Yoneda Lemma
Theorem
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Covariant Functors
Let $C$ be a locally small category.
Let $\mathbf{Set}$ be the category of sets.
Bijection
Let $F: C \to \mathbf {Set}$ be a covariant functor.
Let $A \in C$ be an object.
Let $I_A$ be its identity morphism.
Let $h^A = \map {\operatorname {Hom} } {A, -}$ be its covariant hom-functor.
The class of natural transformations $\map {\operatorname {Nat} } {h^A, F}$ is a small class, and:
- $\alpha: \map {\operatorname {Nat} } {h^A, F} \to \map F A: \eta \mapsto \map {\eta_A} {I_A}$
- $\beta: \map F A \to \map {\operatorname {Nat} } {h^A, F}: u \mapsto \paren {X \mapsto \paren {f \mapsto \map {\paren {\map F f} } u} }$
Naturality
Let $\sqbrk {C, \mathbf {Set} }$ be the covariant functor category.
Let $C \times \sqbrk {C, \mathbf {Set} }$ be the product category.
Let $C \times \sqbrk {C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A, F} \mapsto \map {\operatorname {Nat} } {h^A, F}$ be the covariant functor defined as the composition of the hom bifunctor and the product of the contravariant Yoneda functor $h^-$ and the identity functor $\operatorname{id}_{\sqbrk {C, \mathbf {Set} } }$.
Let $\operatorname{ev} : C \times \sqbrk {C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A, F} \mapsto \map F A$ be the functor evaluation functor.
Then $\Phi_{\tuple {A, F} } : \map {\operatorname {Nat} } {h^A, F} \to \map F A: \eta \mapsto \map {\eta_A} {\operatorname {id}_A}$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.
Contravariant Functors
Let $C$ be a locally small category.
Let $\mathbf{Set}$ be the category of sets.
Bijection
Bijection in Yoneda Lemma for Contravariant Functors
Naturality
Let $[C^{\operatorname{op}}, \mathbf{Set}]$ be the contravariant functor category.
Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] $ be the product category.
Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto \operatorname{Nat}(h_A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the opposite of the covariant Yoneda functor $h_-$ and the identity functor $\operatorname{id}_{[C^{\operatorname{op}}, \mathbf{Set}]}$.
Let $\operatorname{ev} : C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto F(A)$ be the contravariant functor evaluation functor.
Then $\Phi_{(A, F)} : \operatorname{Nat}(h_A, F) \to F(A) : \eta \mapsto \eta_A(\operatorname{id}_A)$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.
Also see
Source of Name
This entry was named for Nobuo Yoneda.