10.3 Sheaves in algebraic topology

Sheaves are mathematical tools that connect local and global properties of spaces. They attach algebraic data to open sets, allowing us to study how local information fits together on a larger scale.

In algebraic topology, sheaves bridge the gap between local and global perspectives. They're used to analyze cohomology theories, fiber bundles, and covering spaces, providing insights into the structure of topological spaces.

Definition of sheaves

• Sheaves are mathematical objects that allow for the study of local-to-global properties of spaces
• They provide a way to attach algebraic data (rings, modules, etc.) to open sets of a topological space
• Sheaves can be thought of as a generalization of the concept of a fiber bundle

Presheaves vs sheaves

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• A presheaf is a contravariant functor from the category of open sets of a topological space to a target category (sets, rings, modules, etc.)
• Sheaves satisfy additional "gluing" conditions that ensure the local data can be uniquely patched together
• The sheaf conditions guarantee that the local data is compatible and can be extended to larger open sets
• Example: The assignment of continuous functions to open sets of a topological space forms a sheaf (of rings)

Sheaf conditions

• The first sheaf condition states that if two sections agree on the intersection of their domains, then they must be restrictions of a unique section defined on the union of their domains
• The second sheaf condition requires that any compatible family of local sections can be glued together to form a global section
• These conditions ensure that the local data encoded by the sheaf can be consistently extended to larger open sets
• Example: The sheaf of differentiable functions on a manifold satisfies the sheaf conditions

Sheaves of abelian groups

• A sheaf of abelian groups assigns an abelian group to each open set of a topological space, with restriction maps between them
• The restriction maps are group homomorphisms and satisfy the sheaf conditions
• Sheaves of abelian groups form an abelian category, which allows for the use of homological algebra techniques
• Example: The sheaf of locally constant functions with values in a fixed abelian group

Sheaves of rings

• A sheaf of rings assigns a ring to each open set of a topological space, with restriction maps that are ring homomorphisms
• The restriction maps must satisfy the sheaf conditions
• Sheaves of rings can be used to study the local structure of schemes in algebraic geometry
• Example: The sheaf of regular functions on an affine algebraic variety

Sheaves of modules

• Given a sheaf of rings $\mathcal{R}$ on a topological space $X$, a sheaf of $\mathcal{R}$-modules assigns an $\mathcal{R}(U)$-module to each open set $U \subseteq X$
• The restriction maps are module homomorphisms and satisfy the sheaf conditions
• Sheaves of modules provide a framework for studying local properties of vector bundles and coherent sheaves in algebraic geometry
• Example: The sheaf of sections of a vector bundle over a manifold

Sheaves on topological spaces

• Sheaves can be defined on various types of topological spaces, each with their own specific properties and applications
• The structure of the underlying topological space often determines the behavior and characteristics of the sheaves defined on it
• Studying sheaves on different classes of topological spaces allows for a deeper understanding of the interplay between topology and algebra

Sheaves on Hausdorff spaces

• Hausdorff spaces are topological spaces in which distinct points have disjoint neighborhoods
• Sheaves on Hausdorff spaces have particularly nice properties, such as the existence of a unique maximal extension for any section defined on a closed set
• The Hausdorff condition ensures that the stalks of a sheaf (the germs at each point) are well-behaved
• Example: The sheaf of continuous functions on a Hausdorff topological space

Sheaves on paracompact spaces

• Paracompact spaces are Hausdorff spaces that admit partitions of unity subordinate to any open cover
• Sheaves on paracompact spaces have a rich theory, including the existence of fine resolutions and the validity of the de Rham theorem
• Paracompactness is a crucial property for the development of sheaf cohomology and its applications
• Example: The sheaf of smooth differential forms on a paracompact smooth manifold

Sheaves on locally compact spaces

• Locally compact spaces are topological spaces in which every point has a compact neighborhood
• Sheaves on locally compact spaces have applications in harmonic analysis and the study of compactly supported cohomology
• The local compactness property allows for the construction of a compactly supported version of sheaf cohomology
• Example: The sheaf of compactly supported continuous functions on a locally compact Hausdorff space

Restriction and extension of sheaves

• Given a sheaf $\mathcal{F}$ on a topological space $X$ and a subspace $Y \subseteq X$, the restriction of $\mathcal{F}$ to $Y$ is a sheaf $\mathcal{F}|_Y$ on $Y$
• The extension problem asks whether a sheaf defined on a subspace can be extended to a sheaf on the ambient space
• The extension problem is closely related to the notion of sheaf cohomology and the obstruction theory
• Example: Extending a sheaf from a closed subspace to the entire space using the extension by zero functor

