13.2 The Yoneda lemma and its consequences

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 Yoneda embedding. 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

• Establishes an isomorphism between the set of natural transformations from a representable functor to any other functor and the set of elements of that functor applied to the representing object
  • For a locally small category $C$, a functor $F: C \to Set$, and an object $A$ in $C$, 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 $\Phi: Nat(Hom_C(A, -), F) \to F(A)$:
     • For a natural transformation $\alpha: Hom_C(A, -) \to F$, let $\Phi(\alpha) = \alpha_A(id_A)$
  2. Define the inverse $\Psi: F(A) \to Nat(Hom_C(A, -), F)$:
     • For an element $x \in F(A)$, define a natural transformation $\Psi(x): Hom_C(A, -) \to F$ by:
       • For each object $B$ in $C$ and morphism $f: A \to B$, let $(\Psi(x))_B(f) = F(f)(x)$
  3. Prove that $\Phi$ and $\Psi$ 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 $Y: C \to [C^{op}, Set]$ 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 $A, B$ in $C$, 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 $F: C \to Set$ is a representable functor, then any two representing objects for $F$ 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 $C$, a functor $F: C^{op} \to Set$, and an object $A$ in $C$, 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)