10.1 Sheaves

Sheaves generalize functions on topological spaces, assigning algebraic structures to open sets. They bridge topology and algebra, allowing local data to be glued into global information. This concept is crucial in modern mathematics, particularly in algebraic geometry and complex analysis.

Sheaves differ from presheaves by satisfying locality and gluing conditions. These ensure consistent combination of local sections into global ones. Sheaf morphisms, stalks, and operations like sheafification provide tools for studying and manipulating these structures, forming the foundation for sheaf cohomology.

Definition of sheaves

• Sheaves are mathematical objects that generalize the concept of functions on a topological space, allowing for the assignment of algebraic structures (groups, rings, modules) to open sets in a way that is compatible with restrictions
• Sheaves provide a framework for studying local-to-global properties, capturing the idea that local data can be glued together to obtain global information
• Sheaves play a fundamental role in modern algebraic geometry, complex analysis, and other areas of mathematics, serving as a bridge between topology and algebra

Presheaves vs sheaves

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• A presheaf is a assignment of algebraic structures to open sets of a topological space, along with restriction maps between these structures, without requiring any compatibility conditions
• A sheaf is a presheaf that satisfies additional conditions, ensuring that local data can be glued together uniquely to obtain global sections
• The sheaf conditions (locality and gluing) distinguish sheaves from presheaves, making sheaves more suitable for capturing the local-to-global properties of a space

Sheaf conditions

• Locality condition: If two sections of a sheaf agree on the overlaps of their domains, then they must be restrictions of a unique section defined on the union of their domains
• Gluing condition: If sections are defined on an open cover of a set and agree on the overlaps, then there exists a unique section on the entire set that restricts to the given sections
• These conditions ensure that local data (sections) can be glued together consistently to obtain global information

Stalks of a sheaf

• The stalk of a sheaf at a point $x$ is the direct limit (colimit) of the sections over all open neighborhoods of $x$, capturing the local behavior of the sheaf near $x$
• Stalks provide a way to study the local properties of a sheaf, as they contain all the information about the sheaf in an arbitrarily small neighborhood of a point
• The stalks of a sheaf form a bundle over the underlying topological space, with the projection map sending each stalk to its corresponding point

Morphisms of sheaves

• Morphisms of sheaves are the natural transformations between sheaves, preserving the sheaf structure and commuting with the restriction maps
• A morphism of sheaves $f: \mathcal{F} \to \mathcal{G}$ consists of a collection of morphisms $f_U: \mathcal{F}(U) \to \mathcal{G}(U)$ for each open set $U$, compatible with restrictions
• Sheaf morphisms allow for the comparison and relation of different sheaves on the same topological space

Definition of morphisms

• A morphism of sheaves $f: \mathcal{F} \to \mathcal{G}$ is a collection of morphisms $f_U: \mathcal{F}(U) \to \mathcal{G}(U)$ for each open set $U$, such that for any open sets $U \subseteq V$, the following diagram commutes:

$$\begin{CD} \mathcal{F}(V) @>{f_V}>> \mathcal{G}(V) \\ @V{\rho_{V,U}}VV @VV{\rho_{V,U}}V \\ \mathcal{F}(U) @>>{f_U}> \mathcal{G}(U) \end{CD}$$

• In other words, the morphisms $f_U$ must be compatible with the restriction maps $\rho_{V,U}$ of the sheaves $\mathcal{F}$ and $\mathcal{G}$

Composition of morphisms

• Sheaf morphisms can be composed, making the category of sheaves on a topological space a well-defined category
• Given sheaf morphisms $f: \mathcal{F} \to \mathcal{G}$ and $g: \mathcal{G} \to \mathcal{H}$, their composition $g \circ f: \mathcal{F} \to \mathcal{H}$ is defined by $(g \circ f)_U = g_U \circ f_U$ for each open set $U$
• Composition of sheaf morphisms is associative and has identity morphisms, satisfying the axioms of a category

Isomorphisms of sheaves

• An isomorphism of sheaves is a sheaf morphism $f: \mathcal{F} \to \mathcal{G}$ that has an inverse morphism $g: \mathcal{G} \to \mathcal{F}$, such that $g \circ f = \text{id}_\mathcal{F}$ and $f \circ g = \text{id}_\mathcal{G}$
• Isomorphic sheaves have the same structure and properties, and can be considered equivalent in the category of sheaves
• Isomorphisms of sheaves preserve the local and global behavior of the sheaves, as well as their cohomological invariants

