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.

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J.F. Jardine <>