nLab sheaf on a topological space

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 “$Sh(Op(X))$”:

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

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:

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

(MacLaneMoerdijk, theorem IX.3 1) or as (Johnstone, lemma C.1.2.2)

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:

• The Bohr topos of a C*-algebra

• The category of hypergraphs

• The Sierpinski topos

Related concepts

• étendue

• localic topos, spatial locale

• abelian sheaf, sheaf of abelian groups

• sheaves on a simplicial topological space

References

• Peter Johnstone, part C (volume 2) Sketches of an elephant: a topos theory compendium.

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic