9.1 Sites and Grothendieck topologies

Sites and Grothendieck topologies expand on the idea of open covers in topology. They provide a framework for studying geometric and algebraic objects in a unified way, allowing us to define "local" properties in diverse mathematical contexts.

The key components of a site include an underlying category and a collection of covering families. These define local structure and enable the study of sheaves, which encapsulate local-to-global data. This approach is crucial for understanding modern algebraic geometry.

Sites and Grothendieck Topologies

Concept and examples of sites

• Site definition combines category $C$ with Grothendieck topology generalizing topological spaces
• Examples of sites encompass small category of open sets in topological space, Zariski site in algebraic geometry, Étale site for schemes, Crystalline site in p-adic cohomology
• Key components of a site include underlying category and collection of covering families defining local structure
• Sites provide framework for studying geometric and algebraic objects in unified manner
• Applications extend to sheaf theory, cohomology theories, and algebraic geometry

Axioms of Grothendieck topology

• Grothendieck topology assigns "covering families" to objects in category formalizing local-to-global principle
• Axioms ensure consistency and compatibility of covering families:
  1. Identity axiom requires singleton family containing identity morphism to be a covering
  2. Stability under base change allows pullback of coverings along morphisms
  3. Transitivity (local character) enables composition of coverings
• Properties include refinement of coverings allowing finer local descriptions and intersection of coverings capturing common local information
• Generalizes open cover concept from classical topology to abstract categorical setting
• Enables study of "local" properties in diverse mathematical contexts (algebraic varieties, schemes)

Construction of sheaf categories

• Presheaf on site defined as contravariant functor from site to Set category encapsulating local-to-global data
• Sheaf condition requires local data to glue consistently with respect to covering families
• Construction process involves:
  1. Starting with category of presheaves
  2. Identifying presheaves satisfying sheaf condition
  3. Forming full subcategory of sheaves
• Yoneda embedding for sites represents objects as presheaves preserving site structure
• Sheafification functor transforms presheaves into sheaves acting as left adjoint to inclusion
• Allows transition from local to global perspective in geometric and algebraic contexts

Sheaves as topos formation

• Topos defined as category equivalent to category of sheaves on a site providing rich structural framework
• Key properties to verify for topos structure:
  • Existence of finite limits and colimits (pullbacks, pushouts)
  • Subobject classifier (truth object) generalizing characteristic functions
  • Power objects representing "set" of subobjects
• Proof outline demonstrates category of sheaves is complete, cocomplete, and possesses necessary categorical structures
• Site-dependent constructions highlight flexibility of topos theory in capturing diverse mathematical contexts
• Consequences of topos structure include internal logic for reasoning and geometric morphisms for relating toposes
• Provides unified approach to geometry, algebra, and logic in abstract categorical setting