Separable closure in categories

Abstract

Grothendieck characterized those categories which are equivalent to a category of continuous G-sets for a profinite group G in [3]. This result has been significantly improved upon by Barr in [l] who gives 2 characterization of the dual of the category of transitive G-sets. Unfortunately there is a gap in Barr’s proof as the functor M which he constructs is, in general, not well defined on morphisms. In Barr’s paper he constructs t functor A4 from Cop to Sets by setting M(A ) = Morf A, N) for N a normal cover of A. If e is another normal object with Mor(A, 0) nonempty, then Mor(A, N) is in one-to-one correspondence with Mor(A, Q). However, this correspondence depends on the choice of a map from N to Q, thus for the action of M on morphisms to be well defined, it is necesssary to make a coherent choice of morphisms (the map from N to (2 composed with the chosen map from e to S must be the chosen map from N to S). That such an array of coherent choices is possibie follows from Thcorcm 3.1. This repairs the gap in Barr’s proof. One of the most familier examples of the categories we consider is the category of finite separable extensions of a field. In this case the resulting group is the Profinite Galois Group (or the Absolute Galois Group) of the field. Analogous Galois Groups can be obtained in this way for many other casts. In this paper we further construct the categorical ana%ogue of the separable closure as a pro-object. Note that the condition (QRC) is equivalent to (RMC) in [I ] given Normal Covers, but is somewhat easier to use. Throughout this paper we implicitly use The Axiom of Choice and the results of Section 4 of [l]. The author is grateful to the reviewer for providing an alternate proof of Theorem 3.1 and to D. Kenoyer for most of the proof to Claim 2 in Theorem 3.2.