10.3 Applications to algebraic geometry and model theory
2 min read•july 25, 2024
Classifying topoi bridge algebraic geometry and model theory, unifying diverse mathematical structures. They encode geometric objects like schemes and algebraic spaces as models of theories, revealing deep connections between algebra, geometry, and logic.
These topoi provide a powerful framework for studying mathematical theories and their models. By representing theories as categories of sheaves, they offer new insights into the nature of mathematical structures and their relationships.
Classifying Topoi in Algebraic Geometry and Model Theory
Schemes as classifying topoi
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- Schemes generalize algebraic varieties unify affine and projective varieties
- Locally ringed spaces glue together affine schemes (SpecA, OSpecA) where A is a commutative ring
- Geometric theory of schemes involves locally ringed spaces satisfying specific axioms (gluing, local nature)
- Zariski topology on schemes arises from prime ideals of rings reflects algebraic properties geometrically
- Classifying topos of a scheme encodes points as models of its geometric theory (Spec )
Algebraic spaces as classifying topoi
- Algebraic spaces generalize schemes allow for quotients by group actions not representable by schemes
- Étale topology refines Zariski topology captures local properties more effectively (formal étaleness)
- Sheaves on the étale of a scheme define algebraic spaces (étale equivalence relations)
- Geometric theory of algebraic spaces involves étale descent conditions and local properties
- Classifying topos of an algebraic space represents models of its associated geometric theory
Existence of classifying topoi
- Coherent theories use finitary logic allow finite conjunctions, disjunctions, and existential quantification
- Geometric theories extend coherent theories include infinitary disjunctions capture more general properties
- Syntactic site construction:
- Form category of formulas and provable implications
- Define Grothendieck topology using covering families
- Take sheaves on this site as the classifying topos
- Universal model in classifying topos represents generic model of the theory
- Completeness theorem: every model of the theory corresponds to a point of the classifying topos
- Soundness theorem: logical consequences in the theory reflected by geometric morphisms between topoi
Classifying topoi in geometry vs model theory
- Geometric morphisms between classifying topoi correspond to interpretations between theories
- Varieties as models of geometric theories (equations, inequations in polynomial rings)
- Zariski spectra of rings form classifying topoi for theories of local rings with given global sections
- Categoricity in model theory relates to properties of classifying topoi (e.g., two-valued)
- Stability theory connects to topos-theoretic notions (e.g., coherent objects in the classifying topos)
- Stone duality generalizes to classify Boolean algebras, Heyting algebras, and certain topoi
Unifying role of classifying topoi
- Topos theory provides common language for algebraic geometry and model theory (sites, sheaves, geometric morphisms)
- Grothendieck topologies generalize classical topological spaces unify various notions of "space" (étale, fppf, crystalline)
- Categorical logic interprets mathematical theories within topoi allows for alternative foundations
- Synthetic differential geometry develops smooth infinitesimal analysis in certain topoi (well-adapted models)
- Applications extend to algebraic topology (homotopy types as ∞-topoi), functional analysis (C*-algebras as ringed topoi)
Key Terms to Review (18)
Categorical equivalence: Categorical equivalence refers to a relationship between two categories where there exist functors that establish a one-to-one correspondence between their objects and morphisms, preserving structure. This concept is crucial because it allows mathematicians to treat different categories as if they are the same in terms of their structural properties, enabling the transfer of knowledge and results between seemingly distinct mathematical frameworks.
Categorical topos: A categorical topos is a category that behaves like the category of sets and possesses certain properties, allowing for the definition of concepts such as limits, colimits, and exponentials within it. This structure not only extends set-theoretic concepts but also facilitates connections between different mathematical fields, such as algebraic geometry and model theory, by providing a framework in which these concepts can be studied and compared.
Cohomology: Cohomology is a mathematical concept that studies the properties of spaces through algebraic invariants derived from their topological structure. This approach captures important features such as holes or gaps in the space, allowing mathematicians to classify spaces and relate them to algebraic objects. Cohomology has significant applications in various fields, including algebraic geometry and model theory, where it helps connect geometric properties with algebraic solutions.
Colimits: Colimits are a fundamental concept in category theory that generalize the idea of 'gluing together' objects and morphisms to form a new object. They allow for the construction of an object that captures the collective behavior of a diagram of objects, including their relationships defined by morphisms. Colimits can be thought of as a way to encapsulate the data from various objects and morphisms into a single entity, making them essential in many areas like algebraic topology and sheaf theory.