Sheaf cohomology

• Sheaf cohomology is a powerful tool for studying global properties of sheaves on topological spaces
• It provides a way to measure the obstruction to the existence of global sections and the deviation from the local-to-global principle
• Sheaf cohomology has applications in various areas of mathematics, including algebraic topology, complex analysis, and algebraic geometry

Čech cohomology of sheaves

• Čech cohomology is a cohomology theory for sheaves based on open covers of the topological space
• Given an open cover $\mathcal{U}$ of a space $X$ and a sheaf $\mathcal{F}$, the Čech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$ is defined using the sections of $\mathcal{F}$ on finite intersections of open sets in $\mathcal{U}$
• The Čech cohomology groups $\check{H}^i(X, \mathcal{F})$ are the cohomology groups of the Čech complex
• Čech cohomology satisfies the axioms of a cohomology theory and is closely related to other cohomology theories, such as sheaf cohomology and singular cohomology

Derived functors approach

• Sheaf cohomology can also be defined using the language of derived functors in homological algebra
• The global sections functor $\Gamma(X, -)$ is a left exact functor from the category of sheaves on $X$ to the category of abelian groups
• The right derived functors of $\Gamma(X, -)$ define the sheaf cohomology groups $H^i(X, \mathcal{F})$
• The derived functors approach provides a systematic way to study the relation between sheaf cohomology and other cohomology theories

Comparison of cohomology theories

• There are various cohomology theories for topological spaces and sheaves, such as singular cohomology, de Rham cohomology, and Čech cohomology
• The comparison theorems establish isomorphisms between different cohomology theories under certain conditions
• For example, on a paracompact Hausdorff space, the Čech cohomology of a constant sheaf is isomorphic to the singular cohomology of the space with coefficients in the abelian group
• These comparison theorems highlight the deep connections between topology and algebra

Cohomology with supports

• Cohomology with supports is a variant of sheaf cohomology that takes into account a family of supports (typically closed subsets) on the topological space
• Given a family of supports $\Phi$ on a space $X$ and a sheaf $\mathcal{F}$, the cohomology groups with supports $H^i_\Phi(X, \mathcal{F})$ measure the cohomological information captured by the sections of $\mathcal{F}$ supported on the sets in $\Phi$
• Cohomology with supports is particularly useful in the study of local cohomology and the cohomology of locally compact spaces
• Example: The cohomology with compact supports of a sheaf on a locally compact Hausdorff space

Operations on sheaves

• Various operations can be performed on sheaves, allowing for the construction of new sheaves from existing ones
• These operations often have interesting geometric and algebraic interpretations and are essential tools in the study of sheaves and their applications
• The functorial properties of these operations provide a rich structure to the category of sheaves on a topological space

Direct and inverse image sheaves

• Given a continuous map $f: X \to Y$ between topological spaces and a sheaf $\mathcal{F}$ on $X$, the direct image sheaf $f_*\mathcal{F}$ on $Y$ is defined by $f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}(V))$ for open sets $V \subseteq Y$
• The inverse image sheaf $f^{-1}\mathcal{G}$ of a sheaf $\mathcal{G}$ on $Y$ is the sheafification of the presheaf $U \mapsto \varinjlim_{f(U) \subseteq V} \mathcal{G}(V)$ on $X$
• The direct and inverse image sheaves are related by adjunction: $\text{Hom}_{\mathcal{O}_Y}(f_*\mathcal{F}, \mathcal{G}) \cong \text{Hom}_{\mathcal{O}_X}(\mathcal{F}, f^{-1}\mathcal{G})$
• Example: The pushforward and pullback of sheaves in the context of morphisms of ringed spaces

Sheaf Hom and tensor product

• Given sheaves $\mathcal{F}$ and $\mathcal{G}$ of $\mathcal{O}_X$-modules on a ringed space $(X, \mathcal{O}_X)$, the sheaf Hom $\mathcal{H}om_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is the sheaf of local homomorphisms between $\mathcal{F}$ and $\mathcal{G}$
• The tensor product sheaf $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ is the sheafification of the presheaf $U \mapsto \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{G}(U)$
• The sheaf Hom and tensor product satisfy various functorial properties and are related by adjunction: $\text{Hom}_{\mathcal{O}_X}(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}, \mathcal{H}) \cong \text{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{H}om_{\mathcal{O}_X}(\mathcal{G}, \mathcal{H}))$
• Example: The sheaf of differential operators on a smooth manifold as a sheaf Hom

