Finite preorders and topological descent. II: étale descent
[For part I, see ibid. 175, 187-205 (2002; Zbl 1018.18004).] This paper provides a characterization of effective étale-descent morphisms of finite topological spaces, equivalently of finite preordered sets. More precisely, let \(p:E\to B\) be a continuous map of finite topological spaces, \({\mathcal E}(E)\) and \({\mathcal E}(B)\) the categories of finite étale bundles over \(E\) and \(B\) respectively, and \(p^*\): \({\mathcal E}(B)\to{\mathcal E} (E)\) the pullback functor along \(p\). It is proved that \(p\) is an effective étale-descent morphism, that is \(p\) is a monadic functor, if and only if the three following properties are fullfilled: (1) The map \(p:E\to p(E)\) is a regular quotient of topological spaces. (2) The universal category \({\mathcal Z} (Eq(p))\) associated to the double category associated to the equivalent relation \(Eq(p)\) on \(E\) determined by \(p\), is a preorder. (3) The map \(p\) is essentially surjective, i.e., up to specialization and generization.
scientific article; zbMATH DE number 1864811