nLab
sheaf on a topological space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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

Sh(𝒞)AAA[𝒞 op,Set]. 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 XX is a sheaf on the site Op(X)Op(X) whose underlying category is the category of open subsets of XX, and whose coverage consists of the open covers of topological spaces. It is usual to write Sh(X)Sh(X) as shorthand for the resulting category of sheaves, but the more systematic name is “Sh(Op(X))Sh(Op(X))”:

Sh(X)Sh(Op(X))AAA[Op(X) 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 XX is a locale, there is its frame of opens Op(X)Op(X) and one obtains the sheaf topos Sh(X)Sh(Op(X))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():SoberTopologicalSpaceAATopos. 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:

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.