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13.2 The Yoneda lemma and its consequences

3 min read•july 23, 2024

The Yoneda Lemma is a fundamental result in category theory that establishes a deep connection between objects and functors. It shows how we can understand an object by looking at all the morphisms going into it, providing a powerful tool for analyzing categorical structures.

This lemma has far-reaching consequences, from proving uniqueness of representing objects to enabling the . It's a cornerstone for understanding natural transformations and representable functors, making it essential for grasping advanced concepts in category theory.

The Yoneda Lemma

Yoneda lemma statement and proof

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Establishes an between the set of natural transformations from a to any other and the set of elements of that functor applied to the representing object
- For a locally small category $C$ $C$ , a functor $F: C \to Set$ $F : C \to S e t$ , and an object $A$ $A$ in $C$ $C$ , there is an isomorphism:
  - $Nat(Hom_C(A, -), F) \cong F(A)$
- Isomorphism is natural in both $A$ and $F$ (functorial in $A$ and $F$ )
Proof:
1. Define the isomorphism $\Phi: Nat(Hom_C(A, -), F) \to F(A)$ $Φ : N a t (Ho m_{C} (A, -), F) \to F (A)$ :
  - For a $\alpha: Hom_C(A, -) \to F$ , let $\Phi(\alpha) = \alpha_A(id_A)$
2. Define the inverse $\Psi: F(A) \to Nat(Hom_C(A, -), F)$ $Ψ : F (A) \to N a t (Ho m_{C} (A, -), F)$ :
  - For an element $x \in F(A)$ $x \in F (A)$ , define a natural transformation $\Psi(x): Hom_C(A, -) \to F$ $Ψ (x) : Ho m_{C} (A, -) \to F$ by:
    - For each object $B$ in $C$ and $f: A \to B$ , let $(\Psi(x))_B(f) = F(f)(x)$
3. Prove that $\Phi$ $Φ$ and $\Psi$ $Ψ$ are inverses:
  - $\Phi(\Psi(x)) = x$ for all $x \in F(A)$
  - $\Psi(\Phi(\alpha)) = \alpha$ for all $\alpha \in Nat(Hom_C(A, -), F)$

Applications of Yoneda lemma

Proves uniqueness of certain natural transformations
- If $F, G: C \to Set$ are functors and $A$ is an object in $C$ , then any natural transformation $\alpha: Hom_C(A, -) \to F$ is uniquely determined by its component $\alpha_A(id_A)$
Proves existence of certain natural transformations
- If $F: C \to Set$ is a functor and $A$ is an object in $C$ , then for any element $x \in F(A)$ , there exists a unique natural transformation $\alpha: Hom_C(A, -) \to F$ such that $\alpha_A(id_A) = x$
Used to solve problems involving natural transformations between functors (e.g. proving isomorphisms, finding inverses)

Yoneda lemma vs Yoneda embedding

Yoneda embedding is a full and faithful functor $Y: C \to [C^{op}, Set]$ $Y : C \to [C^{o p}, S e t]$ defined by:
- For each object $A$ in $C$ , $Y(A) = Hom_C(-, A)$
- For each morphism $f: A \to B$ in $C$ , $Y(f)$ is the natural transformation $Hom_C(-, f): Hom_C(-, A) \to Hom_C(-, B)$
Yoneda lemma implies the Yoneda embedding is full and faithful:
- For any objects $A, B$ $A, B$ in $C$ $C$ , the Yoneda lemma gives an isomorphism:
  - $Hom_{[C^{op}, Set]}(Hom_C(-, A), Hom_C(-, B)) \cong Hom_C(A, B)$
- This isomorphism shows that the Yoneda embedding reflects and preserves morphisms (fully faithful)

Consequences of Yoneda Lemma

Consequences of Yoneda lemma

Implies uniqueness of representing objects up to unique isomorphism
- If $F: C \to Set$ $F : C \to S e t$ is a representable functor, then any two representing objects for $F$ $F$ are uniquely isomorphic
  - Suppose $A$ and $B$ are objects in $C$ such that $F \cong Hom_C(A, -)$ and $F \cong Hom_C(B, -)$
  - The Yoneda lemma gives isomorphisms:
    - $Hom_C(A, B) \cong Nat(Hom_C(A, -), Hom_C(B, -)) \cong Nat(F, F)$
  - The identity natural transformation $id_F$ corresponds to a unique isomorphism $A \to B$
Used to prove the co-Yoneda lemma for contravariant representable functors
- Co-Yoneda lemma: for a locally small category $C$ $C$ , a functor $F: C^{op} \to Set$ $F : C^{o p} \to S e t$ , and an object $A$ $A$ in $C$ $C$ , there is an isomorphism:
  - $Nat(F, Hom_C(-, A)) \cong F(A)$
- Co-Yoneda lemma derived from Yoneda lemma by considering the opposite category $C^{op}$ (dual statement)

Key Terms to Review (17)

Adjunction: Adjunction is a fundamental concept in category theory where two functors stand in a specific relationship, known as an adjoint pair. This relationship allows one functor to be thought of as providing a best approximation of the other, capturing the essence of universal properties. Adjunctions are critical in many areas of mathematics, including the study of limits, colimits, and various constructions that connect different categories.

