MARTA BUNGE
Date : January 23, 2007, 2:30 - 4 PM
Title: LOCALLY DISCRETE LOCALES

ABSTRACT


A locale W is said to be locally discrete if for every locale X, the
partial order of Loc(X,W) is discrete. The category LD of locally
discrete locales is a single universe for both the discrete (Loc_dis)
and the 0-dimensional (Loc_0) locales. This single universe is
suitable for discussing the existence of the support of a distribution
with values in locales. There is a notion of (LD)-fibration whose
geometrical significance is not known and may be worth
investigating. On the other hand, the (Loc_dis)-fibrations are the
complete spreads with a locally connected domain, and the
(Loc_0)-fibrations are the complete spreads with a quasi locally
connected domain (to be defined). If Y--->X  is an etale map with X a
0-dimensional locale, Y need not be 0-dimensional (counterexample
courtesy of P.T. Johnstone). This failure is repaired by the embedding
of Loc_0 into LD. 

(This is joint work with J. Funk and part of our paper "Quasi locally
connected toposes", recently submitted, preprint available at this url:
  http://www.math.mcgill.ca/bunge/quasi.ps
)