15.4 Topos theory in algebraic geometry

Topos theory bridges set theory and geometry, providing a powerful framework for studying algebraic structures. It introduces categories that behave like sets, with special properties that enable logical reasoning about geometric objects.

In algebraic geometry, topoi generalize schemes and support abstract study of geometric objects. They allow for logical and categorical tools to express and manipulate geometric properties, opening new avenues for understanding complex mathematical structures.

Topos Theory Fundamentals

Properties of topoi

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• A topos is a category that behaves similarly to the category of sets
  • Cartesian closed has finite limits, exponential objects, and a terminal object (empty set)
  • Has a subobject classifier $\Omega$, a special object that classifies subobjects (true or false)
    • For any object $A$ and subobject $i: S \hookrightarrow A$, there exists a unique morphism $\chi_S: A \to \Omega$ such that $S$ is the pullback of $\text{true}: 1 \to \Omega$ along $\chi_S$
  • Validates internal logic allows reasoning about objects and morphisms using intuitionistic higher-order logic (propositional and predicate calculus)
    • Enables the interpretation of logical connectives (and, or, not) and quantifiers (for all, there exists) as categorical constructions
    • Facilitates the study of geometric objects using logical methods (proof theory)

Topoi in algebraic geometry

• Topoi provide a general framework for studying geometric objects and their properties
  • Generalize the notion of schemes, which are locally ringed spaces with a sheaf of rings (affine schemes, projective schemes)
    • Schemes can be viewed as topoi equipped with a sheaf of rings satisfying certain conditions (structure sheaf)
  • Allow the development of algebraic geometry in a more abstract and conceptual setting
    • Enables the study of geometric objects that may not be easily described using classical methods (algebraic spaces, stacks)
• Topoi support the use of logical and categorical tools in algebraic geometry
  • Internal logic allows expressing geometric properties using logical formulas (open subsets, closed subsets)
  • Categorical constructions, such as limits (fiber products) and colimits (gluing), can be used to construct and manipulate geometric objects

Examples and Applications

Examples of geometric topoi

• The étale topos $\text{Ét}(X)$ of a scheme $X$
  • Objects are étale sheaves on $X$, which are sheaves for the étale topology (étale coverings)
    • Étale topology is generated by étale morphisms, which are flat (preserves exact sequences) and unramified (no ramification)
  • Morphisms are morphisms of sheaves (natural transformations)
  • Captures the étale cohomology of $X$, an important invariant in algebraic geometry (étale fundamental group)
• The Zariski topos $\text{Zar}(X)$ of a scheme $X$
  • Objects are Zariski sheaves on $X$, which are sheaves for the Zariski topology (open subsets)
    • Zariski topology is generated by open immersions (open embeddings)
  • Morphisms are morphisms of sheaves (restriction maps)
  • Related to the study of coherent sheaves (locally finitely presented) and quasi-coherent sheaves (locally free) on $X$

Applications of topos theory

• Cohomology can be studied using topos-theoretic methods
  • Étale cohomology of a scheme $X$ can be defined as the cohomology of the étale topos $\text{Ét}(X)$
    • Computed using the global sections functor $\Gamma: \text{Ét}(X) \to \text{Set}$ (taking global sections)
  • Other cohomology theories, such as crystalline cohomology (characteristic p), can also be formulated using topoi
• Descent theory can be developed in the context of topoi
  • Descent data for objects and morphisms can be expressed using the internal logic of a topos (gluing conditions)
  • Enables the study of objects and morphisms that are locally defined but glue together globally (vector bundles, principal bundles)
  • Allows the construction of geometric objects by specifying local data and compatibility conditions (Čech cocycles)