Category of presheaves: The category of presheaves on a category $ C $ consists of contravariant functors from $ C $ to the category of sets, along with natural transformations between them. This category captures the idea of assigning a set to each object in $ C $ while respecting the morphisms between those objects, providing a framework for working with collections of data that vary according to the structure of $ C $. The concept is closely tied to the Yoneda lemma, which establishes a powerful relationship between presheaves and the objects in the category.

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.

F. William Lawvere: F. William Lawvere is a prominent mathematician known for his significant contributions to category theory, particularly for developing the foundational aspects of topos theory and the Yoneda lemma. His work has greatly influenced the understanding of mathematical structures and relationships, showcasing how categorical concepts can provide deep insights into various fields, including algebra and logic.

Functor: A functor is a mapping between categories that preserves the structure of those categories, specifically the objects and morphisms. It consists of two main components: a function that maps objects from one category to another, and a function that maps morphisms in a way that respects composition and identity morphisms.

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.

Isomorphism: An isomorphism is a morphism between two objects in a category that establishes a structure-preserving equivalence between them, allowing for a one-to-one correspondence. It indicates that the objects are essentially the same from the perspective of the category, despite potentially differing in their actual representation or underlying elements.

Limit: In category theory, a limit is a universal construction that captures the idea of taking a 'best' way to combine a diagram of objects and morphisms into a single object. It allows us to formally represent the notion of convergence and completeness across various structures, connecting diverse concepts like commutative diagrams, functors, and adjunctions.

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.

N. T. Johnstone: N. T. Johnstone is a prominent mathematician known for his significant contributions to category theory, particularly the formulation and development of the Yoneda lemma. This lemma serves as a foundational result in category theory, establishing a deep connection between objects and morphisms within a category and providing insights into the nature of functors and natural transformations.

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.

Presheaf: A presheaf is a functor that assigns data to the open sets of a topological space in a way that respects the inclusion of open sets. It provides a structured way to understand local data and how it relates to larger contexts, bridging concepts in topology, category theory, and algebraic geometry.

Representability Theorem: The representability theorem is a fundamental result in category theory that establishes a connection between functors and objects in a category. Specifically, it states that a functor from a category to the category of sets is representable if and only if it is naturally isomorphic to the hom-functor associated with an object in that category. This theorem provides a powerful way to understand how certain functors can be 'represented' by objects, leading to deep insights in both mathematics and theoretical computer science.

Representable Functor: A representable functor is a type of functor that is naturally isomorphic to the hom-functor, meaning it can be expressed as the set of morphisms from a fixed object in a category to any object in that category. This concept connects deeply with the Yoneda embedding, allowing us to understand functors in terms of their action on objects through morphisms. Representable functors reveal insights about the structure of categories and provide a powerful framework for reasoning about natural transformations and limits.

Yoneda embedding: The Yoneda embedding is a functor that maps a category to a presheaf category, capturing the essence of objects in terms of their morphisms. This embedding allows us to understand how objects relate to one another through morphisms, emphasizing the representability of functors and establishing a foundation for the Yoneda lemma, which reveals deep insights into the structure of categories and functors.

Yoneda's Isomorphism: Yoneda's Isomorphism is a crucial result in category theory that demonstrates a natural equivalence between the hom-sets of functors and the morphisms of objects within a category. This concept is closely tied to the Yoneda Lemma, which states that any functor from a category to the category of sets can be fully represented by its behavior on morphisms, emphasizing that an object is determined by the way it interacts with all other objects in the category.

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Practice QuizGlossary

Practice Quiz Glossary

13.2 The Yoneda lemma and its consequences

The Yoneda Lemma

Yoneda lemma statement and proof

Top images from around the web for Yoneda lemma statement and proof

Top images from around the web for Yoneda lemma statement and proof

Applications of Yoneda lemma

Yoneda lemma vs Yoneda embedding

Consequences of Yoneda Lemma

Consequences of Yoneda lemma

Key Terms to Review (17)

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