Every étendue comes from a local equivalence relation
A topos \({\mathcal E}\) is an étendue iff there is an object \(U\) in \({\mathcal E}\) covering the terminal object such that the topos \({\mathcal E}/U\) is a localic topos, in other words equivalent to a topos of sheaves on a locale. By an equivalence relation on a locale \(L\) one means a sublocale \(R\subseteq L\times L\) satisfying the expected properties of reflexivity, symmetry and transitivity, together with the requirement that the projections \(R\to L\) are both open maps of locales. The equivalence relations on \(L\) form a sheaf and a global section of this sheaf is referred to as a local equivalence relation. Given such a local equivalence relation \(r\), there is the notion of an \(r\)-sheaf and one obtains a category \(\text{sh}(L,r)\) of \(r\)-sheaves on \(L\). If \(r\) is a locally simply connected local equivalence relation, then \(\text{sh}(L,r)\) is an étendue and moreover there is the main result of the article, namely that every étendue \({\mathcal E}\) is equivalent to a topos of the form \(\text{sh}(L,r)\), where \(r\) is locally simply connected.
scientific article; zbMATH DE number 166141
scientific article