🍃Sheaf Theory Unit 1 – Presheaves and sheaves

Key Concepts and Definitions

• Presheaves generalize the concept of functions on open sets of a topological space
• Sheaves formalize the idea of local data that can be consistently glued together
• Stalks consist of germs, which are equivalence classes of sections at a point
• Restrictions maps allow for the comparison of sections on different open sets
• Sheafification converts a presheaf into a sheaf by adding missing local data
  • Involves a universal property and a canonical morphism
• Étalé spaces provide a geometric perspective on sheaves as fiber bundles
• Sheaf cohomology extends the notion of cohomology to sheaves
  • Measures the obstruction to the existence of global sections

Presheaves: Structure and Properties

• Presheaves are contravariant functors from the category of open sets of a topological space to a target category (often sets, rings, or modules)
• The contravariance captures the idea that restrictions are "reversing" inclusions
• Presheaves assign data to each open set and provide restriction maps between them
• Morphisms between presheaves are natural transformations of the underlying functors
• Presheaves form a category, with composition given by the usual composition of natural transformations
• The category of presheaves is complete and cocomplete
  • Limits and colimits are computed pointwise
• Presheaves can be restricted and extended along continuous maps between topological spaces

From Presheaves to Sheaves

• Sheaves are presheaves that satisfy additional conditions, ensuring local data can be glued together consistently
• The sheaf axioms (locality and gluing) guarantee the existence and uniqueness of global sections given compatible local data
• Locality states that sections can be determined locally
  • If two sections agree on an open cover, they agree on the entire set
• Gluing allows for the construction of a global section from compatible local sections
• The sheafification functor transforms a presheaf into a sheaf
  • Adds the "missing" local data needed to satisfy the sheaf axioms
• The sheafification is left adjoint to the forgetful functor from sheaves to presheaves

Sheaf Axioms and Characteristics

• Sheaves satisfy two key axioms: locality (or uniqueness) and gluing (or existence)
• The locality axiom ensures that sections are determined by their values on an open cover
  • If $s,t \in F(U)$ agree on each $V_i$ in an open cover of $U$, then $s = t$
• The gluing axiom allows for the construction of global sections from compatible local data
  • If $\{s_i \in F(V_i)\}$ agree on overlaps, there exists a unique $s \in F(U)$ restricting to each $s_i$
• Sheaves are determined by their stalks (the germs at each point) and the restriction maps between them
• The étalé space of a sheaf is a topological space that encodes the sheaf's local behavior
  • Provides a geometric perspective on sheaves as fiber bundles over the base space

Examples and Applications

• The constant sheaf assigns the same set or group to each open set, with identity restrictions
• The sheaf of continuous functions assigns the set of continuous real-valued functions to each open set
  • Restriction maps are given by function restriction
• The sheaf of smooth functions on a manifold assigns the set of smooth real-valued functions to each open set
• The sheaf of holomorphic functions on a complex manifold assigns the set of holomorphic functions to each open set
• The structure sheaf of a ringed space (a topological space with a sheaf of rings) captures the local ring structure
  • Important in the study of schemes in algebraic geometry
• Sheaf cohomology is used to study global properties of sheaves and their associated geometric objects
  • Provides invariants for vector bundles and complex manifolds

Morphisms and Categories

• Morphisms of sheaves are morphisms of the underlying presheaves that are compatible with the sheaf structure
  • Induce morphisms on stalks and local sections
• The category of sheaves on a topological space has a rich structure
  • It is an abelian category, allowing for the study of kernels, cokernels, and exact sequences
• Sheaves can be pushed forward and pulled back along continuous maps between topological spaces
  • These functors are adjoint and preserve various properties of sheaves
• The global section functor is a left exact functor from the category of sheaves to the category of sets (or modules)
  • Its right derived functors are the sheaf cohomology functors
• The category of sheaves has enough injectives, allowing for the construction of injective resolutions and the computation of sheaf cohomology

Advanced Topics and Connections

• Sheaves can be defined on sites, which are categories with a Grothendieck topology
  • Allows for the study of sheaves in more general settings, such as étale cohomology in algebraic geometry
• The Čech nerve of an open cover can be used to compute sheaf cohomology
  • Provides a concrete way to calculate cohomology groups using Čech cochains
• The Serre-Swan theorem establishes an equivalence between vector bundles and locally free sheaves on a compact Hausdorff space
• Sheaves play a central role in the theory of schemes in algebraic geometry
  • The structure sheaf of a scheme encodes its local ring structure
• Perverse sheaves are a special class of sheaves used in the study of intersection cohomology and the decomposition theorem
• The Riemann-Hilbert correspondence relates sheaves on a complex manifold to certain systems of differential equations
• Microlocal sheaf theory studies sheaves on cotangent bundles and their applications to symplectic and contact geometry

Practice Problems and Exercises

• Verify that the sheaf of continuous functions satisfies the sheaf axioms
• Compute the stalks of the constant sheaf and the sheaf of continuous functions on a given topological space
• Construct a sheaf from a given presheaf by sheafification
  • Describe the stalks and restriction maps of the resulting sheaf
• Prove that the category of sheaves on a topological space has enough injectives
• Calculate the sheaf cohomology groups of the constant sheaf on a circle and a torus
  • Interpret the results geometrically
• Show that the pullback and pushforward functors for sheaves are adjoint
• Construct an exact sequence of sheaves and study its properties
  • Relate the cohomology groups of the sheaves in the sequence
• Explore the connection between sheaves and vector bundles on a compact Hausdorff space
  • Prove a special case of the Serre-Swan theorem