4.4 Functor categories and natural transformations
3 min read•july 23, 2024
Functor categories take functors as objects and natural transformations as morphisms. They provide a framework for studying relationships between categories, with examples like ^C and ^C representing functors from a category C to sets or groups.
Natural transformations are the key to understanding functor categories. They map between functors, preserving structure through the naturality condition. Composition of natural transformations, both vertical and horizontal, allows for complex relationships between functors to be explored.
Functor Categories
Functor categories and examples
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A is a category whose objects are functors between two fixed categories C and D
The morphisms in a functor category are natural transformations between these functors
Examples of functor categories include:
SetC: functors from category C to the category of sets (Set)
GrpC: functors from category C to the category of groups (Grp)
[Top](https://www.fiveableKeyTerm:top)C: functors from category C to the category of topological spaces (Top)
[Vectk](https://www.fiveableKeyTerm:vectk)C: functors from category C to the category of vector spaces over a field k (Vectk)
Natural transformations as morphisms
In a functor category, the morphisms between functors F,G:C→D are natural transformations α:F⇒G
A α assigns to each X in C a αX:F(X)→G(X) in D
These morphisms must satisfy the naturality condition: for every morphism f:X→Y in C, the following diagram commutes:
F(X) --F(f)--> F(Y)
| |
αX αY
| |
v v
G(X) --G(f)--> G(Y)
The naturality condition ensures that the morphisms αX are compatible with the functorial actions of F and G
Composition of natural transformations
Natural transformations can be composed in two ways: vertically and horizontally
Vertical composition:
Given natural transformations α:F⇒G and β:G⇒H, their vertical composition β∘α:F⇒H is defined componentwise
For each object X in C, (β∘α)X=βX∘αX
The vertical composition of natural transformations is associative and has an identity (the identity natural transformation)
Horizontal composition:
Given natural transformations α:F⇒G and β:H⇒K where F,G:C→D and H,K:D→E, their horizontal composition β∗α:HF⇒KG is defined componentwise
For each object X in C, (β∗α)X=βG(X)∘H(αX)
The horizontal composition of natural transformations is also associative and has an identity
Relationships between functors
Natural transformations can establish various relationships between functors:
Isomorphism: functors F and G are naturally isomorphic (F≅G) if there exist natural transformations α:F⇒G and β:G⇒F such that β∘α=idF and α∘β=idG
Equivalence: functors F and G are naturally equivalent if they are naturally isomorphic up to isomorphism in the target category
Adjunction: functors F:C→D and G:D→C form an adjunction if there exist natural transformations η:idC⇒GF (unit) and ε:FG⇒idD (counit) satisfying certain conditions (triangle identities)
These relationships capture the idea of functors being "the same" or "similar" in various ways
Natural isomorphisms and equivalences
A is a natural transformation α:F⇒G where each component αX is an isomorphism in the target category D
The inverse of a natural isomorphism α is a natural transformation α−1:G⇒F with components (α−1)X=(αX)−1
Functors F and G are naturally isomorphic (F≅G) if there exists a natural isomorphism between them
Natural isomorphism is an equivalence relation on functors: it is reflexive, symmetric, and transitive
Functors F and G are naturally equivalent if they are naturally isomorphic up to isomorphism in the target category
Equivalence of functors is a weaker notion than natural isomorphism
Naturally equivalent functors share many categorical properties and are often considered "the same" in a broader sense
Key Terms to Review (19)
Commutative Diagram: A commutative diagram is a visual representation in category theory that illustrates how various objects and morphisms relate to one another through a series of paths that yield the same result regardless of the path taken. This concept serves as a powerful tool to express relationships between mathematical structures, showing how different compositions and mappings can lead to consistent outcomes.
Component of a natural transformation: A component of a natural transformation is a specific morphism that relates two functors between categories at a particular object. Each natural transformation consists of a collection of components, one for each object in the source category, which ensures that the transformation is consistent across the entire structure, maintaining the relationships defined by the functors. This concept ties into important ideas about the behavior of functors and how they interact through transformations, showcasing the elegance of categorical relationships.
Contravariant Functor: A contravariant functor is a type of functor that reverses the direction of morphisms when mapping between categories. Instead of mapping arrows in a category to arrows in another category directly, it maps arrows in the opposite direction, reflecting a form of duality that has important implications in various areas of mathematics.
