Geodesic
Internal monotone-light factorization for categories via preorders
Theory and applications of categories, The Carboni Festschrift, Tome 13 (2004), pp. 235-251.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

It is shown that, for a finitely-complete category C with coequalizers of kernel pairs, if every product-regular epi is also stably-regular then there exist the reflections (R)Grphs(C) --> (R)Rel(C), from (reflexive) graphs into (reflexive) relations in C, and Cat(C) --> Preord(C), from categories into preorders in C. Furthermore, such a sufficient condition ensures as well that these reflections do have stable units. This last property is equivalent to the existence of a monotone-light factorization system, provided there are sufficiently many effective descent morphisms with domain in the respective full subcategory. In this way, we have internalized the monotone-light factorization for small categories via preordered sets, associated with the reflection Cat --> Preord, which is now just the special case C = Set.
Classification : 18A32, 12F10
Keywords: (reflexive) graph, (reflexive) relation, category, preorder, factorization system, localization, stabilization, descent theory, Galois theory, monotone-light factorization 
@article{TAC_2004_13_a14,
     author = {Joao Xarez},
     title = {Internal monotone-light factorization for categories via preorders},
     journal = {Theory and applications of categories},
     pages = {235--251},
     publisher = {mathdoc},
     volume = {13},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2004_13_a14/}
}
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Joao Xarez. Internal monotone-light factorization for categories via preorders. Theory and applications of categories, The Carboni Festschrift, Tome 13 (2004), pp. 235-251. http://geodesic.mathdoc.fr/item/TAC_2004_13_a14/