the pointed context by Z. Janelidze. This result is useful to obtain an abstract characterization of the torsion-free and of the hereditarily-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend similar ones obtained by W. Rump in the abelian context.Comment: 30 pages. v2: introduction and references update