2-Filteredness and The Point of Every Galois Topos

A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bilimits of topoi, we show that every Galois topos has a point. introduction. Galois topoi (definition 1.5) arise in Grothendieck’s Galois theory of locally connected topoi. They are an special kind of atomic topoi. It is well known that atomic topoi may be pointless [6], however, in this paper we show that any Galois topos has points. We show how the full subcategory of Galois objects (definition 1.2) in any connected locally connected topos E has an structure of 2-filtered 2-category (in the sense of [3]). Then we show that the assignment, to each Galois object A, of the category DA of connected locally constant objects trivialized by A (definition 3.1), determines a 2-functor into the category of categories. Furthermore, this 2-system becomes a pointed 2-system of pointed sites (considering the topology in which each single arrow is a cover). By the results on 2-filtered bi-limits of topoi [4], it follows that, if E is a Galois topos, then it is the bi-limit of this system, and thus, it has a point. context. Throughout this paper S = Sets denotes the topos of sets. All topoi E are assumed to be Grothendieck topoi (over S), the structure map will be denoted by γ : E → S in all cases. 1. Galois topoi and the 2-filtered 2-category of Galois objects We recall now the definition of Galois object in a topos. The original definition of Galois object given in [5] was relative to a surjective point of the topos: 1.1. Definition. Let E γ −→ S be a topos furnished with a surjective point, that is, a geometric morphism S p −→ E whose inverse image functor reflects isomorphisms. Then, an object A is a Galois object if: i) There exists a ∈ pA such that the map Aut(A) a −→ pA, defined by a(h) = p(h)(a) is a bijection (the same holds then for any other b ∈ pA). ii) A is connected and A → 1 is epimorphic. Notice that in the context of the classical Galois theory (Artin’s interpretation) this definition coincides with the definition of normal extension. It is easy to check that in the presence of a (surjective) point the following unpointed definition is equivalent. 1