INTRODUCTION

Attempting to define a "Weil cohomology" with the formal properties

necessary to establish the Weil conjectures, Grothendieck discovered e*tale

cohomology, a fusion of ordinary sheaf cohomology and Galois cohomology.

The definition required an extension of the concept of sheaf to the idea

of a sheaf on a site - a category equipped with an a priori notion of

covering. Contrary to the topological case, two non-isomorphic sites can

give rise to the same category of sheaves. One is thus led to consider

the category of sheaves as the natural object of study, rather than the

generating site. In this way the notion of a Grothendieck topos - the

concept abstracted from categories of sheaves on sites - was introduced.

A systematic study of sites, sheaves, and topoi was carried out by

M. Artin, A. Grothendieck, and J.L. Verdier in SGA4 (1963/64) [2]. The

theory that arose from this study has proved to be extremely useful in

algebraic geometry, contributing in an essential way to the complete proof

by Deligne of all the Weil conjectures. Of course as Deligne in SGAAh

points out, only a fraction of the general theory is strictly necessary to

establish these results - a remark true of any application of a general

theory to a special case.

The development of the theory of topos was guided by the analogy

between the category of sheaves on a site, and sheaves on an ordinary

topological space. Thus a topos was seen to be a kind of generalized

space. To quote from SGA4 [2],p. 301:

"Comme le terme de topos lui-meme est cense" precisement le suggerer,

il semble raisonnable et legitime aux auteurs du present slminaire

de considerer que l'objet de la Topologie est l'e'tude des topos

(et non des seuls espaces topologiques)."

In this paper we intend to deepen this analogy by establishing a theorem,

which compares, in a precise way, this new concept of space to the

classical notion of topological space. This comparison takes the form of

a two-fold extension of the idea of space.

Firstly, in classical topology a space is a set X equipped with a

topology of open sets 0(X)CS P(X). 0(X) is required to be closed under

arbitrary unions, and finite intersections. At the first level of

extension, we replace the lattice of open sets by an arbitrary complete

lattice satisfying the distributive law:

u A (Vu.) = V (uA u.).

iel x iel

Such a lattice is called a locale. The concept of sheaf on a locale is

clear, and gives rise to a corresponding topos. Morphisms of locales

preserve suprema and finite infima - like the inverse image of open sets

vn