On bornologies, locales and toposes of \(M\)-sets (Q1861453)

Let \(N\) be a nonempty set. Let \(B\) be the locale of all bornologies into \(N\). Let \(M\) be the monoid of all endomaps of \(N\). Let \(\Omega\) be the locale of all ideals of \(M\). The first main result of this paper establishes an isomorphism \(B\cong \Omega_j\) for a topology \(j\) on \(\Omega\). Generalizing this result, the second main result of the paper gives an equivalence between the category of Kolmogorov bornological spaces and bounded maps and the full subcategory of a subtopos \(\mathcal B\) of the topos \(\mathcal M\) of all \(M\)-sets that consists of all \(j\)-sheaves separated for the double negation topology of \(\mathcal B\).