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 $\mathcal{S}$.
For $X$ an object in $\mathcal{E}$, let $Aut_{\mathcal{E}}(X)$ be its automorphism group (in $\mathcal{S}$). Then $\Delta Aut(X)$ is canonically a group object in $\mathcal{E}$.
An inhabited object $X$ (the terminal morphism $X \to *$ is an epimorphism) in $\mathcal{E}$ is called a Galois object if it is a $\Delta Aut(X)$-torsor/principal bundle in $\mathcal{E}$, in that the canonical morphism
is an isomorphism.
Any Galois object is a locally constant object: since $X \to *$ is epi we may take it as a cover $U = X \to *$ and then then above principality condition says that pulled back to this cover $X$ becomes constant.
Often a Galois topos is in addition required to be pointed.
For $\mathcal{E}$ 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)