🍃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.
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 on a topological space X consists of:
For each open set U⊆X, a set F(U) called the sections of F over U
For each inclusion of open sets V⊆U, a restriction map F(U)→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) 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 OX-module on a scheme X
The Sheaf Condition
Sheaf is a presheaf F satisfying the following two axioms:
(Gluability) If {Ui} is an open cover of an open set U and si∈F(Ui) are sections such that si∣Ui∩Uj=sj∣Ui∩Uj for all i,j, then there exists a section s∈F(U) such that s∣Ui=si for all i
(Uniqueness) If s,t∈F(U) are sections such that s∣Ui=t∣Ui 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 Sh(X) of the category of presheaves 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) is left adjoint to the forgetful functor Sh(X)→PSh(X)
This means that sheafification is the best approximation of a presheaf by a sheaf
Sheafification can be constructed in two steps:
Construct the separated presheaf F+ by adding in the uniqueness property
Construct the sheafification aF by adding in the gluing property
Sections of the sheafification aF over an open set U can be described as compatible families of germs of sections of 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) of a sheaf F on a topological space X is a topological space that encodes the local behavior of the sheaf
Points of Et(F) are pairs (x,s) where x∈X and s is a germ of a section of F at x
Topology on Et(F) is generated by basic open sets of the form {(x,s):s∈F(U),x∈U} for open sets U⊆X
Natural projection map π:Et(F)→X sending (x,s) to x is a local homeomorphism
This means that locally, Et(F) looks like a product of an open set in X with a discrete set (the stalks of F)
Sections of F over an open set U correspond bijectively to continuous sections of the projection map π 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 φ:F→G on a topological space X is a collection of maps φ(U):F(U)→G(U) for each open set U⊆X that commute with the restriction maps
Sheaf morphisms form a category 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 ℓ-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