Contents

topos theory

# Contents

## Idea

In general, the concept of sheaf is relative to a choice of site, hence to a choice of small category $\mathcal{C}$ equipped with a coverage. Given this, the category of sheaves on $\mathcal{C}$ is the full subcategory of the category of presheaves $[\mathcal{C}^{op}, Set]$ on those presheaves which satisfy the sheaf condition:

$Sh(\mathcal{C}) \overset{\phantom{AAA}}{\hookrightarrow} [\mathcal{C}^{op}, Set] \,.$

Often sheaves are introduced and discussed in the more restrictive sense of sheaves on a topological space. This is indeed a special case: A sheaf on a topological space $X$ is a sheaf on the site $Op(X)$ whose underlying category is the category of open subsets of $X$, and whose coverage consists of the open covers of topological spaces. It is usual to write $Sh(X)$ as shorthand for the resulting category of sheaves, but the more systematic name is “$Sh(Op(X))$”:

$Sh(X) \;\coloneqq\; Sh(Op(X)) \overset{\phantom{AAA}}{\hookrightarrow} [Op(X)^{op}, X] \,.$

A topos which is equivalent to a category of sheaves on a topological spaces, this way, is called a spatial topos.

## Properties

### Localic reflection

Similarly, if $X$ is a locale, there is its frame of opens $Op(X)$ and one obtains the sheaf topos $Sh(X) \coloneqq Sh(Op(X))$ as above. A topos in the essential image of this construction is called a localic topos. A sober topological space is canonically identified as a locale, and hence the category of sheaves over a sober topological space is a localic topos.

Restricted to sober topological spaces, the construction of forming categories of sheaves is a fully faithful functor to the category Topos of toposes with geometric morphisms between them:

$Sh(-) \;\colon\; SoberTopologicalSpace \overset{ \phantom{AA} }{\hookrightarrow} Topos \,.$

For more see at locale the section Localic reflection.

## Examples

Examples of spatial toposes which are not manifestly spatial, by usual their definition, include the following:

## References

Last revised on July 6, 2018 at 07:39:23. See the history of this page for a list of all contributions to it.