Glossary

4.3 Natural transformations and functor categories

3 min read•august 7, 2024

Natural transformations are the secret sauce of functors, letting us compare them smoothly. They're like bridges between functors, with components that play nice with morphisms. Think of them as the glue that holds the functor world together.

Functor categories take things up a notch, treating functors as objects and natural transformations as morphisms. It's like a meta-category where functors themselves become the stars of the show. This setup lets us study functors in a whole new light.

Natural Transformations

Definition and Components

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  • η:F→G\eta: F \to G consists of a family of morphisms ηX:F(X)→G(X)\eta_X: F(X) \to G(X) for each object XX in the source category
  • Components of a natural transformation are the individual morphisms ηX\eta_X that make up the natural transformation
  • Components must satisfy the for every f:X→Yf: X \to Y in the source category: G(f)∘ηX=ηY∘F(f)G(f) \circ \eta_X = \eta_Y \circ F(f)
  • Naturality condition ensures that the components of the natural transformation are compatible with the morphisms in the source and target categories
  • Commutative diagram can be used to visualize the naturality condition

Natural Isomorphisms and Properties

  • is a natural transformation η:F→G\eta: F \to G where each component ηX\eta_X is an in the target category
  • Inverse of a natural isomorphism is also a natural transformation, denoted as η−1:G→F\eta^{-1}: G \to F
  • Identity natural transformation 1F:F→F1_F: F \to F has components (1F)X=1F(X)(1_F)_X = 1_{F(X)} for each object XX in the source category
  • Composition of natural transformations η:F→G\eta: F \to G and μ:G→H\mu: G \to H is defined componentwise: (μ∘η)X=μX∘ηX(\mu \circ \eta)_X = \mu_X \circ \eta_X
  • Natural transformations can be used to define (if there exist natural isomorphisms between two functors)

Functor Categories

Definition and Objects

  • CD\mathbf{C^D} has functors F:D→CF: \mathbf{D} \to \mathbf{C} as objects
  • Morphisms in the functor category are natural transformations between the functors
  • Identity functor 1D:D→D1_\mathbf{D}: \mathbf{D} \to \mathbf{D} serves as the identity object in the functor category
  • Constant functor ΔX:D→C\Delta_X: \mathbf{D} \to \mathbf{C} sends every object in D\mathbf{D} to a fixed object XX in C\mathbf{C} and every morphism in D\mathbf{D} to the identity morphism 1X1_X

Composition of Natural Transformations

  • Vertical composition of natural transformations η:F→G\eta: F \to G and μ:G→H\mu: G \to H is defined as μ∘η:F→H\mu \circ \eta: F \to H
  • Horizontal composition of natural transformations η:F→G\eta: F \to G in CD\mathbf{C^D} and θ:H→K\theta: H \to K in DE\mathbf{D^E} is a natural transformation η∗θ:F∘H→G∘K\eta * \theta: F \circ H \to G \circ K in CE\mathbf{C^E}
  • Horizontal composition is defined componentwise: (η∗θ)X=ηK(X)∘F(θX)(\eta * \theta)_X = \eta_{K(X)} \circ F(\theta_X)
  • Interchange law holds for vertical and horizontal composition: (μ∘η)∗(θ∘ϕ)=(μ∗θ)∘(η∗ϕ)(\mu \circ \eta) * (\theta \circ \phi) = (\mu * \theta) \circ (\eta * \phi)

Yoneda Lemma

Statement and Consequences

  • states that for any functor F:Cop→SetF: \mathbf{C^{op}} \to \mathbf{Set} and any object XX in C\mathbf{C}, there is a bijection between the set of natural transformations Nat(HomC(X,−),F)\text{Nat}(\text{Hom}_\mathbf{C}(X,-), F) and the set F(X)F(X)
  • Bijection is given by evaluating a natural transformation at the identity morphism of XX
  • Yoneda lemma implies that the functor HomC(X,−)\text{Hom}_\mathbf{C}(X,-) (represented functor) completely determines the object XX up to isomorphism
  • Yoneda embedding C→SetCop\mathbf{C} \to \mathbf{Set^{C^{op}}} sends each object XX to its represented functor HomC(X,−)\text{Hom}_\mathbf{C}(X,-) and each morphism f:X→Yf: X \to Y to the natural transformation HomC(f,−)\text{Hom}_\mathbf{C}(f,-)
  • Yoneda embedding is fully faithful, meaning that it preserves and reflects isomorphisms between objects and morphisms

Key Terms to Review (17)

