Glossary

2.1 Types of functors: covariant, contravariant, and bifunctors

2 min read•july 25, 2024

Functors are the bridges between categories, allowing us to translate structures and relationships. Covariant functors preserve direction, contravariant functors reverse it, and bifunctors take two inputs, each playing a unique role in .

These functor types are crucial for understanding how mathematical structures relate. From preserving group structures to reversing set inclusions and combining vector spaces, functors provide powerful tools for exploring and connecting diverse mathematical concepts.

Understanding Functor Types

Types of functors

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  • Covariant functors map between categories preserving direction of original morphisms maintain composition and identity morphisms denoted as F:C→DF: C \to D ()
  • Contravariant functors reverse direction of morphisms between categories invert composition denoted as F:Cop→DF: C^{op} \to D ()
  • Bifunctors take two arguments from two categories map to a third category denoted as F:C×D→EF: C \times D \to E ()

Examples of functor behavior

  • : Forgetful functor from groups to sets preserves structure of morphisms maps group homomorphisms to functions between underlying sets
  • : Power set functor reverses subset inclusions maps larger sets to smaller power sets
  • : Tensor product of vector spaces takes two vector spaces produces new vector space with dimension equal to product of input dimensions

Properties of functor types

  • Covariant functors preserve direction of morphisms maintain categorical structure respect composition and identities ()
  • Contravariant functors reverse arrows useful for dual constructions turn colimits into limits ()
  • Bifunctors generalize binary operations often used to define tensor products provide framework for ()

Applications in category theory

  • Covariant functors:
    1. Construct free objects
    2. Define adjunctions
    3. Build universal constructions ()
  • Contravariant functors:
    1. Define
    2. Work with opposite categories
    3. Study (Hom functors)
  • Bifunctors:
    1. Define bilinear maps
    2. Construct product and coproduct functors
    3. Study (Tensor product)

Key Terms to Review (20)

