🔢Category Theory Unit 14 – Topoi and Geometric Morphisms

What's This All About?

• Topos theory provides a unified framework for studying logic, geometry, and algebra in a categorical setting
• Topoi (plural of topos) generalize the concept of topological spaces and sheaves, allowing for the study of "variable sets"
• Geometric morphisms are structure-preserving maps between topoi that respect the internal logic and geometry
• Studying topoi and geometric morphisms allows for a deeper understanding of the connections between different branches of mathematics
• Topos theory has applications in various fields, including algebraic geometry, mathematical physics, and theoretical computer science
  • In algebraic geometry, topoi provide a foundation for studying schemes and their cohomology
  • In mathematical physics, topoi are used to formalize quantum mechanics and quantum field theory
  • In theoretical computer science, topoi are used to study type theory and categorical semantics

Key Concepts

• Topos: a category that behaves like the category of sets and functions, with additional structure for logic and geometry
• Subobject classifier: an object $\Omega$ in a topos that generalizes the concept of a subset, allowing for the internalization of logic
• Power object: an object $P(A)$ in a topos that represents the collection of all subobjects of an object $A$
• Internal logic: the logical structure within a topos, which can be intuitionistic or classical depending on the properties of the topos
• Geometric morphism: a pair of adjoint functors $f_* \dashv f^*$ between topoi that preserve the logical and geometric structure
  • The direct image functor $f_*$ preserves colimits and finite limits
  • The inverse image functor $f^*$ preserves all limits and is left exact
• Locale: a complete Heyting algebra, which can be viewed as a generalized topological space without points
• Grothendieck topos: a topos that arises as the category of sheaves on a site (a small category with a Grothendieck topology)

Topoi Unpacked

• A topos is a category $\mathcal{E}$ with the following properties:
  • $\mathcal{E}$ has all finite limits and colimits
  • $\mathcal{E}$ has exponential objects, i.e., for any objects $A$ and $B$, there exists an object $B^A$ representing the morphisms from $A$ to $B$
  • $\mathcal{E}$ has a subobject classifier $\Omega$
• The subobject classifier $\Omega$ allows for the internalization of logic within the topos
  • For any object $A$ and subobject $S \hookrightarrow A$, there is a unique characteristic morphism $\chi_S: A \to \Omega$
  • The subobject classifier generalizes the concept of a two-element set $\{0, 1\}$ in the category of sets
• Topoi can be used to study various mathematical structures, such as rings, groups, and modules, in a unified framework
• The internal logic of a topos can be intuitionistic or classical, depending on the properties of the topos
  • In an intuitionistic topos, the law of excluded middle may not hold, allowing for constructive reasoning
  • In a Boolean topos, the subobject classifier is isomorphic to a two-element set, and the internal logic is classical
• Topoi provide a foundation for studying sheaf theory and cohomology in a general setting
  • Sheaves on a topological space can be viewed as objects in a topos of sheaves
  • Cohomology theories can be defined using the language of topoi and geometric morphisms

Geometric Morphisms Explained

• A geometric morphism $f: \mathcal{F} \to \mathcal{E}$ between topoi consists of a pair of adjoint functors $f_* \dashv f^*$
  • The direct image functor $f_*: \mathcal{F} \to \mathcal{E}$ preserves colimits and finite limits
  • The inverse image functor $f^*: \mathcal{E} \to \mathcal{F}$ preserves all limits and is left exact
• Geometric morphisms preserve the logical and geometric structure of topoi
  • The inverse image functor $f^*$ preserves the subobject classifier and the internal logic
  • The direct image functor $f_*$ preserves the geometric structure, such as exponential objects and power objects
• Geometric morphisms can be composed, forming a category of topoi and geometric morphisms
• Examples of geometric morphisms include:
  • The global sections morphism $\Gamma: \mathcal{E} \to \mathbf{Set}$, where $\Gamma^*$ is the constant sheaf functor and $\Gamma_*$ is the global sections functor
  • The stalk morphism at a point $x: \mathbf{Set} \to \mathcal{E}$, where $x^*$ evaluates a sheaf at the point $x$
• Geometric morphisms can be used to study the relationships between different topoi and their associated mathematical structures
  • For example, a geometric morphism between the topos of sheaves on a topological space and the topos of sets can be used to study the cohomology of the space

