🧮Topos Theory Unit 10 – Geometric Morphisms and Classifying Topoi

Key Concepts and Definitions

• Topos a category with finite limits, exponentials, and a subobject classifier
• Geometric morphism a pair of adjoint functors between topoi that preserve the logical structure
• Inverse image functor the right adjoint of a geometric morphism, preserving finite limits and colimits
• Direct image functor the left adjoint of a geometric morphism, preserving the subobject classifier and exponentials
  • Also known as the direct image functor or the pushforward functor
• Classifying topos a topos that classifies a particular geometric theory or set of axioms
• Subobject classifier an object in a topos that represents the concept of "truth values" and allows for the internalization of logic
• Sheaf a presheaf that satisfies the gluing axioms, allowing for the construction of global objects from local data
• Grothendieck topology a collection of covering sieves that specify when local data can be glued together to form a sheaf

Geometric Morphisms: The Basics

• Geometric morphisms are structure-preserving functors between topoi that generalize the notion of continuous maps between topological spaces
• Every geometric morphism consists of a pair of adjoint functors the inverse image functor and the direct image functor
  • The inverse image functor preserves finite limits, colimits, and the subobject classifier
  • The direct image functor preserves exponentials and the subobject classifier
• Geometric morphisms can be composed, forming a category of topoi with geometric morphisms as arrows
• The identity geometric morphism on a topos consists of the identity functor as both the inverse and direct image functors
• Geometric morphisms can be used to compare and relate different topoi, allowing for the transfer of properties and constructions between them

Properties of Geometric Morphisms

• A geometric morphism is faithful if and only if its inverse image functor is faithful
• A geometric morphism is essentially surjective if and only if its inverse image functor is essentially surjective
• A geometric morphism is an equivalence if and only if it is both faithful and essentially surjective
  • Equivalent topoi have the same logical and categorical properties
• A geometric morphism is a surjection if and only if its direct image functor is full and faithful
• A geometric morphism is an inclusion if and only if its inverse image functor is full and faithful
• A geometric morphism is a local homeomorphism if and only if its inverse image functor has a further left adjoint
  • Local homeomorphisms correspond to étale maps in algebraic geometry
• A geometric morphism is hyperconnected if and only if its inverse image functor preserves the subobject classifier

Classifying Topoi: Introduction

• A classifying topos is a topos that represents a particular geometric theory or set of axioms
• The classifying topos of a theory $T$ is denoted as $Set[T]$ and is defined as the topos of sheaves on the syntactic category of $T$
• For any topos $E$ and geometric morphism $f: E \to Set[T]$, the inverse image functor $f^*$ induces a model of $T$ in $E$
  • This correspondence between geometric morphisms and models is the key property of classifying topoi
• Classifying topoi provide a way to study the semantics of geometric theories in a categorical setting
• The classifying topos of a theory can be seen as a "universal" or "generic" model of that theory
• The existence of classifying topoi for geometric theories is a consequence of the Giraud-Diaconescu theorem

Construction of Classifying Topoi

• The construction of the classifying topos $Set[T]$ for a geometric theory $T$ involves several steps
  1. Define the syntactic category $C_T$ of the theory $T$, which has formulas as objects and provable entailments as arrows
  2. Equip $C_T$ with a Grothendieck topology, called the coherent topology, which specifies when a collection of formulas covers another formula
  3. Define $Set[T]$ as the topos of sheaves on the site $(C_T, J_{coh})$, where $J_{coh}$ is the coherent topology
• The subobject classifier in $Set[T]$ is given by the sheaf of "provably true" formulas in the theory $T$
• The exponentials in $Set[T]$ correspond to the construction of function types in the internal language of the topos
• Models of the theory $T$ in any topos $E$ correspond to geometric morphisms $E \to Set[T]$, with the inverse image functor inducing the interpretation of the language of $T$ in $E$

Applications in Mathematics and Logic

• Classifying topoi provide a bridge between logic and geometry, allowing for the study of theories using categorical and sheaf-theoretic tools
• The classifying topos of a geometric theory can be used to construct generic models and study the semantics of the theory
  • For example, the classifying topos of the theory of rings is the topos of sheaves on the opposite of the category of rings
• Classifying topoi can be used to prove independence results and construct counterexamples in logic and set theory
  • The classifying topos of the theory of well-pointed topoi is not well-pointed, showing that the axiom of choice is independent of the other topos axioms
• Classifying topoi have applications in algebraic geometry, where they provide a sheaf-theoretic approach to the study of schemes and algebraic spaces
• The theory of classifying topoi has connections to higher category theory and homotopy type theory, providing a framework for the study of higher-order logic and dependent type theories

Examples and Problem-Solving

• The classifying topos of the empty theory is the topos of sets, $Set$
  • This corresponds to the fact that models of the empty theory in any topos $E$ are just objects of $E$
• The classifying topos of the theory of objects is the topos of presheaves on the terminal category, $Set^{*}$
  • Models of this theory correspond to objects in a topos, together with a global element
• The classifying topos of the theory of groups is the topos of presheaves on the opposite of the category of groups, $Grp$
  • Geometric morphisms from a topos $E$ to $Set[Grp]$ correspond to internal groups in $E$
• The classifying topos of the theory of rings is the topos of sheaves on the opposite of the category of rings, $Sh(Ring^{op})$
  • This topos classifies the theory of rings and provides a setting for the study of generic rings and algebraic geometry
• Problem Given a geometric theory $T$, construct its classifying topos $Set[T]$ and describe the universal property of this topos in terms of models of $T$

Advanced Topics and Current Research

• The study of classifying topoi for infinitary first-order theories and their connections to accessible categories and locally presentable categories
• The construction of classifying topoi for higher-order theories, such as the calculus of constructions, and their relation to homotopy type theory
• The use of classifying topoi in the study of moduli problems and the construction of moduli spaces in algebraic geometry
  • For example, the classifying topos of the theory of elliptic curves can be used to construct the moduli stack of elliptic curves
• The development of a theory of classifying topoi for geometric theories with additional structure, such as monoidal or braided monoidal theories
• The application of classifying topoi to the study of quantum mechanics and quantum field theory, where they provide a setting for the study of quantum logic and quantum sheaf theory
• The exploration of connections between classifying topoi, higher category theory, and the theory of $\infty$-topoi, which generalize the notion of Grothendieck topoi to higher-categorical settings
• The use of classifying topoi in the study of topos-theoretic approaches to foundations of mathematics, such as the categorical foundations program and the univalent foundations program