The monotone-light factorization for n-categories via n-preorders

Starting with a symmetric monoidal adjunction with certain properties, one derives another symmetric monoidal adjunction with the same properties between the respective categories of all V-categories. If one begins with a reflection of a full replete subcategory, the derived adjunction is also a reflection of the same kind. Semi-left-exactness (also called admissibility in categorical Galois theory) or the stronger stable units property is inherited by the derived reflection. Applying these results, one concludes that the reflection of the category of all n-categories into the category of n-preorders has stable units. Then, it is also shown that this reflection determines a monotone-light factorization system on n-categories, n>=1, and that the light morphisms are precisely the n-functors faithful with respect to n-cells. In order to achieve such results, it was also shown that n-functors surjective both on vertically composable triples of horizontally composable pairs of n-cells, and on horizontally composable triples of vertically composable pairs of n-cells are effective descent morphisms in the category of all n-categories nCat, n>=1

A pretorsion theory for the category of all categories

A pretorsion theory for the category of all categories is presented. The associated prekernels and precokernels are calculated for every functor.Une théorie de prétorsion pour la catégorie de toutes les catégories est présentée. Les prénoyaux et préconoyaux associés sont calculés pour chaque foncteur.publishe

The monotone-light factorization for 2-categories via 2-preorders

It is shown that the reflection 2Cat --> 2PreOrd of the category of all 2-categories into the category of 2-preorders determines a monotone-light factorization system on 2Cat and that the light morphisms are precisely the 2-functors faithful on 2-cells with respect to the vertical structure. In order to achieve such result it was also proved that the reflection 2Cat --> 2Preord has stable units, a stronger condition than admissibility in categorical Galois theory, and that the 2-functors surjective both on horizontally and on vertically composable triples of 2-cells are the effective descent morphisms in 2Cat.publishe

The monotone-light factorization for 2-categories via 2-prorders

It is shown that the reflection 2Cat --> 2Preord of the category of all 2-categories into the category of 2-preorders determines a monotone-light factorization system on 2Cat and that the light morphisms are precisely the 2-functors faithful on 2-cells with respect to the vertical structure. In order to achieve such result it was also proved that the reflection 2Cat --> 2Preord has stable units, a stronger condition than admissibility in categorical Galois theory, and that the 2-functors surjective both on horizontally and on vertically composable triples of 2-cells are the effective descent morphisms in 2Cat

Product preservation and stable units for reflections into idempotent subvarieties

We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the above-mentioned condition holds.publishe

A GALOIS THEORY WITH STABLE UNITS FOR SIMPLICIAL SETS

Abstract. We recall and reformulate certain known constructions, in order to make a convenient setting for obtaining generalized monotone-light factorizations in the sense of A. Carboni, G. Janelidze, G. M. Kelly and R. Paré. This setting is used to study the existence of monotone-light factorizations both in categories of simplicial objects and in categories of internal categories. It is shown that there is a non-trivial monotone-light factorization for simplicial sets, such that the monotone-light factorization for reflexive graphs via reflexive relations is a special case of it, obtained by truncation. More generally, we will show that there exists a monotone-light factorization associated with every full subcategory Mono(Fn), n ≥ 0, consisting of all simplicial sets whose unit morphisms are monic for the localization Fn: Set ∆op → Set ∆op n, which truncates each simplicial set after the object of n-simplices. The monotone-light factorization for categories via preorders is as well derived from the proposed setting. We also show that, for regular Mal’cev categories, the reflection of internal groupoids into internal equivalence relations necessarily produces monotone-light factorizations. It turns out that all these reflections do have stable units, in the sense of C. Cassidy, M. Hébert and G. M. Kelly, giving rise to Galois theories

Well-behaved Epireflections for Kan Extensions

Let K : B -> A be a functor such that the image of the objects in B is a cogenerating set of objects for A. Then, the right Kan extensions adjunction Set(K) (sic) Ran(K) induces necessarily an epireflection with stable units and a monotone-light factorization. This result follows from the one stating that an adjunction produces an epireflection in a canonical way, provided there exists a prefactorization system which factorizes all of its unit morphisms through epimorphisms. The stable units property, for the so obtained epireflections, is thereafter equivalently restated in such a manner it is easily recognizable in the examples. Furthermore, having stable units, there is a strong but quite simple sufficient condition for the existence of an associated monotone-light factorization, which has proved to be fruitful