### Boolean localization, in practice, by J.F. Jardine

The Boolean localization theorem asserts that every Grothendieck topos
can be faithfully imbedded into a topos E that satisfies the axiom of
choice. This theorem is a combination of results of Diaconescu and
Barr, and has been known to have homotopy theoretic consequences since
Van Osdol's proof of the Illusie conjecture in the late 1970's, but it
has only appeared in the literature in fragmentary form until the
recent appearance of the Mac Lane-Moerdijk text. The purpose of this
(mainly expository) paper is to demonstrate how Boolean localization can be
used to show that the categories of simplicial presheaves and
simplicial sheaves on an arbitrary Grothendieck site have proper
closed simplicial model structures. These results have been known by
other means since about 1984; the proof given here for simplicial
sheaves roughly approximates that given by Joyal, but it does not
involve sheaves of homotopy groups.

This version of the paper replaces preprint no. 127 of the archive,
which has been removed. There was a relatively serious (and silly) error in
the proof of Lemma 10.

Addendum:
This paper has been published, see Doc. Math. 1
(1996), 245-275. It has been removed from these archives.

Requires the lams* fonts, available here.

J.F. Jardine <jardine@uwo.ca>