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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
@article{TAC_2007_18_a7,
author = {Marta Bunge and Jonathon Funk},
title = {Quasi locally connected toposes},
journal = {Theory and applications of categories},
pages = {209--239},
publisher = {mathdoc},
volume = {18},
year = {2007},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2007_18_a7/}
}
Marta Bunge; Jonathon Funk. Quasi locally connected toposes. Theory and applications of categories, Tome 18 (2007), pp. 209-239. http://geodesic.mathdoc.fr/item/TAC_2007_18_a7/