5.3 Examples of cartesian closed categories
3 min read•july 25, 2024
Cartesian closed categories are a fundamental concept in category theory, combining terminal objects, binary products, and . They provide a powerful framework for understanding function spaces and in various mathematical contexts.
theory and category theory both showcase cartesian closed structures. The category of sets (Set) and small categories () demonstrate these properties, with unique features like function spaces and functor categories serving as exponential objects.
Cartesian Closed Categories in Set Theory and Category Theory
Cartesian closure of sets
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- Definition of a cartesian encompasses existence of terminal object representing final object in category allowing unique morphism from any object existence of binary products allowing pairing of objects and existence of exponential objects enabling function objects
- Category of sets (Set) exemplifies cartesian closed structure with singleton set as terminal object allowing unique function from any set cartesian product of sets as binary products and function space for sets A and B as exponential objects
- Exponential objects in Set comprise function space for sets A and B establishing bijection between and
- Natural isomorphism for exponential objects formulated as demonstrates correspondence between functions of multiple arguments and curried functions
- Verification of cartesian closed axioms for Set involves proving existence of terminal object binary products and exponential objects along with natural isomorphisms
Cartesian closure of small categories
- Category of small categories (Cat) exhibits cartesian closed structure with terminal object as category containing one object and one morphism
- Binary products in Cat constructed as product of categories with objects as pairs from constituent categories and morphisms as pairs of morphisms from constituent categories
- Exponential objects in Cat manifest as functor category for small categories C and D with objects as functors from D to C and morphisms as natural transformations between functors
- Verification of cartesian closed axioms for Cat involves proving existence of terminal object binary products and exponential objects along with natural isomorphisms
- Natural isomorphism for exponential objects in Cat demonstrates correspondence between functors and natural transformations in product and exponential categories
Examples of cartesian closed categories
- Category of topological spaces () with objects as topological spaces and morphisms as continuous functions
- Cartesian closed subcategories of Top include category of compactly generated Hausdorff spaces and category of locally compact Hausdorff spaces
- Category of vector spaces over a field exhibits cartesian closed structure (finite-dimensional vector spaces)
- Category of modules over a commutative ring demonstrates cartesian closed properties (finitely generated projective modules)
- Cartesian closed categories in computer science encompass category of domains (continuous lattices) and category of complete partial orders (CPOs)
- Topos theory connects to cartesian closed categories with elementary toposes as cartesian closed categories possessing subobject classifier
Properties of exponential objects
- General properties of exponential objects include adjunction between product and exponential functors and currying/uncurrying operations
- Exponential objects in Set manifest as set of functions between two sets with evaluation and co-evaluation maps
- Exponential objects in Cat take form of functor categories with natural transformations as morphisms
- Exponential objects in topological spaces utilize compact-open topology on function spaces ensuring continuity of evaluation map
- Universal property of exponential objects guarantees uniqueness up to isomorphism and preservation under equivalence of categories
- Exponential preserving functors maintain structure of exponential objects between categories relating to cartesian closed functors
- Applications of exponential objects appear in higher-order functions in programming languages and internal language of cartesian closed categories
Key Terms to Review (18)
Adjoint Functor Theorem: The Adjoint Functor Theorem states that a functor between two categories has a left adjoint if and only if it preserves all small limits, and it has a right adjoint if and only if it preserves all small colimits. This theorem connects the concepts of limits and colimits in category theory with the existence of adjoint functors, providing a powerful framework for understanding how different mathematical structures relate to one another.
Cat: In category theory, a 'cat' (short for category) is a mathematical structure consisting of objects and morphisms (arrows) between those objects that satisfy specific composition and identity laws. This structure allows for the representation of relationships and transformations within various mathematical contexts, including cartesian closed categories where products and exponentials can be defined.
Closed Category: A closed category is a type of category that has a certain structure allowing for the existence of internal hom-objects. Specifically, it means that for any two objects A and B in the category, there is an object denoted as [A, B], which represents the morphisms from A to B. This concept is crucial in understanding cartesian closed categories, where product and exponential structures coexist, enabling a rich framework for functional programming and type theory.
Colimit: A colimit is a universal construction in category theory that generalizes the concept of taking a limit of a diagram of objects and morphisms, allowing for the 'gluing' together of objects in a category. It serves as a way to define the 'largest' object that can be mapped into all objects in a given diagram, capturing the idea of combining various structures in a coherent way.
