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
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