Toposes pour les nuls

The \generalized universe of sets" aspect of toposes is relatively easy to understand and is well documented in the literature: start with Goldblatt [1] and proceed via Mac Lane and Moerdijk [5], or MacLarty [4], to Johnstone [2]. The basic trick is to use categorical properties to characterize set-theoretic constructions in the category of sets, and thence to transfer them to other categories that are su ciently similar. The generalized spaces, on the other hand, though present in ideas of toposes right from their introduction by Grothendieck, are somewhat mysterious. Much of this is because the generalized universes of sets are not direct expressions of the spatial idea but represent it by a mathematical duality. My aim here is at least to present a clear picture of how intuitions of generalized spaces t into a mathematical framework of generalized categories of sets. To try to be clear, I shall use the word topos only for the view as generalized space. When I consider it as a generalized universe of sets, I shall call it a G-frame (standing for Giraud/Grothendieckframe). By the duality which I shall explain, the G-frame is used to represent the topos.