On bornologies, locales and toposes of M-sets

Let M be the monoid of all endomaps of a non-empty set N, Ω the locale of all ideals of M, and let be the topos of all M-sets. The core of this paper is formed by a locale B, a subtopos and two theorems, where B is the locale of all bornologies defined on subsets of N and is the topos of j-sheaves for a topology j : Ω→Ω. The first theorem shows a morphism of locales B→Ω with nucleus j which induces an isomorphism of locales between B and the sublocaleΩj Ω. The second theorem, which generalizes the first one, gives an equivalence between the category of Kolmogorov bornological spaces and bounded maps, and the full subcategory formed by all j-sheaves which are separated for the double negation topology of .