Sheafification of presheaves

• Given a presheaf $\mathcal{F}$ on a topological space $X$, the sheafification (or associated sheaf) $\mathcal{F}^+$ is the "closest" sheaf to $\mathcal{F}$ in a precise sense
• The sheafification is obtained by a universal property: it is the best approximation of $\mathcal{F}$ by a sheaf
• The stalks of the sheafification $\mathcal{F}^+$ are isomorphic to the stalks of the presheaf $\mathcal{F}$
• Example: The sheaf of continuous functions on a topological space as the sheafification of the presheaf of locally bounded functions

Pullback and pushforward of sheaves

• The pullback and pushforward of sheaves are functorial operations that allow for the transfer of sheaves between different spaces
• Given a continuous map $f: X \to Y$ and a sheaf $\mathcal{F}$ on $Y$, the pullback sheaf $f^*\mathcal{F}$ on $X$ is defined as the inverse image sheaf $f^{-1}\mathcal{F}$ tensored with the pullback of the structure sheaf
• The pushforward sheaf $f_*\mathcal{G}$ of a sheaf $\mathcal{G}$ on $X$ is simply the direct image sheaf
• Pullbacks and pushforwards are related by adjunction: $\text{Hom}_{\mathcal{O}_Y}(f_*\mathcal{G}, \mathcal{F}) \cong \text{Hom}_{\mathcal{O}_X}(\mathcal{G}, f^*\mathcal{F})$
• Example: The pullback of the sheaf of differential forms under a smooth map between manifolds

Applications in algebraic topology

• Sheaves have numerous applications in algebraic topology, providing a bridge between local and global properties of spaces
• They offer a powerful language to study cohomology theories, fiber bundles, and covering spaces
• Sheaf-theoretic methods have led to important results and insights in algebraic topology

Constant and locally constant sheaves

• A constant sheaf on a topological space $X$ with value in an abelian group $A$ is a sheaf that assigns the group $A$ to every open set and has identity maps as restriction maps
• A locally constant sheaf (or local system) on $X$ is a sheaf that is locally isomorphic to a constant sheaf
• Locally constant sheaves are closely related to representations of the fundamental group of the space
• Example: The orientation sheaf on a manifold, which is a locally constant sheaf with stalks isomorphic to $\mathbb{Z}$

Orientation sheaves

• An orientation sheaf on a manifold $M$ is a locally constant sheaf with stalks isomorphic to $\mathbb{Z}$ and with restriction maps given by the sign of the Jacobian determinant of the transition functions
• The orientation sheaf encodes information about the orientability of the manifold
• The cohomology of the orientation sheaf is related to the orientability and the fundamental class of the manifold
• Example: The orientation sheaf on a Möbius strip is a non-trivial locally constant sheaf

Sheaves and covering spaces

• Sheaves can be used to study covering spaces and their properties
• Given a covering space $p: E \to B$, the sheaf of sections of $p$ is a locally constant sheaf on the base space $B$
• The category of locally constant sheaves on $B$ is equivalent to the category of covering spaces over $B$
• This equivalence allows for the application of sheaf-theoretic methods to the study of covering spaces and their cohomology
• Example: The sheaf of sections of the universal cover of a topological space

Sheaves and fiber bundles

• Sheaves provide a natural language to study fiber bundles and their associated structures
• Given a fiber bundle $\pi: E \to B$ with fiber $F$, the sheaf of sections of $\pi$ is a sheaf on the base space $B$ that encodes the local trivializations of the bundle
• The cohomology of the sheaf of sections is related to the classification and obstruction theory of fiber bundles
• Sheaf-theoretic methods can be used to study vector bundles, principal bundles, and their characteristic classes
• Example: The sheaf of smooth sections of a vector bundle over a manifold

Sheaf theory and homological algebra

• Sheaf theory and homological algebra are closely intertwined, with sheaves providing a geometric context for homological constructions
• Homological algebra techniques, such as resolutions and derived functors, play a central role in the study of sheaves and their cohomology
• The interplay between sheaf theory and homological algebra has led to important advances in algebraic geometry and representation theory

Injective and flasque resolutions

• Injective and flasque resolutions are important tools in the computation of sheaf cohomology
• An injective resolution of a sheaf $\mathcal{F}$ is a long exact sequence of sheaves $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots$, where each $\mathcal{I}^i$ is an injective sheaf
• A flasque resolution is a similar construction, where the sheaves are required to be flasque (a weaker condition than injectivity)
• The cohomology of $\mathcal{F}$ can be computed using the global sections of the injective or flasque resolution
• Example: The Godement resolution of a sheaf, which is a canonical flasque resolution