ABSTRACT In this paper we compare, in a precise way, the concept of Grothendieck topos to the classical notion of topological space. The 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)QP(X). Here, we replace 0(X) by an arbitrary complete lattice satisfying the distributive law u A ( \/ u.) iel 1 = \/ (uAu.). Such a lattice is called a locale. The concept of sheaf iel x on a locale is clear, and gives rise to a corresponding topos. The category of (extended) spaces and continuous maps is the dual of the category of locales. We study this category systematically, developing particularly the concept of open mapping. Secondly, we show that the difference between an arbitrary Grothendieck topos and our new notion of space lies in the possibility of action by a spatial groupoid. That is, if G, + G~ is a groupoid in the category of (extended) spaces, then the general notion of Grothendieck topos is captured by considering sheaves on Gn with a continuous action by G,. This is an extension of Grothendieck's interpretation of classical Galois theory. The basic technique is descent theory for morphisms of locales, developed in the general set theory of an arbitrary elementary topos. vi
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