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
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