In this paper, we discuss various “general nonsense” aspects of the geometry of semi-graphs of profinite groups [cf. [Mzk3], Appendix], by applying the language of anabelioids introduced in [Mzk4]. After proving certain basic properties concerning various commensurators associated to a semi-graph of anabelioids, we show that the geometry of a semi-graph of anabelioids may be recovered from the category-theoretic structure of certain naturally associated categories — e.g., “temperoids” [in essence, the analogue of a Galois category for the “tempered fundamental groups” of [André]] and “categories of localizations”. Finally, we apply these techniques to obtain certain results in the absolute anabelian geometry [cf. [Mzk3], [Mzk8]] of tempered fundamental groups associated to hyperbolic curves over p-adic local fields.

Abstract. 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