Sheafification

• Sheafification is the process of turning a presheaf into a sheaf by enforcing the sheaf conditions (locality and gluing) while preserving the essential structure of the presheaf
• The sheafification of a presheaf is the "closest" sheaf to the presheaf, in the sense that it satisfies a universal property
• Sheafification allows for the extension of constructions and results from sheaves to presheaves, making it a fundamental tool in sheaf theory

Presheaf to sheaf construction

• Given a presheaf $\mathcal{F}$, its sheafification $\mathcal{F}^+$ is constructed by first defining the stalks $\mathcal{F}^+_x$ as the colimit of the sections of $\mathcal{F}$ over open neighborhoods of $x$
• The sections of $\mathcal{F}^+$ over an open set $U$ are then defined as the continuous functions $s: U \to \bigsqcup_{x \in U} \mathcal{F}^+_x$ such that for each $x \in U$, $s(x)$ is an element of the stalk $\mathcal{F}^+_x$
• The restriction maps of $\mathcal{F}^+$ are induced by the restriction maps of $\mathcal{F}$ and the universal property of colimits

Universal property of sheafification

• The sheafification $\mathcal{F}^+$ of a presheaf $\mathcal{F}$ satisfies the following universal property: for any sheaf $\mathcal{G}$ and any presheaf morphism $\varphi: \mathcal{F} \to \mathcal{G}$, there exists a unique sheaf morphism $\varphi^+: \mathcal{F}^+ \to \mathcal{G}$ such that $\varphi = \varphi^+ \circ \iota$, where $\iota: \mathcal{F} \to \mathcal{F}^+$ is the natural presheaf morphism
• This universal property characterizes the sheafification as the "best approximation" of a presheaf by a sheaf, and ensures its uniqueness up to isomorphism

Examples of sheafification

• The sheafification of the presheaf of continuous functions on a topological space is the sheaf of continuous functions, as continuous functions already satisfy the sheaf conditions
• The sheafification of the presheaf of holomorphic functions on an open subset of $\mathbb{C}$ is the sheaf of holomorphic functions, as holomorphic functions also satisfy the sheaf conditions
• The sheafification of the constant presheaf with value in a group $G$ is the locally constant sheaf with stalks isomorphic to $G$, which assigns to each connected component of an open set a copy of $G$

Sheaves on topological spaces

• Sheaves on topological spaces provide a way to study the local and global properties of functions, sections, and other algebraic structures associated with the space
• The category of sheaves on a topological space $X$ is denoted by $\text{Sh}(X)$, and it includes several important examples and constructions
• Sheaves on topological spaces form the foundation for the application of sheaf theory in various branches of mathematics, such as algebraic geometry, complex analysis, and topology

Continuous functions as sheaves

• The assignment of the set of continuous real-valued functions to each open set of a topological space $X$ defines a sheaf $\mathcal{C}_X$ on $X$
• For an open set $U \subseteq X$, $\mathcal{C}_X(U)$ is the set of continuous functions $f: U \to \mathbb{R}$, with restriction maps given by the usual restriction of functions
• The sheaf $\mathcal{C}_X$ encodes the local and global properties of continuous functions on $X$, and is a fundamental example in sheaf theory

Locally constant sheaves

• A sheaf $\mathcal{F}$ on a topological space $X$ is called locally constant if for every point $x \in X$, there exists an open neighborhood $U$ of $x$ such that the restriction $\mathcal{F}|_U$ is isomorphic to a constant sheaf
• Locally constant sheaves are characterized by the property that their stalks are isomorphic to a fixed algebraic structure (group, ring, module) on each connected component of $X$
• Locally constant sheaves play a crucial role in the study of covering spaces and fundamental groups in algebraic topology

Sheaf of sections

• Given a continuous map $f: Y \to X$ between topological spaces, the assignment of the set of continuous sections of $f$ over each open set $U \subseteq X$ defines a sheaf on $X$, called the sheaf of sections of $f$
• For an open set $U \subseteq X$, the sections of $f$ over $U$ are the continuous maps $s: U \to Y$ such that $f \circ s = \text{id}_U$
• The sheaf of sections captures the local and global behavior of the map $f$, and is a key concept in the study of fiber bundles and vector bundles in topology and geometry

Operations on sheaves

• Several operations can be performed on sheaves, allowing for the construction of new sheaves from existing ones
• These operations, such as direct sums, tensor products, pullbacks, and pushforwards, provide a rich algebraic structure on the category of sheaves
• Operations on sheaves are essential tools in the study of sheaf cohomology and the application of sheaves in various fields of mathematics

