🍃Sheaf Theory Unit 2 – Sheafification and sheaf space

Key Concepts and Definitions

• Sheaf theory studies the local-to-global passage of data attached to the open sets of a topological space
• Presheaf consists of data attached to each open set of a topological space along with restriction maps between the data whenever one open set is contained in another
• Sheaf is a presheaf satisfying the gluability and unique gluing axioms which allow local data to be uniquely glued together into global data
  • Gluability axiom states that if compatible data is given on a collection of open sets covering an open set, then this data can be glued together to give data on the larger open set
  • Unique gluing axiom ensures that the glued data is unique
• Sheafification is the process of turning a presheaf into a sheaf by adding in the necessary gluing data
• Stalks are the germs of the data attached to a point by a (pre)sheaf
• Étalé space is a topological space associated to a (pre)sheaf that encodes its local behavior
• Sheaf cohomology extends the idea of cohomology to sheaves and captures global properties of the sheaf

Motivation and Historical Context

• Sheaf theory originated in the 1940s and 1950s with the work of Jean Leray, Henri Cartan, and Jean-Pierre Serre
• Initially developed as a tool in algebraic topology and complex analysis to study global properties of manifolds and complex spaces
• Grothendieck recognized sheaves as a fundamental concept in algebraic geometry and used them to develop the theory of schemes
  • Schemes are a generalization of algebraic varieties that allow for more general spaces to be studied using algebraic and geometric techniques
• Sheaves provide a way to study local-to-global properties in various mathematical contexts such as differential geometry, algebraic geometry, and complex analysis
• The language of sheaves allows for the formulation of many important results and concepts such as the Riemann-Hilbert correspondence, Hodge theory, and the Grothendieck-Riemann-Roch theorem
• Sheaf cohomology has become a central tool in modern algebraic and complex geometry
• Sheaf theory has applications in other areas such as mathematical physics (quantum field theory), logic (topos theory), and theoretical computer science (contextual semantics)

Presheaves and Their Properties

• Presheaf $\mathcal{F}$ on a topological space $X$ consists of:
  • For each open set $U \subseteq X$, a set $\mathcal{F}(U)$ called the sections of $\mathcal{F}$ over $U$
  • For each inclusion of open sets $V \subseteq U$, a restriction map $\mathcal{F}(U) \to \mathcal{F}(V)$ satisfying the composition and identity axioms
• Restriction maps encode how the data assigned to larger open sets relates to the data assigned to smaller open sets
• Presheaves form a category $\mathbf{PSh}(X)$ where the objects are presheaves on $X$ and morphisms are natural transformations between them
• Presheaves have a natural notion of kernel, cokernel, image, and exact sequences
• Operations such as direct sums, tensor products, and sheaf hom can be defined for presheaves
• Constant presheaf assigns the same set or group to each open set with identity maps as the restriction maps
• Presheaves of abelian groups, rings, modules, and algebras are important in algebraic geometry and lead to the notion of an $\mathcal{O}_X$-module on a scheme $X$

The Sheaf Condition

• Sheaf is a presheaf $\mathcal{F}$ satisfying the following two axioms:
  1. (Gluability) If $\{U_i\}$ is an open cover of an open set $U$ and $s_i \in \mathcal{F}(U_i)$ are sections such that $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i,j$, then there exists a section $s \in \mathcal{F}(U)$ such that $s|_{U_i} = s_i$ for all $i$
  2. (Uniqueness) If $s,t \in \mathcal{F}(U)$ are sections such that $s|_{U_i} = t|_{U_i}$ for all $i$, then $s = t$
• Gluability axiom allows compatible local sections to be glued into a global section
• Uniqueness axiom ensures that the glued section is unique
• Sheaf condition can be expressed in terms of equalizer diagrams involving the sections over an open cover and their pairwise intersections
• Sheaves form a full subcategory $\mathbf{Sh}(X)$ of the category of presheaves $\mathbf{PSh}(X)$
• Many natural examples of presheaves are actually sheaves, such as:
  • Continuous functions on a topological space
  • Smooth functions on a manifold
  • Holomorphic functions on a complex manifold
  • Regular functions on an algebraic variety
• Sheaf condition is local in nature and allows for the study of local-to-global properties

Constructing Sheaves from Presheaves

• Sheafification is the process of turning a presheaf into a sheaf by adding in the necessary gluing data
• Sheafification functor $\mathbf{a} : \mathbf{PSh}(X) \to \mathbf{Sh}(X)$ is left adjoint to the forgetful functor $\mathbf{Sh}(X) \to \mathbf{PSh}(X)$
  • This means that sheafification is the best approximation of a presheaf by a sheaf
• Sheafification can be constructed in two steps:
  1. Construct the separated presheaf $\mathcal{F}^+$ by adding in the uniqueness property
  2. Construct the sheafification $\mathbf{a}\mathcal{F}$ by adding in the gluing property
• Sections of the sheafification $\mathbf{a}\mathcal{F}$ over an open set $U$ can be described as compatible families of germs of sections of $\mathcal{F}$
• Sheafification preserves many properties of presheaves such as being abelian groups, rings, modules, or algebras
• Sheafification is an exact functor and preserves finite limits and colimits
• Many constructions for sheaves can be defined by first constructing them for presheaves and then sheafifying, such as:
  • Kernel, cokernel, and image presheaves
  • Direct sums and tensor products of presheaves
  • Presheaf hom and sheaf hom
• Sheafification can be used to construct important sheaves such as:
  • Sheaf of differentiable functions on a manifold
  • Sheaf of holomorphic functions on a complex manifold
  • Structure sheaf of a scheme

