#
A Galois theory with stable units for simplicial sets

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Joao J. Xarez

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(F_n)*, *n >= 0*, consisting of all simplicial
sets whose unit morphisms are monic for the localization
$F_n:\mathbf{Set}^{\Delta^{op}}\rightarrow\mathbf{Set}^{\Delta^{op}_n}$,
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.

Keywords:
simplicial object, simplicial set, internal category,
internal preorder, regular category, Mal'cev category, descent
theory, Galois theory, reflection with stable units,
monotone-light factorization, Kan extension, elementary topos,
geometric morphism

2000 MSC:
18A32, 18A40, 18G30, 12F10, 55U10,
08B05, 18B25

*Theory and Applications of Categories,*
Vol. 15, CT2004,
No. 7, pp 178-193.

http://www.tac.mta.ca/tac/volumes/15/7/15-07.dvi

http://www.tac.mta.ca/tac/volumes/15/7/15-07.ps

http://www.tac.mta.ca/tac/volumes/15/7/15-07.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/15/7/15-07.dvi

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