Direct sums of sheaves

• Given sheaves $\mathcal{F}$ and $\mathcal{G}$ on a topological space $X$, their direct sum $\mathcal{F} \oplus \mathcal{G}$ is the sheaf defined by $(\mathcal{F} \oplus \mathcal{G})(U) = \mathcal{F}(U) \oplus \mathcal{G}(U)$ for each open set $U \subseteq X$, with restriction maps given by the direct sum of the restriction maps of $\mathcal{F}$ and $\mathcal{G}$
• The direct sum of sheaves is the coproduct in the category of sheaves, and it satisfies a universal property similar to that of the direct sum of modules or vector spaces
• Direct sums of sheaves are used in the construction of flasque resolutions and the computation of sheaf cohomology

Tensor products of sheaves

• Given sheaves $\mathcal{F}$ and $\mathcal{G}$ of modules over a sheaf of rings $\mathcal{R}$ on a topological space $X$, their tensor product $\mathcal{F} \otimes_\mathcal{R} \mathcal{G}$ is the sheaf defined by $(\mathcal{F} \otimes_\mathcal{R} \mathcal{G})(U) = \mathcal{F}(U) \otimes_{\mathcal{R}(U)} \mathcal{G}(U)$ for each open set $U \subseteq X$, with restriction maps induced by the tensor product of the restriction maps of $\mathcal{F}$ and $\mathcal{G}$
• The tensor product of sheaves satisfies a universal property similar to that of the tensor product of modules, and it provides a way to combine sheaves of modules over a common sheaf of rings
• Tensor products of sheaves are used in the study of sheaf cohomology and the construction of derived functors in the category of sheaves

Pullbacks and pushforwards

• Given a continuous map $f: X \to Y$ between topological spaces and a sheaf $\mathcal{F}$ on $Y$, the pullback of $\mathcal{F}$ along $f$ is the sheaf $f^*\mathcal{F}$ on $X$ defined by $(f^*\mathcal{F})(U) = \mathcal{F}(f(U))$ for each open set $U \subseteq X$, with restriction maps induced by the restriction maps of $\mathcal{F}$
• The pullback operation $f^*$ is a functor from the category of sheaves on $Y$ to the category of sheaves on $X$, and it preserves sheaf operations such as direct sums and tensor products
• Given a sheaf $\mathcal{G}$ on $X$, the pushforward of $\mathcal{G}$ along $f$ is the sheaf $f_*\mathcal{G}$ on $Y$ defined by $(f_*\mathcal{G})(V) = \mathcal{G}(f^{-1}(V))$ for each open set $V \subseteq Y$, with restriction maps induced by the restriction maps of $\mathcal{G}$
• The pushforward operation $f_*$ is a functor from the category of sheaves on $X$ to the category of sheaves on $Y$, and it is the right adjoint of the pullback functor $f^*$
• Pullbacks and pushforwards are essential tools in the study of sheaf cohomology and the construction of derived functors, as they allow for the transfer of sheaves between different topological spaces

Sheaf cohomology

• Sheaf cohomology is a powerful tool for studying the global properties of sheaves on a topological space, extending the ideas of cohomology from algebraic topology to the setting of sheaves
• There are several approaches to defining sheaf cohomology, including Čech cohomology and the derived functors approach, each with its own advantages and applications
• Sheaf cohomology plays a fundamental role in algebraic geometry, complex analysis, and other areas of mathematics, providing invariants and obstructions for various geometric and analytic problems

Čech cohomology of sheaves

• Čech cohomology is a combinatorial approach to sheaf cohomology, based on the construction of a cochain complex from an open cover of the topological space
• Given a sheaf $\mathcal{F}$ on a topological space $X$ and an open cover $\mathcal{U} = \{U_i\}_{i \in I}$ of $X$, the Čech cochain complex $C^\bullet(\mathcal{U}, \mathcal{F})$ is defined by $C^p(\mathcal{U}, \mathcal{F}) = \prod_{i_0, \ldots, i_p \in I} \mathcal{F}(U_{i_0} \cap \ldots \cap U_{i_p})$, with coboundary maps given by alternating sums of restriction maps
• The Čech cohomology groups $\check{H}^p(\mathcal{U}, \mathcal{F})$ are the cohomology groups of the Čech cochain complex, and the Čech cohomology groups of $\mathcal{F}$ are defined as the direct limit of $\check{H}^p(\mathcal{U}, \mathcal{F})$ over all open covers $\mathcal{U}$ of $X$
• Čech cohomology is particularly useful for computing the cohomology of sheaves on paracompact Hausdorff spaces, as it agrees with other cohomology theories in this setting

Derived functors approach

• The derived functors approach to sheaf cohomology is based on the idea of deriving the global sections functor, which is left exact but not right exact
• Given a sheaf $\mathcal{F}$ on a topological space $X$, the derived functors of the global sections functor $\Gamma(X, -)$ are defined by taking an injective resolution $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \