🧮Topos Theory Unit 12 – Topos Theory: Algebraic and Geometric Views

Key Concepts and Definitions

• Topos defined as a category with certain properties that make it behave like a generalized space of sets
• Objects in a topos can be thought of as generalized sets, while morphisms between objects represent functions between these sets
• Subobject classifier is a special object $\Omega$ in a topos that plays the role of the set of truth values
• Internal logic of a topos determined by the structure of its subobject classifier, allowing for intuitionistic or constructive reasoning
  • Heyting algebra structure of $\Omega$ enables intuitionistic logic, where the law of excluded middle may not hold
• Sheaves are fundamental objects in a topos, generalizing the notion of functions defined on open sets of a topological space
  • Sheaf consists of a functor $F: \mathcal{O}(X)^{op} \to \mathbf{Set}$, where $\mathcal{O}(X)$ is the category of open sets of a topological space $X$
• Geometric morphisms are structure-preserving functors between topoi that respect the logical and spatial properties of the topoi
• Points of a topos are geometric morphisms from the topos of sets $\mathbf{Set}$ to the given topos, providing a notion of generalized elements

Historical Context and Development

• Topos theory emerged in the 1960s as a unification of ideas from algebraic geometry and categorical logic
• Alexander Grothendieck introduced the concept of a topos in his work on algebraic geometry, particularly in the study of étale cohomology
  • Grothendieck's topoi were categories of sheaves on a site, generalizing the notion of sheaves on a topological space
• William Lawvere and Myles Tierney independently developed the axiomatic definition of an elementary topos, emphasizing its logical and set-theoretic properties
• Lawvere's work on categorical logic, particularly the theory of hyperdoctrines, provided a foundation for understanding the internal logic of topoi
• The connection between sheaf theory and forcing in set theory, established by Lawvere and Tierney, revealed the deep logical aspects of topos theory
• Further developments by mathematicians such as Peter Johnstone, Saunders Mac Lane, and Michael Barr expanded the scope and applications of topos theory
• Topos theory has since found applications in various areas of mathematics, including algebraic geometry, mathematical logic, and theoretical computer science

Algebraic Foundations

• Topos theory relies heavily on the language and tools of category theory, particularly the notions of limits, colimits, and adjunctions
• Subobject classifier $\Omega$ in a topos is a generalization of the two-element set $\{0, 1\}$ in $\mathbf{Set}$, representing the set of truth values
  • Morphisms from an object $A$ to $\Omega$ correspond to subobjects of $A$, analogous to characteristic functions of subsets in set theory
• Power object $P(A)$ in a topos is a generalization of the power set, representing the collection of all subobjects of an object $A$
  • Exponential object $\Omega^A$ is isomorphic to the power object $P(A)$, establishing a connection between subobjects and morphisms to $\Omega$
• Topos has finite limits and colimits, allowing for the construction of products, coproducts, equalizers, and coequalizers
  • Existence of finite limits and colimits is crucial for the internal logic and set-theoretic reasoning in a topos
• Cartesian closed structure of a topos, given by the existence of exponential objects, enables the interpretation of higher-order logic and lambda calculus

Geometric Perspectives

• Topos theory provides a generalized framework for studying spaces and their properties using categorical and logical tools
• Grothendieck topologies on a category $\mathcal{C}$ specify a notion of covering sieves, generalizing the concept of open covers in topology
  • Sheaves on a site $(\mathcal{C}, J)$, where $J$ is a Grothendieck topology, form a topos $\mathbf{Sh}(\mathcal{C}, J)$
• Étale topoi, arising from étale morphisms in algebraic geometry, capture intrinsic properties of schemes and algebraic spaces
  • Étale cohomology, defined using sheaves on the étale site, provides a powerful tool for studying algebraic varieties over arbitrary fields
• Classifying topoi, such as the classifying topos of a geometric theory $\mathbb{T}$, represent generalized spaces of models of $\mathbb{T}$
  • Geometric morphisms from a topos $\mathcal{E}$ to the classifying topos of $\mathbb{T}$ correspond to $\mathbb{T}$-models in $\mathcal{E}$
• Topos-theoretic approach to topology, where topological spaces are studied via their categories of sheaves, offers new insights and proof techniques
  • Comparison lemma relates the topos of sheaves on a topological space to the topos of étale sheaves on a related scheme

