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.