Covariant Functor: A covariant functor is a mapping between categories that preserves the structure of the categories by associating each object in one category to an object in another category and each morphism in the first category to a morphism in the second category, maintaining the direction of morphisms. This concept is fundamental as it allows for the systematic translation of structures and relationships from one context to another, helping to illustrate connections across different mathematical frameworks.
Diagram Category: A diagram category is a type of category that represents a diagram, which consists of objects and morphisms arranged in a specified pattern. This framework allows for the organization and study of relationships among various objects in a structured way, facilitating the exploration of concepts like limits, colimits, and functors in category theory. Diagram categories are crucial for understanding how different categories can interact with one another through functorial mappings.
Functor Category: A functor category is a category whose objects are functors between two fixed categories and whose morphisms are natural transformations between these functors. This concept allows for a structured way to study collections of functors and their relationships, providing insight into how different categories can interact through mappings. It also plays a crucial role in understanding contravariant functors, representable functors, and presheaves, as it allows us to encapsulate the behavior of these mathematical structures in a categorical framework.
Functoriality of limits: Functoriality of limits refers to the property that allows limits in one category to be transformed into limits in another category through a functor. This means that if you have a diagram in one category and apply a functor to it, the limit of the diagram will correspond to the limit of the image of that diagram in the target category. This concept is crucial for understanding how structures and properties are preserved across different mathematical contexts.
Grp: In category theory, a 'grp' represents the category of groups, where the objects are groups and the morphisms are group homomorphisms. This concept serves as a fundamental example of a category that encapsulates both algebraic structures and their relationships, connecting to various mathematical contexts such as functors, limits, and colimits.
Morphism: A morphism is a structure-preserving map between two objects in a category, reflecting the relationships between those objects. Morphisms can represent functions, arrows, or transformations that connect different mathematical structures, serving as a foundational concept in category theory that emphasizes relationships rather than individual elements.
Natural Isomorphism: A natural isomorphism is a special type of natural transformation between two functors that establishes a one-to-one correspondence that is 'natural' in a specific sense. This means that it respects the structure of the categories involved, allowing the isomorphism to commute with morphisms, which is key in understanding relationships between different categories through functors.
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 family of morphisms that connect the objects in one category to their images in another category, ensuring that the relationships between the objects are maintained across different mappings. This concept ties together various important aspects of category theory, allowing mathematicians to relate different structures in a coherent manner.
Object: In category theory, an object is a fundamental entity that can be thought of as a point of focus within a category. Objects can represent various mathematical structures, such as sets, groups, or topological spaces, and they interact through morphisms that define relationships between them. Understanding objects is essential, as they form the building blocks of categories and facilitate the exploration of mathematical concepts in a unified way.
Preserve Composition: To preserve composition means that a functor maintains the structure of morphisms between categories, specifically ensuring that the image of the composition of two morphisms is the composition of their images. This concept is fundamental to understanding how functors operate between categories, as it guarantees that the relationships between objects and morphisms in one category are reflected in another.
Preserve identity morphisms: To preserve identity morphisms means that a functor, when mapping objects and morphisms between categories, must map each identity morphism in the source category to the corresponding identity morphism in the target category. This property is crucial because it ensures that the structure of identities is maintained across different categories, allowing for meaningful comparisons and transformations between them.
Samuel Eilenberg: Samuel Eilenberg was a prominent mathematician known for his foundational contributions to category theory, particularly through the development of key concepts that shape the field. His work laid the groundwork for understanding mathematical structures and their relationships, influencing areas like algebraic topology, algebra, and logic.
Saunders Mac Lane: Saunders Mac Lane was a prominent American mathematician, best known for his foundational work in category theory and for co-authoring the influential book 'Categories for the Working Mathematician.' His contributions helped to shape category theory as a unifying language for various mathematical disciplines and established the framework that connects diverse mathematical concepts.
Set: A set is a well-defined collection of distinct objects, considered as an object in its own right. In category theory, sets serve as the fundamental building blocks for constructing more complex mathematical structures, allowing for the exploration of relationships and mappings between different sets through functions.
Top: In category theory, a 'top' typically refers to a terminal object in a category, which is an object such that there is a unique morphism from any object in the category to this terminal object. The existence of terminal objects helps in defining limits and colimits, playing a crucial role in understanding the structure of categories.
Vect_k: The category vect_k is the category of vector spaces over a field k and linear transformations between them. This category serves as a fundamental example in category theory, highlighting the structure of objects and morphisms while illustrating various mathematical concepts such as functors, natural transformations, and monoidal categories.