9.3 Comparison with elementary topoi

Grothendieck and elementary topoi are powerful mathematical structures that generalize the notion of space and set theory. They share key properties like having a subobject classifier and being cartesian closed, enabling internal logic and higher-order functions.

However, they differ in size, completeness, and existence of generators. Grothendieck topoi are typically larger and more complete, while elementary topoi are more flexible in size. These differences impact their applications in mathematics and logic.

Grothendieck Topoi vs Elementary Topoi

Properties of Grothendieck vs elementary topoi

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• Grothendieck topoi
  • Categories equivalent to categories of sheaves on a site defined as mathematical structures generalizing notion of topological space (étale topology)
  • Properties:
    • Possess all small limits and colimits enabling wide range of categorical constructions
    • Have a subobject classifier allowing for internal logic and set-theoretic operations
    • Are cartesian closed supporting function spaces and higher-order functions
• Elementary topoi
  • Categories satisfying axioms resembling properties of category of sets formulated to capture essential features of set theory categorically
  • Properties:
    • Have finite limits and colimits supporting basic categorical operations (products, equalizers)
    • Possess a subobject classifier enabling internal logic similar to Grothendieck topoi
    • Are cartesian closed allowing for function spaces and higher-order functions
• Key differences
  • Size considerations
    • Grothendieck topoi typically large categories containing infinite objects (sheaves on infinite spaces)
    • Elementary topoi can be small or large encompassing finite and infinite structures
  • Completeness
    • Grothendieck topoi complete and cocomplete allowing for arbitrary limits and colimits
    • Elementary topoi only require finite limits and colimits restricting some constructions
  • Existence of generators
    • Grothendieck topoi have a set of generators enabling representation of objects in terms of simpler ones
    • Elementary topoi do not necessarily have generators lacking this structural property

Examples of non-overlapping topoi

• Grothendieck topoi not elementary topoi
  • Category of sheaves on non-trivial topological space (manifolds, algebraic varieties)
  • Category of G-sets for infinite group G (representations of infinite symmetry groups)
  • Category of sheaves on non-trivial Grothendieck site (étale site in algebraic geometry)
• Elementary topoi not Grothendieck topoi
  • Effective topos modeling realizability in computer science
  • Category of finite sets lacking infinite objects required for Grothendieck topoi
  • Category of nominal sets used in theoretical computer science for name-binding

Axiom of choice in topoi distinction

• Grothendieck topoi
  • Always holds due to existence of enough projectives ensuring choice functions
  • Enables powerful results in algebraic geometry and topology
• Elementary topoi
  • May or may not hold depending on specific topos
  • Allows construction of counterexamples to properties valid in Grothendieck topoi (non-well-orderable objects)
• Implications for model theory
  • Grothendieck topoi model intuitionistic higher-order logic with choice supporting classical mathematics
  • Elementary topoi model intuitionistic higher-order logic without choice enabling constructive approaches

Foundations and Logic

Implications for mathematics and logic

• Foundations of mathematics
  • Generalized notion of space extending beyond traditional topological spaces
  • Framework for synthetic differential geometry allowing infinitesimals
  • Constructive mathematics development without law of excluded middle
• Higher-order logic
  • Models for intuitionistic type theory supporting dependent types
  • Internal languages of topoi enabling categorical logic
  • Categorical logic study formalizing mathematical reasoning
• Applications in other areas
  • Algebraic geometry: étale cohomology revolutionizing study of schemes
  • Sheaf theory: generalized sheaves on sites beyond topological spaces
  • Category theory: generalizing set-theoretic foundations to categorical setting
• Philosophical implications
  • Questions set theory primacy as foundation suggesting alternatives
  • Structural approach to mathematics emphasizing relationships over objects
  • Highlights category theory importance in foundations unifying diverse areas