2-Filteredness and The Point of Every Galois Topos

Eduardo J. Dubuc
2008 Applied Categorical Structures  
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
more » ... 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 D A 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.
doi:10.1007/s10485-008-9145-4 fatcat:zec7lz544jad3ap32maudirehq