The monotone-light factorization for categories via orders and preorders
by
João J. Xarez
University of Aveiro, Portugal

Monotone-light factorization of morphisms in an abstract category C, with respect to a full reflective subcategory X, was studied by A. Carboni, G. Janelidze, G. M. Kelly, and R. Paré [1]. According to [1], the existence of such factorization requires strong additional conditions on the reflection C --> X, which however hold in several classical examples of categorical Galois theory including the (Galois theory of the) adjunction between compact Hausdorff and Stone spaces, needed to make the classical monotone-light factorization of S. Eilenberg [3] and G. T. Whyburn [4] a special case of the categorical one.

We show that monotone-light factorization also does exist for C = Cat, the category of all categories, and X being either the category Ord of (partially) ordered sets, or the category Preord of preorders. A crucial observation here is that the reflections Cat --> Ord and Cat --> Preord have stable units in the sense of C. Cassidy, M. Hébert, and G. M. Kelly [2]. This gives two different factorization systems on Cat, and it turns out that the light morphisms (=coverings) with respect to Preord are precisely faithful functors. Therefore Galois theories of categories via orders and preorders are much richer, i.e., have more covering morphisms, than the ``standard'' one (via sets, regarded as discrete categories).

We also give explicit descriptions of normal, separable, and other types of morphisms that occur in Galois theory of categories via orders and preorders.

References

[1] A. Carboni, G. Janelidze, G. M. Kelly, and R. Paré: On localization and stabilization for factorization systems, Applied Categorical Structures 5, 1997, 1-58

[2] C. Cassidy, M. Hébert, and G. M. Kelly: Reflective subcategories, localizations and factorization systems. J. Austral. Math. Soc. Ser. A 38, no. 3, 1985, 287-329

[3] S. Eilenberg: Sur les transformations continues d'espaces métriques compacts, Fundamenta Mathematicae 22, 1934, 292-296

[4] G. T. Whyburn, Non-alternating transformations, American Journal of Mathematics 56, 1934, 294-302

Date received: May 23, 2002