Decidable (= separable) objects and morphisms in lextensive categories
In a category \({\mathcal C}\) with finite limits and finite sums which are disjoint and universal [a lextensive category in the sense of \textit{A. Carboni}, \textit{S. Lack} and \textit{R. F. C. Walters}, J. Pure Appl. Algebra 84, No. 2, 145-158 (1993; Zbl 0784.18001)], the authors provide an extensive study of, in topos-theoretic terminology, decidable objects, i.e., of those objects \(D\) whose diagonal \(D\to D \times D\) is a coproduct injection, and of decidable morphisms, i.e., of decidable objects in the slices of the category \({\mathcal C}\). In the opposite of the category of commutative rings, \(f:A \to B\) is decidable iff \(A\) is a separable \(B\)-algebra. Like topoi [cf. \textit{M. Barr} and \textit{R. Diaconescu}, Cah. Topologie Géom. Différ. 22, 301-314 (1981; Zbl 0472.18002)], lextensive categories allow for the definition of finite covering, and the authors give a powerful characterization of those decidable morphisms with connected codomains which are finite coverings. They apply this result to two fundamental cases; for \({\mathcal C}\) the opposite of the category of commutative rings, it provides a simple proof of the fact that so-called strongly separable algebras are finite coverings in \({\mathcal C}\); for \({\mathcal C}\) the category \textit{étale}\((B)\) it gives the classical topological result that a local homeomorphism \(f:A \to B\) of connected Hausdorff spaces is a finite covering in the classical sense if \(B\) is path-connected, \(f\) has the path-lifting property, and at least one fibre of \(f\) is compact.
scientific article; zbMATH DE number 896573
scientific article