GALOIS THEORY IX theory of elementary topoi, which provided the set-theoretical aspect of topos theory, expressed in categorical terms. Thus, we take the point of view that an arbitrary, but fixed, (elementary) topos is our basic universe S of sets. We work in this universe as we would in naive set theory, except that, of course, we cannot use the axiom of choice, or the law of the excluded middle. This has been shown to be correct by various authors (e.g. Boileau-Joyal [7]). As an illustration of the effectiveness of this method, we might cite, for example, chapter VI, §3, where it is shown that if A is a locale in 3, then the category of A-modules in S is equivalent to the category of sup-lattices in the topos of sheaves on A, a result that cannot even be stated without adopting this position.

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