Contents

topos theory

# Contents

## Definition

###### Definition

A site is called a local site if

• it has a terminal object $*$;

• the only covering family of $*$ is the trivial cover.

This appears as (Johnstone example C.3.6.3 (d)).

## Properties

###### Proposition

The category of sheaves $Sh(C)$ on a local site $C$ is a local topos.

###### Proof

Since $C$ has a terminal object, the global section functor $Sh(C) \to Set$ is given by evaluation on that object, hence is precomposition of sheaves with the inclusion $* \to C$. At the level of presheaves this has a right Kan extension functor, given by sending a set $S$ to the presheaf

$\nabla S : U \mapsto S^{C(*,U)} \,.$

This is indeed a sheaf if $*$ is covered only by the trivial cover.

and

## References

The definition appears as example C.3.6.3 (d) in

[!redirects local sites]

Last revised on January 1, 2012 at 01:33:15. See the history of this page for a list of all contributions to it.