Adjoint Functors: Adjoint functors are pairs of functors that establish a relationship between two categories, where one functor is left adjoint to the other and vice versa. This relationship captures how certain structures in one category can be transformed into structures in another, highlighting the interplay between concepts like limits and colimits. They are pivotal in various mathematical contexts, such as representing solutions to certain problems or bridging different areas of mathematics.
Category Theory: Category theory is a branch of mathematics that deals with abstract structures and the relationships between them, providing a unifying framework for understanding different mathematical concepts. It emphasizes the notion of objects and morphisms, where objects can represent mathematical structures, and morphisms represent the relationships or transformations between these structures. In the context of natural transformations and functor categories, category theory plays a crucial role in formalizing how different categories relate to each other through functors and how these functors can be transformed while preserving structure.
Component of a natural transformation: A component of a natural transformation is a specific morphism that connects the objects of two categories via the transformation itself, facilitating the relationship between two functors. Each component corresponds to an object in the source category, ensuring that the transformation is consistent across the structure of the categories involved. This consistency is key to understanding how different functors relate to each other through a natural transformation.
Contravariant Functor: A contravariant functor is a type of mapping between categories that reverses the direction of morphisms. It takes an object from one category and maps it to an object in another category while also reversing the arrows, meaning if there is a morphism from object A to object B, the functor will map this to a morphism from the image of B back to the image of A. This concept is crucial in understanding relationships between structures and plays a significant role in topics like natural transformations and derived functors.
Covariant Functor: A covariant functor is a type of mapping between categories that preserves the direction of morphisms. This means that if there is a morphism from object A to object B in one category, the functor maps this morphism to a morphism from the image of A to the image of B in another category. Covariant functors are essential in establishing relationships between different mathematical structures, and they play a key role in defining natural transformations and derived functors.
Equivalence of Categories: Equivalence of categories is a concept in category theory that describes when two categories are, in a certain sense, structurally the same. This notion is not just about having the same objects and morphisms but instead emphasizes the existence of functors between the two categories that create a correspondence preserving the relationships between their objects and morphisms. Essentially, if two categories are equivalent, they can be considered interchangeable for purposes such as studying properties of mathematical structures.
Functor Category: A functor category is a category where the objects are functors between two categories, and the morphisms are natural transformations between those functors. This concept helps organize and relate different functorial mappings, allowing for a structured way to study transformations and relationships in category theory. Functor categories play a crucial role in understanding how various mathematical structures interact through functorial relationships.
Hom-set transformation: A hom-set transformation is a specific type of function between hom-sets in category theory that connects the morphisms between two objects in different categories. It plays a critical role in defining natural transformations, which are essential for understanding the relationship between functors and how they interact with the structure of categories. Hom-set transformations help to formalize the idea of transforming one morphism into another while preserving certain properties.
Identity Transformation: The identity transformation is a specific type of natural transformation that acts as the identity on objects and morphisms in a category, mapping each object to itself and each morphism to itself. This concept is foundational in category theory and emphasizes the idea that every object and morphism has a unique transformation that leaves it unchanged, serving as a fundamental building block for understanding more complex transformations.
Isomorphism: An isomorphism is a mapping between two mathematical structures that establishes a one-to-one correspondence, preserving the operations and relations of the structures. It shows that two structures are fundamentally the same in terms of their properties, even if they may appear different at first glance. This concept connects deeply with how we understand relationships in category theory and other mathematical frameworks.
Isomorphism of Natural Transformations: An isomorphism of natural transformations occurs when two natural transformations between the same functors can be considered equivalent, meaning there exists a pair of natural transformations that are inverses of each other. This relationship reflects a deeper structure within category theory, showing how transformations can preserve the categorical structure and function similarly across different contexts. Understanding these isomorphisms helps in the study of functor categories and their interactions with natural transformations.
Monomorphism: A monomorphism is a morphism that is left-cancellable, meaning if two morphisms composed with it yield the same result, then the two morphisms must be the same. This concept is fundamental in category theory as it generalizes the notion of injective functions from set theory, highlighting how certain structures can be embedded into others while preserving distinctness and structure.
Morphism: A morphism is a structure-preserving map between two objects in a category, serving as a central concept in category theory. Morphisms can represent functions, transformations, or relationships and play a crucial role in defining how objects interact within mathematical structures. They help to create frameworks for understanding mathematical concepts like exact sequences, functors, and transformations between categories.
Natural Isomorphism: A natural isomorphism is a type of isomorphism between functors that preserves the structure in a coherent way, meaning that the isomorphisms can be chosen 'naturally' with respect to the morphisms of the categories involved. This concept connects deeply with how functors relate to each other and allows for the transfer of properties across categories while maintaining their relationships. Understanding natural isomorphisms is crucial for comprehending transformations, adjoint functors, derived functors, and axioms in homological algebra.
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 images of objects under two different functors, ensuring that the transformation commutes with all morphisms in the categories. This concept is essential for understanding how different mathematical structures can be interconnected through mappings that respect their inherent properties.
Naturality Condition: The naturality condition is a principle in category theory that ensures a natural transformation between two functors behaves consistently with respect to the morphisms in their respective categories. It asserts that for any morphism between objects, the transformation commutes with the action of these morphisms, ensuring that the transformation is coherent and systematic across different contexts.
Yoneda Lemma: The Yoneda Lemma is a fundamental result in category theory that establishes a deep relationship between objects and morphisms in a category through functors. It states that for any category, an object can be fully characterized by the set of morphisms that originate from it or target it, allowing for a powerful way to relate different objects and their mappings. This concept connects naturally to various ideas, such as functors that can either preserve or reverse arrows in a category, the notion of natural transformations that facilitate comparisons between functors, and the concept of adjoint functors that link pairs of functors together in a coherent way.
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