Separable Morphisms of Simplicial Sets

We show that the class of separable morphisms in the sense of G. Janelidze and W. Tholen in the case of Galois structure of second order coverings of simplicial sets due to R. Brown and G. Janelidze coincides with the class of covering maps of simplicial sets. Introduction Separable morphisms were introduced in [3] by A. Carboni and G. Janelidze for lextensive categories. In the way of [3] one can consider separable morphism in a lextensive category Fam(A), the category of families of objects in a category A. What we call Γ1 below can be seen as a special case of this situation, a characterization of separable morphisms for which is given by Theorem 2.1. G. Janelidze and W. Tholen [8] defined separable morphisms in a category C for a given pointed endofunctor of C. Given an adjunction I,H : X C one can consider separable morphisms with respect to the induced monad. Then, in a special case, an appropriate adjunction I,H : Sets Fam(A) between Fam(A) and the category of sets gives the same notion of separable morphism in a lextensive category Fam(A) as [3]. Definition 1.1 in this paper is essentially of [8], the difference being that in place of an adjunction we consider a Galois structure, which together with a pair of adjoint functors I,H : X C consists of specified classes of morphisms F and F ′, called fibrations, in C and X respectively (see G. Janelidze [7], the earlier reference is G. Janelidze [6]), and we require separable morphisms to be fibrations. Our purpose is to describe the class of separable morphisms for the Galois structure introduced by R. Brown and G. Janelidze in [2] (Γ2 below). Theorem 2.4 states that for this Galois structure separable morphisms are exactly the Kan fibrations which are covering maps of simplicity sets. 1. Separable morphisms In this section C is a finitely complete category. Let X be a full reflective subcategory of C with the inclusion H : X → C. Suppose the reflection I : C → X and its unit η : 1→ HI are chosen in a such way that the counit is an identity IH = 1. Received May 10, 2006, revised June 17, 2006; published on July 10, 2006. 2000 Mathematics Subject Classification: 55U10, 18A20, 18A40, 18G55, 57M10.