The notion of Galois topos formalizes the collection of locally constant sheaves that are classified by Galois theory in the connected and locally connected case.
Let
be a topos sitting by its global section geometric morphism over a base .
For an object in , let be its automorphism group (in ). Then is canonically a group object in .
An inhabited object (the terminal morphism is an epimorphism) in is called a Galois object if it is a -torsor/principal bundle in , in that the canonical morphism
is an isomorphism.
Any Galois object is locally constant object: since is epi we may take it as a cover and then then above principality condition says that pulled back to this cover becomes constant.
Often a Galois topos is in addition required to be pointed.
For connected and locally connected, the full subcategory generated by locally constant objects is a Galois topos.
This appears as (Dubuc, theorem 5.2.4).
The definition appears in
Alexander Grothendieck, SGA 1 (1960-61), Springer Lecture Notes in Mathematics 224 (1971).
Ieke Moerdijk, Prodiscrete groups and Galois toposes Proc. Kon. Nederl. Akad. van Wetens. Series A, 92-2 (1989)
Eduardo Dubuc, On the representation theory of Galois and Atomic Topoi (arXiv:0208222)