topos theory

# 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

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

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 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

###### 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)