7.3 Presheaf topoi and functor categories
2 min readโขjuly 25, 2024
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
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- Presheaf maps objects and morphisms in small category C to sets and functions in Set preserves composition and identity morphisms in opposite direction (functors )
- Presheaf topos encompasses presheaves on C as objects and natural transformations between presheaves as morphisms (category )
- 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
- 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:
- Construct finite products componentwise
- Form exponential objects using natural transformations
- Identify terminal object as constant functor
- Build equalizers pointwise
- 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 ฮฉ in 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 โ ฮฉ maps terminal object to maximal sieve represents "truth" in the topos
- Characteristic morphism ฯ_m: X โ ฮฉ for subobject m: Y โ X encodes membership information for each element x in X(c) ฯ_m(c)(x) yields sieve of all f: d โ 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 set-based topoi in axiomatic set theory and topos-theoretic approaches to quantum mechanics
Key Terms to Review (18)
Adjunctions in Topoi: Adjunctions in topoi refer to a pair of functors between two categories that are connected by a natural isomorphism, which allows for a deep relationship between them. This concept plays a vital role in understanding how different topoi can be related through the lens of functor categories and presheaf topoi, establishing a framework for the interaction between objects and morphisms in these structures.
Algebraic varieties as sheaves: Algebraic varieties as sheaves refer to the mathematical objects that encapsulate the concept of solutions to polynomial equations, representing geometric shapes in algebraic geometry. These varieties can be studied using the language of sheaves, which provide a way to systematically handle local data associated with the points of the variety and their relations. This connection allows for a deeper understanding of the structure of varieties by applying tools from category theory and topology, particularly through the framework of presheaf topoi and functor categories.
Categorical logic: Categorical logic is a type of logic that focuses on the relationships between categories rather than individual objects. It provides a framework for reasoning about objects and their properties using diagrams and categorical statements, which connect different categories through functors and natural transformations. This concept plays a crucial role in understanding the structure of presheaf topoi and their relationship to functor categories, as well as how these ideas compare with elementary topoi.
Coherent Sheaves: Coherent sheaves are a specific type of sheaf that are both finitely generated and have the property that every finitely generated submodule of their stalks is coherent. This concept is crucial in the study of algebraic geometry and topos theory, as it allows for the generalization of properties from algebraic varieties to more abstract settings. They are particularly important when working with schemes and the notion of local properties, as they ensure that geometric intuition can be translated into algebraic terms.
Colimits in functor categories: Colimits in functor categories are universal constructions that generalize the idea of 'gluing together' objects and morphisms in a category through a functor. They allow us to build new objects from existing ones while preserving structure, similar to how limits work but in a dual way. This concept is essential for understanding how various constructions like coequalizers, coproducts, and more arise when working with presheaf topoi and functor categories.
Equivalence of Categories: Equivalence of categories is a concept that describes when two categories are 'the same' in a certain formal sense, meaning there exists a pair of functors that are inverses up to natural isomorphism. This notion connects deeply with natural transformations, adjunctions, and more complex structures like presheaf topoi and Grothendieck topologies, allowing mathematicians to translate problems and results between different categorical frameworks while preserving their essential properties.
Finite Sheaves: Finite sheaves are mathematical structures that associate a finite set of data to each open subset of a topological space, providing a way to manage local information globally. These sheaves play a significant role in algebraic geometry and topology by allowing us to work with sections over various open sets while ensuring that the data behaves well with respect to restrictions and gluing conditions.
Functor Category: A functor category is a category whose objects are functors from one category to another, and whose morphisms are natural transformations between these functors. This concept plays a crucial role in connecting different areas of category theory, particularly in understanding how structures behave under transformation and relating them through naturality. Functor categories provide a framework for applying the Yoneda lemma, which allows us to study objects in terms of their relationships with other objects.
Grothendieck's construction: Grothendieck's construction is a method used to create a category from a functor that assigns to each object of one category a fiber category over that object. This concept is crucial for understanding how different categories can be related through functors and helps in building presheaf topoi by associating sheaves with varying base categories. It serves as a bridge between category theory and the study of sheaves, enabling the manipulation and understanding of structures in algebraic geometry and beyond.
Homotopy Theory: Homotopy theory is a branch of algebraic topology that studies spaces up to continuous deformations, called homotopies. It focuses on understanding the properties of topological spaces that are invariant under such deformations, leading to insights about their structure and relationships. This concept connects deeply with various mathematical structures, providing a framework for analyzing completeness in categories, functorial relationships in presheaf topoi, and the interactions between differential geometry and topology.
Limits in presheaf topoi: Limits in presheaf topoi refer to the construction of universal objects that represent the common behavior of diagrams of presheaves over a given category. These limits allow mathematicians to analyze and connect various presheaves, providing insight into their structure and relationships. This concept is essential for understanding how different presheaves can be combined or compared in a categorical context.
Lurie's Theory of Higher Topoi: Lurie's Theory of Higher Topoi is a framework in category theory that extends classical topos theory by introducing higher-dimensional structures, focusing on the relationships between categories and their higher categorical analogs. This theory enables mathematicians to understand complex structures by treating them as 'spaces' of morphisms between objects, providing a deeper insight into both homotopy theory and sheaf theory, particularly through the notion of presheaf topoi.
Morphism of topoi: A morphism of topoi is a structure-preserving map between two topoi, which consists of a pair of functors that relate their underlying categories while preserving the categorical properties and the structure of the sheaves. This concept is crucial when studying how different topoi interact and allows for the transfer of properties and constructions between them, making it an essential part of the framework in category theory.
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.
Set-valued presheaf topos: A set-valued presheaf topos is a category formed by the collection of set-valued functors defined on a small category, together with natural transformations between these functors. This concept is crucial in understanding how local data can be organized globally within a topological space, reflecting a structure that allows for the manipulation and analysis of sets associated with each object in the category. It establishes a bridge between categorical logic and topology, enabling the study of sheaves and their applications.
Sheaf topos: A sheaf topos is a category of sheaves over a topological space or a site, where objects are sheaves and morphisms are natural transformations between them. This structure allows for the study of local properties and provides a way to connect topological ideas with categorical concepts, leading to powerful results in both algebraic geometry and logic. Sheaf topoi play a vital role in understanding the behavior of functions and spaces through the lens of categories.
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.
Topological Spaces as Presheaves: Topological spaces can be viewed as presheaves over the category of open sets, where each open set is assigned a set of sections (often functions or continuous maps) that behave well under restriction. This perspective highlights how the structure of a topological space can be understood through the lens of category theory, particularly in relation to presheaf topoi and functor categories, enabling a richer understanding of continuity and limits.