Category Theory Connections

• Topos theory is deeply rooted in category theory, with topoi being particular kinds of categories with additional structure
• Adjunctions play a central role in topos theory, relating topoi to other categories and providing a means of constructing new topoi from existing ones
  • Geometric morphisms between topoi are adjoint pairs of functors satisfying certain conditions
• Grothendieck topologies and sheaves on a site are inherently categorical notions, relying on the concepts of sieves and presheaves
• Yoneda lemma, a fundamental result in category theory, is used extensively in topos theory to study the relationship between objects and morphisms
  • Yoneda embedding provides a fully faithful functor from a category to the category of presheaves on that category
• Monadicity and descent theory, which study the relationship between adjunctions and monads, have important applications in topos theory
  • Effective descent morphisms in a topos correspond to geometric morphisms that are monadic
• 2-categorical structure of the category of topoi, with geometric morphisms as 1-cells and natural transformations as 2-cells, enriches the study of topoi

Applications in Mathematics

• Topos theory has found significant applications in various branches of mathematics, providing new perspectives and unifying concepts
• In algebraic geometry, étale cohomology and the study of étale topoi have revolutionized the field, allowing for the development of schemes and algebraic spaces
  • Étale fundamental group of a scheme, defined using the étale topos, captures arithmetic and geometric properties of the scheme
• Synthetic differential geometry, which studies smooth spaces using the internal logic of a topos, offers an alternative approach to classical differential geometry
  • Kock-Lawvere axiom in synthetic differential geometry states that the set of infinitesimal elements in the real line is a nilpotent subobject
• Topos-theoretic methods have been applied to the study of modal and intuitionistic logic, providing semantic models and completeness results
  • Grothendieck topologies and sheaves have been used to construct forcing models of intuitionistic set theories
• In theoretical computer science, topoi have been used to study the semantics of programming languages and type theories
  • Realizability topoi, constructed from partial combinatory algebras, provide models for higher-order logic and lambda calculus

Advanced Topics and Current Research

• Homotopy topos theory, which combines ideas from homotopy theory and topos theory, has emerged as an active area of research
  • $\infty$-topoi, a higher categorical generalization of topoi, are used to study homotopy-theoretic properties of spaces and categories
• Derived algebraic geometry, which studies spaces and schemes using derived categories and homotopical algebra, relies heavily on topos-theoretic methods
  • Derived schemes and stacks are defined using the language of $\infty$-topoi and sheaves of simplicial rings
• Topos theory has been applied to the study of quantum physics and quantum logic, providing a framework for understanding quantum systems and their properties
  • Quantum topoi, which incorporate non-commutative and non-distributive structures, have been proposed as a foundation for quantum theory
• Categorical logic and the study of higher-order categorical structures, such as 2-topoi and $(\infty, 1)$-topoi, continue to be active areas of research
  • Univalent foundations and homotopy type theory, which aim to provide a constructive and computationally-friendly foundation for mathematics, have connections to topos theory

Problem-Solving Techniques

• When working with topoi, it is essential to understand the interplay between the logical and spatial aspects of the topos
  • Consider how the internal logic of the topos, determined by its subobject classifier, affects the reasoning and construction of objects
• Utilize the categorical tools available in a topos, such as limits, colimits, and exponential objects, to solve problems and prove statements
  • Construct objects and morphisms using universal properties, such as products, coproducts, and pullbacks
• Employ the Yoneda lemma and the Yoneda embedding to study the relationship between objects and morphisms in a topos
  • Use the Yoneda lemma to show that certain morphisms are unique or to establish isomorphisms between objects
• Exploit the connection between sheaves and geometric morphisms to translate problems between different topoi
  • Use the adjunctions between topoi, such as the direct and inverse image functors, to move objects and morphisms between topoi
• Consider the use of classifying topoi and the interpretation of geometric theories to study models and their properties
  • Construct classifying topoi for specific geometric theories and use geometric morphisms to study the models of these theories
• Utilize the internal language and logic of a topos to reason about objects and morphisms in a set-theoretic or type-theoretic manner
  • Interpret logical formulas and constructions in the internal language of a topos to prove statements and construct objects
• Draw upon the vast array of examples and counterexamples in topos theory to gain intuition and insight into problem-solving strategies
  • Study examples of topoi arising from algebraic geometry, topology, and logic to understand the diverse range of applications and techniques available