Sheaf Theory

🍃Sheaf Theory Unit 2 – Sheafification and sheaf space

Sheafification and sheaf space are fundamental concepts in sheaf theory, bridging local and global data on topological spaces. This unit explores how presheaves become sheaves through sheafification, and how sheaves can be represented geometrically as étalé spaces. These concepts are crucial for understanding the local-to-global behavior of mathematical structures. Sheafification allows us to glue compatible local data, while étalé spaces provide a geometric perspective on sheaves, enabling the use of topological techniques in their study.

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 F\mathcal{F} on a topological space XX consists of:
    • For each open set UXU \subseteq X, a set F(U)\mathcal{F}(U) called the sections of F\mathcal{F} over UU
    • For each inclusion of open sets VUV \subseteq U, a restriction map F(U)F(V)\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 PSh(X)\mathbf{PSh}(X) where the objects are presheaves on XX 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 OX\mathcal{O}_X-module on a scheme XX

The Sheaf Condition

  • Sheaf is a presheaf F\mathcal{F} satisfying the following two axioms:
    1. (Gluability) If {Ui}\{U_i\} is an open cover of an open set UU and siF(Ui)s_i \in \mathcal{F}(U_i) are sections such that siUiUj=sjUiUjs_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j} for all i,ji,j, then there exists a section sF(U)s \in \mathcal{F}(U) such that sUi=sis|_{U_i} = s_i for all ii
    2. (Uniqueness) If s,tF(U)s,t \in \mathcal{F}(U) are sections such that sUi=tUis|_{U_i} = t|_{U_i} for all ii, then s=ts = 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 Sh(X)\mathbf{Sh}(X) of the category of presheaves PSh(X)\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 a:PSh(X)Sh(X)\mathbf{a} : \mathbf{PSh}(X) \to \mathbf{Sh}(X) is left adjoint to the forgetful functor Sh(X)PSh(X)\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 F+\mathcal{F}^+ by adding in the uniqueness property
    2. Construct the sheafification aF\mathbf{a}\mathcal{F} by adding in the gluing property
  • Sections of the sheafification aF\mathbf{a}\mathcal{F} over an open set UU can be described as compatible families of germs of sections of F\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 Et(F)\mathbf{Et}(\mathcal{F}) of a sheaf F\mathcal{F} on a topological space XX is a topological space that encodes the local behavior of the sheaf
  • Points of Et(F)\mathbf{Et}(\mathcal{F}) are pairs (x,s)(x,s) where xXx \in X and ss is a germ of a section of F\mathcal{F} at xx
  • Topology on Et(F)\mathbf{Et}(\mathcal{F}) is generated by basic open sets of the form {(x,s):sF(U),xU}\{(x,s) : s \in \mathcal{F}(U), x \in U\} for open sets UXU \subseteq X
  • Natural projection map π:Et(F)X\pi : \mathbf{Et}(\mathcal{F}) \to X sending (x,s)(x,s) to xx is a local homeomorphism
    • This means that locally, Et(F)\mathbf{Et}(\mathcal{F}) looks like a product of an open set in XX with a discrete set (the stalks of F\mathcal{F})
  • Sections of F\mathcal{F} over an open set UU correspond bijectively to continuous sections of the projection map π\pi over UU
  • 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 XX
    • Sheaf is soft (sections extend from any closed set to the whole space) iff its étalé space is a fiber bundle over XX
  • 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 φ:FG\varphi : \mathcal{F} \to \mathcal{G} on a topological space XX is a collection of maps φ(U):F(U)G(U)\varphi(U) : \mathcal{F}(U) \to \mathcal{G}(U) for each open set UXU \subseteq X that commute with the restriction maps
  • Sheaf morphisms form a category Sh(X)\mathbf{Sh}(X) where the objects are sheaves on XX 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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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