Separable closure in categories
\textit{M. Barr} [J. Pure Appl. Algebra 19, 21-42 (1980; Zbl 0441.18007)] defined axioms for a category \({\mathcal A}\) which imply that \({\mathcal A}\) is equivalent to the dual of the category of transitive discrete G-sets for a profinite group G. He did so by defining \(\Pi{\mathcal A}\), the category whose objects are formal finite products of objects of \({\mathcal A}\), and showing that (\(\Pi{\mathcal A})^{opp}\) is a ``Galois category'' in the sense of \textit{A. Grothendieck} [Revêtements étales et groupe fondamental (SGA 1) (Lect. Notes Math. 224) (1971; Zbl 0234.14002)] by defining a functor \(M: (\Pi{\mathcal A})^{opp}\to\) (finite sets) which preserves finite limits and colimits: for an object A of \(\Pi{\mathcal A}\), \(MA=Mor(A,B)\) for B a normal object admitting a monomorphism from each direct factor of A in \({\mathcal A}.\) However, the author observes here that there is a gap in the definition of M: the independence of MA on the normal object B is not natural, and thus for M to be well-defined on morphisms it is necessary to show that there is a coherent choice of morphisms among normal objects of \({\mathcal A}\). The purpose of this paper is to show that for a category satisfying Barr's axioms, such a coherent choice of morphisms exists and is unique up to an equivalence which implies that the functor M is unique up to isomorphism.
scientific article; zbMATH DE number 4019295