nLab
local site

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Definition

Definition

A site is called a local site if

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

Properties

Proposition

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

Proof

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

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

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

See (Johnstone example C.3.6.3 (d)).

Examples

and

References

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

[!redirects local sites]

Revised on January 1, 2012 01:33:15 by Dmitri Pavlov (195.131.219.66)