nLab
Galois topos

Context

Topos Theory

Could not include topos theory - contents

Contents

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

(ΔΓ):ΓΔ𝒮 (\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 𝒮\mathcal{S}.

Definition

For XX an object in \mathcal{E}, let Aut (X)Aut_{\mathcal{E}}(X) be its automorphism group (in 𝒮\mathcal{S}). Then ΔAut(X)\Delta Aut(X) is canonically a group object in \mathcal{E}.

An inhabited object XX (the terminal morphism X*X \to * is an epimorphism) in \mathcal{E} is called a Galois object if it is a ΔAut(X)\Delta Aut(X)-torsor/principal bundle in \mathcal{E}, in that the canonical morphism

(Id,ρ):X×ΔAut(X)X×X (Id,\rho) : X \times \Delta Aut(X) \stackrel{}{\to} X \times X

is an isomorphism.

Remark

Any Galois object is locally constant object: since X*X \to * is epi we may take it as a cover U=X*U = X \to * and then then above principality condition says that pulled back to this cover XX becomes constant.

Definition

A Galois topos is a topos that is

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

Revised on March 18, 2013 03:12:05 by Bas Spitters (192.16.204.218)