Curry and Uncurry: Curry and uncurry are concepts that relate to the manipulation of functions in the context of cartesian closed categories, particularly involving transformations between different forms of function representation. Curry transforms a function that takes multiple arguments into a sequence of functions, each taking a single argument, while uncurry converts a curried function back into a function that takes a tuple of arguments. These operations highlight the structure and behavior of functions within the framework of category theory.
Currying: Currying is a mathematical concept where a function that takes multiple arguments is transformed into a series of functions, each taking a single argument. This technique allows for partial application of functions, enabling the creation of new functions by fixing some arguments while leaving others free for later specification. Currying connects deeply with the structure of cartesian closed categories, allowing for a flexible handling of morphisms and their arguments.
Diagonal Functor: A diagonal functor is a specific type of functor that maps an object from a category to a product of that object with itself, effectively representing the 'diagonal' inclusion of that object into a Cartesian product. This functor is important in establishing the structure of Cartesian closed categories, as it allows for the interpretation of certain morphisms and relations between objects, supporting key concepts such as exponential objects and natural transformations.
Evaluation morphism: An evaluation morphism is a specific type of morphism in category theory that captures the idea of applying a function to an argument. In the context of cartesian closed categories, it relates to exponential objects and provides a way to 'evaluate' these functions at specific points, effectively linking inputs and outputs within the structure of the category.
Exponential Objects: Exponential objects are a key concept in category theory that generalizes the notion of function spaces. Specifically, if a category has products, an exponential object represents the space of morphisms from one object to another, capturing the idea of 'functions' between objects in that category. This concept is foundational for understanding adjunctions, Cartesian closed categories, and presheaves, as it allows for a structured way to discuss how objects can relate to each other through mapping and transformation.
Hom-functor: The hom-functor is a type of functor that maps pairs of objects from a category to the morphisms between them. This construction helps in understanding relationships between objects, as it allows us to consider morphisms as elements in a set, making it easier to analyze transformations and adjunctions. Additionally, the hom-functor plays a crucial role in the structure of cartesian closed categories by connecting objects with their corresponding morphisms, facilitating the exploration of their properties and interactions.
Limit: In category theory, a limit is a universal construction that captures the idea of 'convergence' of objects and morphisms. It formalizes how objects can be combined or related through diagrams, providing a way to describe the most efficient or optimal way to represent a collection of objects and their relationships.
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.
Product morphism: A product morphism is a specific type of morphism in category theory that arises from the construction of products in a category. This morphism represents a way to combine objects and their relationships within the category, allowing for the simultaneous treatment of multiple objects as a single entity. In the context of Cartesian closed categories, product morphisms play a crucial role in illustrating how these categories handle the pairing of objects and the structure of morphisms.
Samuel Eilenberg: Samuel Eilenberg was a prominent mathematician known for his foundational work in category theory, topology, and algebra, particularly in the context of algebraic topology and topos theory. His contributions significantly advanced the understanding of categories, functors, and adjunctions, which are crucial concepts in modern mathematics.
Set: A set is a well-defined collection of distinct objects, considered as an object in its own right. In various mathematical contexts, sets can represent elements or points within categories, and they form the foundational building blocks for many structures, including functions and relations. Understanding sets is crucial for grasping more complex concepts such as morphisms, functors, and object properties within categorical frameworks.
Top: In category theory, a 'top' typically refers to a terminal object within a category, which is an object such that for every other object in the category, there exists a unique morphism leading to it. This concept is fundamental in understanding the structure of categories, as terminal objects play a crucial role in both the formation of functors and the construction of exponential objects, influencing how we interpret subobjects and their characteristic functions.
Tupling: Tupling is the process of combining two or more objects into a single composite object, often represented as an ordered pair or tuple. This concept is essential in understanding how morphisms in categories can be structured, particularly in cartesian closed categories, where tuples represent products that enable the construction of functions and relations between objects.
Yoneda Lemma: The Yoneda Lemma is a foundational result in category theory that relates functors to natural transformations, stating that every functor from a category to the category of sets can be represented by a set of morphisms from an object in that category. This lemma highlights the importance of morphisms and allows for deep insights into the structure of categories and functors.