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Model structures for homotopy of internal categories

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T. Everaert, R.W. Kieboom and T. Van der Linden

The aim of this paper is to describe Quillen model category structures
on the category CatC of internal categories and functors in a given
finitely complete category C. Several non-equivalent notions of
internal equivalence exist; to capture these notions, the model
structures are defined relative to a given Grothendieck topology on
C.

Under mild conditions on C, the regular epimorphism topology
determines a model structure where we is the class of weak
equivalences of internal categories (in the sense of Bunge and Pare).
For a Grothendieck topos C we get a structure that, though different
from Joyal and Tierney's, has an equivalent homotopy category. In case
C is semi-abelian, these weak equivalences turn out to be homology
isomorphisms, and the model structure on CatC induces a notion of
homotopy of internal crossed modules. In case C is the category
Gp of groups and homomorphisms, it reduces to the case of crossed
modules of groups.

The trivial topology on a category C determines a model structure on
CatC where we is the class of strong equivalences (homotopy
equivalences), fib the class of internal functors with the homotopy
lifting property, and cof the class of functors with the homotopy
extension property. As a special case, the ``folk'' Quillen model
category structure on the category Cat = CatSet of small categories
is recovered.

Keywords:
internal category, Quillen model category, homotopy, homology

2000 MSC:
Primary 18G55 18G50 18D35; Secondary 20J05 18G25 18G30

*Theory and Applications of Categories,*
Vol. 15, CT2004,
No. 3, pp 66-94.

http://www.tac.mta.ca/tac/volumes/15/3/15-03.dvi

http://www.tac.mta.ca/tac/volumes/15/3/15-03.ps

http://www.tac.mta.ca/tac/volumes/15/3/15-03.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/15/3/15-03.dvi

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