Definable sets: Definable sets are collections of mathematical objects that can be specified or described using a particular language or logical framework. They play a crucial role in understanding the structure of mathematical theories, particularly in model theory and algebraic geometry, as they allow for the identification and classification of sets based on specific properties or relationships.
Elementary Topos: An elementary topos is a category that behaves like the category of sets, providing a framework for doing set theory in a categorical context. It possesses certain properties such as having all finite limits and colimits, and a subobject classifier that allows for a notion of 'truth' in the topos. This structure connects deeply with universal properties, functors, algebraic geometry, and logical frameworks, making it a cornerstone concept in various mathematical applications.
Functor: A functor is a mathematical mapping between categories that preserves the structure of those categories, meaning it maps objects to objects and morphisms to morphisms in a way that respects the composition and identity of the categories. Functors play a crucial role in connecting different mathematical structures and help in defining various concepts such as natural transformations and limits.
Grothendieck topos: A Grothendieck topos is a category that behaves like the category of sheaves on a topological space, providing a general framework for sheaf theory in algebraic geometry and beyond. It captures the notion of 'space' and 'sheaf' in a categorical way, linking various areas of mathematics such as geometry, logic, and model theory through universal properties and representable functors.
Internal logic: Internal logic refers to the system of logical reasoning that operates within a topos, allowing one to interpret and reason about objects and morphisms in a way that is consistent with the categorical structure of the topos. This concept connects the external properties of a topos with its internal relationships, revealing how mathematical truths can be established within its framework.
Limits: In category theory, limits provide a way to generalize the concept of 'convergence' found in calculus, allowing one to find a universal object that represents the 'best approximation' of a diagram of objects and morphisms. This notion connects closely with colimits and their properties, offering insights into how structures can be constructed and analyzed within categories.
Natural Transformation: A natural transformation is a way of transforming one functor into another while preserving the structure of the categories involved. It consists of a collection of morphisms that relate the outputs of two functors at each object in the source category, ensuring coherence across all morphisms in that category. This concept links various areas of category theory, such as functor categories and representable functors, through its universal properties and its application in understanding limits and colimits.
Peter Johnstone: Peter Johnstone is a prominent mathematician known for his influential work in category theory and topos theory, which have shaped the understanding of concepts like adjoint functors and subobject classifiers. His contributions provide deep insights into the structure of topoi and their applications in various mathematical fields, including algebraic geometry and logic.
Presheaf Topos: A presheaf topos is a category of presheaves on a small category, which serves as a generalized framework for discussing sheaves in topology and algebraic geometry. This concept allows mathematicians to study the relationships between different mathematical structures and the morphisms between them in a flexible and powerful way. The presheaf topos plays a crucial role in understanding how geometric morphisms operate and has important applications in various areas, including model theory.
Sheaf: A sheaf is a mathematical tool that captures local data attached to the open sets of a topological space and allows for the gluing of this data to form global sections. Sheaves play a crucial role in connecting local properties of spaces to global properties, and they serve as a foundational concept in various areas such as algebraic geometry, topology, and logic.
Sheaf Semantics: Sheaf semantics is a framework that uses sheaves to give a precise meaning to logical languages and structures, especially in the context of categorical logic and topos theory. It allows for the interpretation of logical formulas in a way that reflects how local data can be consistently patched together to form global data. This concept connects deeply with various areas such as algebraic geometry, model theory, internal languages of topoi, and applications in computer science and logic.
Site: In the context of topos theory, a site is a category equipped with a Grothendieck topology, which allows the definition of sheaves and their associated properties. A site serves as a framework for understanding the relationships between various categories and can be thought of as a generalized space where morphisms and coverings dictate how local data can be glued together to form global objects. This concept is essential for establishing connections between various mathematical disciplines, particularly in the study of sheaf theory, algebraic geometry, and higher categorical structures.
Topos of sheaves: A topos of sheaves is a category that arises from the study of sheaves on a topological space, capturing the behavior of sheaves and their relationships in a structured way. It provides a framework that generalizes set theory, allowing for the manipulation of sheaves similar to how one would handle sets. This concept is crucial for understanding the interplay between topology and algebraic structures, particularly in the realms of geometry and logic.
William Lawvere: William Lawvere is a prominent mathematician known for his foundational work in category theory and topos theory, which has profoundly influenced modern mathematics. His contributions include the development of the concept of elementary toposes and the introduction of categorical logic, bridging the gap between abstract mathematics and practical applications in various fields.