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Notions of flatness relative to a Grothendieck topology

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Panagis Karazeris

Completions of (small) categories under certain kinds
of colimits and exactness conditions have been studied extensively in
the literature. When the category that we complete is not left exact
but has some weaker kind of limit for finite diagrams, the universal
property of the completion is usually stated with respect to functors
that enjoy a property reminiscent of flatness. In this fashion notions
like that of a left covering or a multilimit merging functor have
appeared in the literature. We show here that such notions coincide
with flatness when the latter is interpreted relative to (the internal
logic of) a site structure associated to the target category. We
exploit this in order to show that the left Kan extensions of such
functors, along the inclusion of their domain into its completion, are
left exact. This gives in a very economical and uniform manner the
universal property of such completions. Our result relies heavily on
some unpublished work of A. Kock from 1989. We further apply this to
give a pretopos completion process for small categories having a weak
finite limit property.

Keywords:
flat functor, postulated colimit, geometric logic, exact completion, pretopos
completion, left exact Kan extension

2000 MSC:
18A35, 03G30, 18F10

*Theory and Applications of Categories,*
Vol. 12, 2004,
No. 5, pp 225-236.

http://www.tac.mta.ca/tac/volumes/12/5/12-05.dvi

http://www.tac.mta.ca/tac/volumes/12/5/12-05.ps

http://www.tac.mta.ca/tac/volumes/12/5/12-05.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/12/5/12-05.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/12/5/12-05.ps

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