7.3 Presheaf topoi and functor categories

Presheaf topoi and functor categories are powerful tools in category theory. They generalize the concept of sets and functions, allowing us to work with more complex mathematical structures. These constructions provide a rich framework for studying relationships between objects and morphisms.

Presheaf topoi offer a way to model dynamic information, while functor categories extend this idea to arbitrary categories. Both possess key properties of elementary topoi, including subobject classifiers and internal logic, making them versatile for various mathematical applications.

Presheaf Topoi and Functor Categories

Definition of presheaf topoi

• Presheaf maps objects and morphisms in small category C to sets and functions in Set preserves composition and identity morphisms in opposite direction (functors $F: C^{op} \rightarrow Set$)
• Presheaf topos encompasses presheaves on C as objects and natural transformations between presheaves as morphisms (category $Set^{C^{op}}$)
• Properties of presheaf topos include cartesian closed structure allows internal function objects supports all limits and colimits enables universal constructions possesses subobject classifier provides rich logical framework (Heyting algebra)

Functor categories as elementary topoi

• Functor category comprises functors as objects and natural transformations as morphisms generalizes presheaf topoi to arbitrary source and target categories
• Elementary topos properties demonstrated in functor categories:
  1. Construct finite products componentwise
  2. Form exponential objects using natural transformations
  3. Identify terminal object as constant functor
  4. Build equalizers pointwise
  5. Define subobject classifier using sieves
• Proof establishes functor categories as models of intuitionistic type theory with dependent types

Subobject classifier in presheaf topoi

• Subobject classifier Ω in $Set^{C^{op}}$ assigns set of sieves on c to each object c in C (sieves: collections of morphisms with codomain c closed under precomposition)
• True morphism t: 1 → Ω maps terminal object to maximal sieve represents "truth" in the topos
• Characteristic morphism χ_m: X → Ω for subobject m: Y → X encodes membership information for each element x in X(c) χ_m(c)(x) yields sieve of all f: d → c such that X(f)(x) lies in image of m(d)
• Construction generalizes power set operation in Set captures notion of "subset" in presheaf context

Presheaf vs set-based topoi

• Similarities include elementary topos structure subobject classifiers cartesian closed property support for internal logic
• Differences:
  • Presheaf objects: functors encoding both static and dynamic information (sheaves)
  • Set-based objects: sets with additional structure (topological spaces)
  • Subobject classifier: complex in presheaves (sieves) simpler in sets (two-element set)
  • Internal logic: intuitionistic in presheaves classical in sets
• Presheaf topoi offer nuanced representation of mathematical structures (schemes in algebraic geometry) while set-based topoi provide concrete models (smooth manifolds in differential geometry)
• Applications: presheaf topoi in sheaf theory and categorical logic set-based topoi in axiomatic set theory and topos-theoretic approaches to quantum mechanics