#
Internal monotone-light factorization for categories via preorders

##
Joao Xarez

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**.

Keywords:
(reflexive) graph, (reflexive) relation, category,
preorder, factorization system, localization, stabilization,
descent theory, Galois theory, monotone-light factorization

2000 MSC:
18A32, 12F10

*Theory and Applications of Categories,*
Vol. 13, 2004,
No. 15, pp 235-251.

http://www.tac.mta.ca/tac/volumes/13/15/13-15.dvi

http://www.tac.mta.ca/tac/volumes/13/15/13-15.ps

http://www.tac.mta.ca/tac/volumes/13/15/13-15.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/13/15/13-15.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/13/15/13-15.ps

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