Connections and Applications

• Topos theory provides a unified language for studying various branches of mathematics, including:
  • Algebraic geometry: topoi can be used to study schemes and their cohomology
  • Algebraic topology: topoi provide a framework for studying homotopy theory and generalized cohomology theories
  • Mathematical logic: topoi allow for the study of different logical systems and their relationships
  • Category theory: topoi are a central object of study in category theory, providing insights into the nature of mathematical structures
• Applications of topos theory include:
  • Synthetic differential geometry: a topos-theoretic approach to studying smooth manifolds and infinitesimals
  • Quantum mechanics: topoi can be used to provide a categorical foundation for quantum mechanics and quantum logic
  • Computer science: topoi have applications in type theory, domain theory, and the semantics of programming languages
• Topos theory has connections to other areas of mathematics, such as:
  • Model theory: topoi can be used to construct models of various logical theories
  • Number theory: the study of Grothendieck topoi arising from schemes is closely related to arithmetic geometry
  • Mathematical physics: topoi provide a framework for studying quantum field theories and their symmetries

Examples and Exercises

• Example: The category of sets $\mathbf{Set}$ is a topos
  • The subobject classifier in $\mathbf{Set}$ is the two-element set $\{0, 1\}$
  • The power object of a set $A$ is the power set $\mathcal{P}(A)$
  • The internal logic of $\mathbf{Set}$ is classical
• Example: The category of sheaves $\mathbf{Sh}(X)$ on a topological space $X$ is a topos
  • The subobject classifier in $\mathbf{Sh}(X)$ is the sheaf of open sets
  • The exponential object $B^A$ is the sheaf of continuous functions from $A$ to $B$
  • The internal logic of $\mathbf{Sh}(X)$ is intuitionistic, unless $X$ is a discrete space
• Exercise: Show that the category of presheaves $\mathbf{PSh}(C)$ on a small category $C$ is a topos
  • Construct the subobject classifier and power objects in $\mathbf{PSh}(C)$
  • Determine the internal logic of $\mathbf{PSh}(C)$
• Exercise: Given a geometric morphism $f: \mathcal{F} \to \mathcal{E}$ between topoi, show that the inverse image functor $f^*$ preserves the subobject classifier and the internal logic
  • Use the adjunction $f_* \dashv f^*$ to prove the preservation of the subobject classifier
  • Show that $f^*$ preserves the logical connectives and quantifiers

Common Pitfalls

• Confusing topoi with topological spaces: while topoi generalize some properties of topological spaces, they are distinct concepts with different axioms and structures
• Overlooking the importance of the subobject classifier: the subobject classifier is a crucial component of a topos, enabling the internalization of logic and the study of subobjects
• Misunderstanding the role of geometric morphisms: geometric morphisms are not just functors between topoi, but they preserve the logical and geometric structure through the adjunction $f_* \dashv f^*$
• Assuming that all topoi have classical logic: the internal logic of a topos can be intuitionistic or classical, depending on the properties of the topos and the subobject classifier
• Neglecting the connection between topoi and sheaves: many important examples of topoi arise from categories of sheaves on a site, and understanding this connection is crucial for applying topos theory to other areas of mathematics
• Overcomplicating the concepts: while topos theory can be abstract and technical, focusing on the core ideas and examples can help build intuition and understanding
• Ignoring the categorical perspective: topos theory is deeply rooted in category theory, and understanding the categorical concepts and techniques is essential for working with topoi and geometric morphisms effectively

Further Reading

• "Sheaves in Geometry and Logic: A First Introduction to Topos Theory" by Saunders Mac Lane and Ieke Moerdijk
  • A classic textbook that provides a gentle introduction to topos theory, focusing on the connection between sheaves and logic
• "Topoi: The Categorial Analysis of Logic" by Robert Goldblatt
  • A comprehensive textbook that covers the foundations of topos theory, including the internal logic, subobject classifiers, and geometric morphisms
• "Sketches of an Elephant: A Topos Theory Compendium" by Peter T. Johnstone
  • A multi-volume treatise that provides an in-depth exploration of topos theory, covering a wide range of topics and applications
• "The Elephant in the Topos" by Olivia Caramello
  • A series of lecture notes that introduce topos theory from a categorical perspective, emphasizing the connections between topoi, sites, and geometric theories
• "Topos Theory" by Peter T. Johnstone
  • A concise introduction to topos theory, focusing on the categorical aspects and the connections to mathematical logic
• "Theories, Sites, Toposes: Relating and Studying Mathematical Theories Through Topos-Theoretic 'Bridges'" by Olivia Caramello
  • A research monograph that explores the relationships between mathematical theories using topos-theoretic techniques, such as the theory of classifying topoi and geometric morphisms
• "From Sets to Types to Categories to Sets" by John L. Bell
  • An article that provides an overview of the connections between set theory, type theory, and category theory, with a focus on the role of topoi in unifying these different perspectives