Sheaf Space and Étalé Space

• Étalé space $\mathbf{Et}(\mathcal{F})$ of a sheaf $\mathcal{F}$ on a topological space $X$ is a topological space that encodes the local behavior of the sheaf
• Points of $\mathbf{Et}(\mathcal{F})$ are pairs $(x,s)$ where $x \in X$ and $s$ is a germ of a section of $\mathcal{F}$ at $x$
• Topology on $\mathbf{Et}(\mathcal{F})$ is generated by basic open sets of the form $\{(x,s) : s \in \mathcal{F}(U), x \in U\}$ for open sets $U \subseteq X$
• Natural projection map $\pi : \mathbf{Et}(\mathcal{F}) \to X$ sending $(x,s)$ to $x$ is a local homeomorphism
  • This means that locally, $\mathbf{Et}(\mathcal{F})$ looks like a product of an open set in $X$ with a discrete set (the stalks of $\mathcal{F}$)
• Sections of $\mathcal{F}$ over an open set $U$ correspond bijectively to continuous sections of the projection map $\pi$ over $U$
• Étalé space provides a useful geometric perspective on sheaves and allows for the use of topological techniques to study sheaves
• Many properties of sheaves can be translated into properties of their étalé spaces, such as:
  • Sheaf is flasque (sections extend uniquely from any open set to the whole space) iff its étalé space is a disjoint union of open sets in $X$
  • Sheaf is soft (sections extend from any closed set to the whole space) iff its étalé space is a fiber bundle over $X$
• Étalé spaces can be used to define operations on sheaves such as:
  • Pullback of a sheaf along a continuous map
  • Pushforward of a sheaf along a local homeomorphism
  • Tensor product and sheaf hom of sheaves

Morphisms of Sheaves

• Morphism of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ on a topological space $X$ is a collection of maps $\varphi(U) : \mathcal{F}(U) \to \mathcal{G}(U)$ for each open set $U \subseteq X$ that commute with the restriction maps
• Sheaf morphisms form a category $\mathbf{Sh}(X)$ where the objects are sheaves on $X$ and the morphisms are sheaf morphisms
• Sheaf morphisms can be characterized in several equivalent ways:
  • As a morphism of presheaves that respects the sheaf condition
  • As a continuous map between the étalé spaces of the sheaves that commutes with the projection maps
  • Locally, as a collection of stalk maps that are compatible with the restriction maps
• Sheaf morphisms have many nice properties:
  • Composition of sheaf morphisms is a sheaf morphism
  • Identity map of a sheaf is a sheaf morphism
  • Sheaf isomorphisms correspond to homeomorphisms of the étalé spaces that commute with the projection maps
• Many constructions for sheaves are functorial with respect to sheaf morphisms, such as:
  • Kernel, cokernel, and image sheaves
  • Direct sums and tensor products of sheaves
  • Sheaf hom and sheaf cohomology
• Sheaf morphisms can be used to define important constructions such as:
  • Exact sequences of sheaves
  • Sheaves of modules over a sheaf of rings
  • Pullback and pushforward of sheaves along a continuous map
• Sheaf morphisms provide a way to compare and relate different sheaves on a space and are an essential part of the theory of sheaves

Applications and Examples

• Sheaves of continuous, smooth, holomorphic, or regular functions on a space provide a way to study the local and global properties of these functions
  • Sheaf of continuous functions on a topological space
  • Sheaf of smooth functions on a manifold
  • Sheaf of holomorphic functions on a complex manifold
  • Sheaf of regular functions on an algebraic variety
• Sheaves of abelian groups, rings, modules, and algebras play a fundamental role in algebraic geometry and the theory of schemes
  • Structure sheaf of a scheme encodes the algebraic and geometric properties of the scheme
  • Quasi-coherent sheaves on a scheme generalize the notion of modules over a ring
  • Coherent sheaves on a scheme are analogous to finitely generated modules over a ring and are used to study properties such as dimension and cohomology
• Sheaf cohomology extends the idea of cohomology to sheaves and provides a way to study global properties of sheaves
  • Čech cohomology of a sheaf is defined using open covers and the sheaf condition
  • Derived functor cohomology of a sheaf is defined using injective or flasque resolutions
  • Sheaf cohomology plays a key role in the study of algebraic and complex geometry, such as in the proofs of the Riemann-Roch theorem and the Hodge decomposition theorem
• Sheaves and their cohomology have applications in various areas of mathematics and physics, such as:
  • Algebraic topology (sheaves on a topological space and their cohomology)
  • Complex analysis (sheaves of holomorphic functions and their cohomology)
  • Differential geometry (sheaves of smooth functions and their cohomology)
  • Mathematical physics (sheaves in quantum field theory and string theory)
  • Number theory (étale and $\ell$-adic sheaves in arithmetic geometry)
• Sheaves provide a unifying language and framework for studying local-to-global properties in various mathematical contexts and have become an essential tool in modern mathematics