Monotone-light factorisation systems and torsion theories

Abstract

Given a torsion theory (Y, X) in an abelian category C, the reflector I: C → X 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. Paré 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.

Conditional Flatness, Fiberwise Localizations, and Admissible Reflections

“…is normal. This was called condition (N) in [15]. The fact that (N) is equivalent to condition (1) easily follows from Proposition 2.1 by choosing the quotient E/T(K) as E in diagram (4-2).…”

“…in Grp(Top), by taking the kernel t K : T(K) → K of the unit η K of any localization, one obtains a characteristic subgroup T(K) of K (see, for instance, Example 2.2 in [15]). Accordingly, the subgroup T(K) is also normal in E, and hence condition (N) in Remark 4.6 holds, as desired.…”

“…In any homological category, a property equivalent to properties and used in Proposition 4.5 consists of requiring that the monomorphism is normal. This was called condition in [15]. The fact that is equivalent to condition easily follows from Proposition 2.1 by choosing the quotient as in diagram (4-2).…”

“…Consider a localization functor for which each coaugmentation morphism of the localization is a normal epimorphism ( open surjective group homomorphism). Given any short exact sequence in , by taking the kernel of the unit of any localization, one obtains a characteristic subgroup of K (see, for instance, Example in [15]). Accordingly, the subgroup is also normal in E , and hence condition in Remark 4.6 holds, as desired.…”