2.2 Étalé space

Étalé spaces are a key concept in sheaf theory, providing a way to study local properties of spaces. They consist of a topological space and a continuous map that's locally invertible, allowing for the construction of sheaves on a given space.

Étalé spaces generalize covering maps, offering more flexibility in studying local properties. They're used to create sheaves of sections and germs, construct étalé spaces from presheaves, and associate sheaves to presheaves. These applications make étalé spaces a powerful tool in sheaf theory.

Étalé spaces

• Étalé spaces are a fundamental concept in sheaf theory that allow for the study of local properties of a space
• Consist of a topological space $E$ along with a continuous map $p: E \to X$ to another topological space $X$ such that $p$ is a local homeomorphism
• Provide a way to construct sheaves on a topological space $X$ by considering the sections or germs of the étalé space over $X$

Étalé maps

Local homeomorphisms as étalé maps

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• A continuous map $p: E \to X$ is called an étalé map if for every point $e \in E$, there exists an open neighborhood $U$ of $e$ such that $p|_U: U \to p(U)$ is a homeomorphism onto an open subset of $X$
• Local homeomorphisms are the defining property of étalé maps and ensure that the map $p$ is locally invertible
• Examples of local homeomorphisms include the projection map from a product space onto one of its factors and the quotient map from a space to its quotient space

Étalé maps vs covering maps

• Étalé maps are a generalization of covering maps, which are continuous surjective maps $p: E \to X$ such that every point $x \in X$ has an open neighborhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets, each of which is mapped homeomorphically onto $U$ by $p$
• While every covering map is an étalé map, not every étalé map is a covering map (the map may not be surjective or the fibers may not be discrete)
• Étalé maps allow for the study of local properties of spaces without the global constraints imposed by covering maps

Sheaves from étalé spaces

Sheaf of sections

• Given an étalé space $p: E \to X$, the sheaf of sections $\mathcal{F}$ associates to each open set $U \subseteq X$ the set $\mathcal{F}(U)$ of continuous sections of $p$ over $U$, i.e., continuous maps $s: U \to E$ such that $p \circ s = \text{id}_U$
• The restriction maps of the sheaf of sections are given by restricting the domain of the sections
• The sheaf of sections captures the local behavior of the étalé space and is a fundamental example of a sheaf on $X$

Sheaf of germs

• The sheaf of germs $\mathcal{G}$ of an étalé space $p: E \to X$ associates to each point $x \in X$ the set (stalk) $\mathcal{G}_x$ of germs of local sections of $p$ at $x$, i.e., equivalence classes of sections defined on neighborhoods of $x$, where two sections are equivalent if they agree on some smaller neighborhood of $x$
• The sheaf of germs is a sheaf on $X$ that captures the local behavior of the étalé space near each point
• There is a natural map from the sheaf of sections to the sheaf of germs that sends a section to its germ at each point, and this map is an isomorphism of sheaves

Constructions using étalé spaces

Étalé space of a presheaf

• Given a presheaf $\mathcal{F}$ on a topological space $X$, its étalé space $E(\mathcal{F})$ is constructed as the disjoint union of the stalks $\mathcal{F}_x$ for all $x \in X$, equipped with a topology generated by the sets $\{(x, s_x) \mid s \in \mathcal{F}(U), x \in U\}$ for open sets $U \subseteq X$ and sections $s \in \mathcal{F}(U)$
• The projection map $p: E(\mathcal{F}) \to X$ sending $(x, s_x)$ to $x$ is an étalé map, and the sheaf of sections of this étalé space is canonically isomorphic to the sheafification of $\mathcal{F}$
• The étalé space construction provides a way to associate an étalé space to any presheaf and study its sheafification

Associated sheaf via étalé spaces

• Given a presheaf $\mathcal{F}$ on a topological space $X$, the associated sheaf $\mathcal{F}^+$ can be constructed using the étalé space $E(\mathcal{F})$
• The associated sheaf $\mathcal{F}^+$ is defined as the sheaf of sections of the étalé space $E(\mathcal{F})$, i.e., $\mathcal{F}^+(U) = \Gamma(U, E(\mathcal{F}))$ for open sets $U \subseteq X$
• This construction provides an alternative way to sheafify a presheaf and highlights the connection between presheaves, sheaves, and étalé spaces

Properties of étalé spaces

Separation properties

• Étalé spaces inherit separation properties from their base spaces: if $X$ is Hausdorff (respectively, regular, normal, paracompact), then so is any étalé space over $X$
• However, an étalé space may have stronger separation properties than its base space (an étalé space over a non-Hausdorff space can be Hausdorff)
• Separation properties of étalé spaces are important in the study of sheaf cohomology and the classification of sheaves

Compactness and paracompactness

• If $X$ is a compact (respectively, paracompact) topological space and $p: E \to X$ is an étalé map, then $E$ is also compact (respectively, paracompact)
• Compactness and paracompactness of étalé spaces are crucial in the study of sheaf cohomology, as they ensure the existence of partitions of unity and the vanishing of higher cohomology groups
• Many important results in sheaf theory, such as the de Rham theorem and the Čech-to-derived spectral sequence, rely on the compactness or paracompactness of the spaces involved

Morphisms of étalé spaces

Continuous maps of étalé spaces

• A morphism of étalé spaces from $p_1: E_1 \to X$ to $p_2: E_2 \to X$ over the same base space $X$ is a continuous map $f: E_1 \to E_2$ such that $p_2 \circ f = p_1$
• Morphisms of étalé spaces capture the notion of continuous maps between the total spaces that are compatible with the étalé maps
• The composition of two morphisms of étalé spaces is again a morphism, and the identity map is a morphism, making étalé spaces over $X$ into a category

Sheaf morphisms from étalé space maps

• A morphism of étalé spaces $f: E_1 \to E_2$ over $X$ induces a morphism of sheaves $f^*: \mathcal{F}_2 \to \mathcal{F}_1$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are the sheaves of sections of $E_1$ and $E_2$, respectively
• The sheaf morphism $f^*$ is defined by composition with $f$: for an open set $U \subseteq X$ and a section $s \in \mathcal{F}_2(U)$, $f^*(s) = f \circ s$
• This correspondence between morphisms of étalé spaces and morphisms of sheaves is functorial and provides a way to study sheaf morphisms using étalé spaces

Applications of étalé spaces

Classifying sheaves

• Étalé spaces can be used to classify sheaves on a topological space $X$ up to isomorphism
• Every sheaf $\mathcal{F}$ on $X$ is isomorphic to the sheaf of sections of an étalé space over $X$, namely its étalé space $E(\mathcal{F})$
• Two sheaves are isomorphic if and only if their associated étalé spaces are isomorphic as étalé spaces over $X$
• This classification result provides a geometric perspective on sheaves and allows for the study of sheaf theory using topological methods

Sheaf cohomology via étalé spaces

• Étalé spaces provide a geometric approach to the study of sheaf cohomology
• Given a sheaf $\mathcal{F}$ on a topological space $X$, the sheaf cohomology groups $H^i(X, \mathcal{F})$ can be computed using the Čech cohomology of an open cover of $X$ with coefficients in the sheaf of sections of the étalé space $E(\mathcal{F})$
• This approach to sheaf cohomology is particularly useful when the space $X$ is paracompact, as it allows for the use of partitions of unity and the computation of cohomology using acyclic covers
• The étalé space perspective also provides a way to relate sheaf cohomology to other geometric invariants, such as de Rham cohomology and singular cohomology, via the de Rham theorem and the Čech-to-derived spectral sequence