Adjunction: Adjunction is a fundamental concept in category theory that describes a special relationship between two functors, where one functor can be seen as a left adjoint and the other as a right adjoint. This relationship highlights how objects and morphisms in one category correspond to objects and morphisms in another category, allowing for the transfer of structure and properties between them. Adjunctions often reveal deep connections between different mathematical structures and can be instrumental in constructing exponential objects, sheafification processes, and expressing concepts within specialized languages.
Bifunctor: A bifunctor is a mathematical concept that takes two categories and provides a mapping from pairs of objects in those categories to a third category, while also respecting the structure of the categories involved. This means that a bifunctor operates on two different types of inputs simultaneously, allowing for more complex relationships and interactions than a standard functor, which only deals with one category at a time. By providing this duality, bifunctors are essential in various mathematical contexts, particularly in the study of natural transformations and adjunctions.
Bilinear Maps: Bilinear maps are functions that take two inputs from two different vector spaces and produce a single output in a third vector space, satisfying linearity in each argument separately. This means that if you fix one argument and vary the other, the map behaves linearly, and vice versa. Bilinear maps connect closely with concepts like dual spaces and tensor products, making them essential in understanding structures in algebra and geometry.
Cartesian Product: The Cartesian product is a mathematical operation that combines two sets to form a new set containing all possible ordered pairs of elements from the original sets. In the context of functors, it can illustrate how covariant, contravariant, and bifunctors interact with multiple sets or categories, showing the relationships and transformations between them.
Category Theory: Category theory is a mathematical framework that deals with abstract structures and relationships between them, focusing on the concept of objects and morphisms. It provides a way to formalize mathematical concepts across various fields, emphasizing the connections and mappings between different structures rather than their individual components. This abstraction is crucial for understanding complex relationships in mathematics, including transformations through functors, the properties of isomorphisms, and connections to logic and foundational mathematics.
Contravariant Functor: A contravariant functor is a type of mapping between categories that reverses the direction of morphisms, taking objects from one category to another while flipping the arrows. This means that if there is a morphism from object A to object B in the original category, a contravariant functor will map these objects to another morphism going from the image of B back to the image of A. Understanding contravariant functors is crucial for grasping how relationships between different mathematical structures can be modeled and transformed.
Coproduct Functor: The coproduct functor is a construction in category theory that generalizes the notion of a disjoint union or sum of objects within a category. It plays a crucial role in understanding how objects can be combined and interacts with other types of functors, particularly covariant and contravariant functors, to create new objects while preserving the structure and relationships of the original entities.
Covariant Functor: A covariant functor is a type of mapping between categories that preserves the direction of morphisms. In simpler terms, if you have a morphism (or arrow) from one object to another in the first category, a covariant functor will map that morphism to a morphism between the corresponding objects in the second category, keeping the same direction. This concept ties into how we understand morphisms and isomorphisms, as well as how different types of functors interact with natural transformations and help us explore functor categories and the Yoneda lemma.
Duality Principles: Duality principles are fundamental concepts in mathematics and logic that highlight a correspondence between two seemingly different structures or theories, often revealing how properties can be transformed by reversing certain relations. In category theory, particularly when dealing with functors, duality principles demonstrate that for every statement or theorem, there exists a dual statement that is equally valid, which often involves interchanging objects and morphisms. This concept plays a crucial role in understanding the behavior of covariant, contravariant, and bifunctors.
Forgetful Functor: A forgetful functor is a type of functor that 'forgets' some structure or properties of the objects and morphisms it maps between categories, essentially providing a way to relate different categories while losing some information. It often connects categories that have a more complex structure to simpler ones, making it easier to work with and understand the relationships between various mathematical constructs.
Functoriality: Functoriality refers to the property of a functor that maps morphisms in one category to morphisms in another category in a way that preserves the structure of the categories. This means that if there is a morphism between objects in the first category, the functor will produce a corresponding morphism between the images of those objects in the second category, maintaining composition and identity. Functoriality connects various mathematical concepts and structures, illustrating how they interact through mappings.
Hom Functor: The Hom functor is a fundamental concept in category theory that assigns to each pair of objects in a category a set of morphisms between them. It plays a crucial role in understanding relationships between objects and morphisms within categories, establishing a bridge between different structures, especially when considering covariant and contravariant functors.
Homology Functor: The homology functor is a mathematical construct that associates a sequence of abelian groups or modules to a topological space, providing algebraic invariants that capture its shape and structure. This functor plays a critical role in algebraic topology, transforming geometric data into algebraic objects that can be analyzed and compared, and is deeply connected to the concepts of covariant and contravariant functors.
Monoidal Categories: Monoidal categories are a type of category equipped with a tensor product, which allows for the combination of objects and morphisms in a way that respects the category's structure. They include an identity object and a set of natural isomorphisms that express the associative and unital properties of the tensor product. This concept is crucial when exploring functor types, as they can act in structured ways on the objects and morphisms, and play a significant role in understanding adjunctions through their ability to express relationships between different categories.
Naturality: Naturality is a property of certain mathematical constructions, particularly in category theory, where a transformation or a morphism can be shown to commute with other structures in a natural way. It emphasizes that such transformations do not depend on arbitrary choices and behave consistently across different contexts, making them more universally applicable. In the realm of functors, natural transformations highlight how functorial relationships are maintained, while adjunctions illustrate naturality in the context of units and counits, showcasing their integral role in the structure of categories.
Power Set Functor: The power set functor is a mathematical concept that assigns to each set a new set containing all possible subsets of the original set, including the empty set and the set itself. This functor can be viewed as a covariant functor because it preserves the direction of morphisms between sets, reflecting how functions between sets relate to their power sets. The power set functor is an essential example in category theory that illustrates how functors operate, especially in discussions about adjunctions, where it often serves as one half of a pair of adjoint functors.
Product Functor: A product functor is a specific type of functor that takes two categories and produces their product in a way that respects the structure of both categories. It essentially combines objects and morphisms from two categories into a new category where objects are pairs of objects and morphisms are pairs of morphisms, thus creating a categorical product. This concept connects deeply to the ideas of covariant and contravariant functors, as well as adjunctions and exponential objects, showcasing how structures can be built from simpler components.
Representable Functors: Representable functors are a special type of functor that can be expressed in terms of hom-sets, specifically as a set of morphisms from a fixed object in a category to other objects in that category. This property allows for a deeper understanding of the structure of categories and how objects relate to one another through morphisms, linking representable functors to covariant and contravariant functors.
Tensor Product: The tensor product is a construction that combines two vector spaces into a new vector space, encapsulating the idea of bilinearity. It is a way to express relationships between vectors from different spaces, allowing for the creation of multilinear maps. This concept is crucial when dealing with functors, particularly bifunctors, which take two categories as input and yield another category, thus linking the idea of tensor products to various functor types.
Yoneda embedding: The Yoneda embedding is a fundamental concept in category theory that allows one to represent objects of a category as functors from that category to the category of sets. This construction establishes a deep connection between objects and morphisms, illustrating how every object can be viewed in terms of its relationships with all other objects through natural transformations and functors.
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