#
Quasi locally connected toposes

##
Marta Bunge and Jonathon Funk

We have shown that complete spreads (with a locally connected domain) over
a bounded topos **E** (relative to **S**) are `comprehensive' in the
sense that they are precisely the second factor of a factorization
associated with an instance of the comprehension scheme involving
**S**-valued distributions on **E**. Lawvere has asked whether the
`Michael coverings' (or complete spreads with a definable dominance
domain) are comprehensive in a similar fashion. We give here a positive
answer to this question. In order to deal effectively with the
comprehension scheme in this context, we introduce a notion of an
`extensive topos doctrine,' where the extensive quantities (or
distributions) have values in a suitable subcategory of what we call
`locally discrete' locales. In the process we define what we mean by a
quasi locally connected topos, a notion that we feel may be of interest in
its own right.

Keywords:
complete spreads, distributions, zero-dimensional locales, comprehensive
factorization

2000 MSC:
18B25, 57M12, 18C15, 06E15

*Theory and Applications of Categories,*
Vol. 18, 2007,
No. 8, pp 209-239.

http://www.tac.mta.ca/tac/volumes/18/8/18-08.dvi

http://www.tac.mta.ca/tac/volumes/18/8/18-08.ps

http://www.tac.mta.ca/tac/volumes/18/8/18-08.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/8/18-08.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/8/18-08.ps

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