nLab Galois topos

Idea

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.

Definition

Let

$$(\Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \mathcal{S}$$

be a topos sitting by its global section geometric morphism over a base $\mathcal{S}$.

Definition

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

$$(Id,\rho) : X \times \Delta Aut(X) \stackrel{}{\to} X \times X$$

is an isomorphism.

Remark

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.

Definition

A Galois topos is a topos that is

• locally connected;

• connected;

• separated by its Galois objects.

Remark

Often a Galois topos is in addition required to be pointed.

Examples

Proposition

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).

References

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)