Reflective subcategories, localizations and factorizationa systems

J. Knot Theory Ramifications

2014

Abstract. We study and compare two factorisation systems for surjective homomorphisms in the category of quandles. The first one is induced by the adjunction between quandles and trivial quandles, and a precise description of the two classes of morphisms of this factorisation system is given. In doing this we observe that a special class of congruences in the category of quandles always permute in the sense of the composition of relations, a fact that opens the way to some new universal algebraic investigations in the category of quandles. The second factorisation system is the one discovered by E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter. We conclude with an example showing a difference between these factorisation systems.

Section: Proof Consider the Following Commutative Diagram Wherementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On factorization systems for surjective quandle homomorphisms

Even

J. Knot Theory Ramifications

2014

“…Then, we shall show that they correspond, on the one hand, to homological closure operators satisfying the additional property of weak heredity, and, on the other hand, to a special kind of epireflections, called fibered epireflections [6] (also called semi-left-exact reflections in [5,17]). …”

Section: Torsion Theoriesmentioning

confidence: 99%

“…We show in Section 4 that the classical axioms defining a torsion theory keep their full meaning in any homological category, and we then prove several basic properties: for instance, in any homological category, a torsion-free subcategory always determines its torsion subcategory in a unique way. We then prove that a torsion theory is the same thing as a fibered epireflection (Theorem 4.12), i.e., an epireflection such that the left adjoint is a fibration [6,17]. A torsion theory corresponds to a special kind of closure operator on kernels, the weakly hereditary closure operators.…”

Section: Introductionmentioning

confidence: 99%

Torsion theories in homological categories

Bourn

2006

Journal of Algebra

We introduce and study the notion of torsion theory in the non-abelian context of homological categories, and we investigate the properties of the corresponding closure operator. We then consider several new examples of torsion theories in the category of topological groups and, more generally, in any category of topological semi-abelian algebras. We finally characterize the hereditary torsion theories, and we analyse a new example in the homological category of crossed modules.

“…Let us denote the class of morphisms that are locally in M by M * . "Dually", we have that the class M is always stable under pullback (even for an arbitrary (pre)factorisation system (E, M)) while for E this is generally false: it was proved in [9] that E is pullback-stable if, and only if, I preserves arbitrary pullbacks (i.e. if it is a localisation).…”

Section: Introductionmentioning

confidence: 99%

Monotone-light factorisation systems and torsion theories

Bulletin des Sciences Mathématiques

Everaert

2013

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.