Definition:Yoneda Functor

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Definition

Let $C$ be a locally small category.

Let $C^{\operatorname{op}}$ be its opposite category.

Let $\mathbf{Set}$ be the category of sets.

Let $\sqbrk {C^{\operatorname{op} }, \mathbf {Set} }$ be the functor category between them.


Yoneda Embedding

The Yoneda embedding of $C$ is the covariant functor $h_- : C \to \sqbrk {C^{\operatorname{op}}, \mathbf{Set} }$ which sends:

an object $X \in C$ to the contravariant hom-functor $h_X = \map {\operatorname {Hom} } {-, X}$
a morphism $f : X \to Y$ to the postcomposition natural transformation $h_f : \map {\operatorname {Hom} } {-, X} \to \map {\operatorname {Hom} } {-, Y}$


Contravariant Yoneda Functor

The contravariant Yoneda functor of $C$ is the contravariant functor $h^- : C \to \sqbrk {C, \mathbf {Set} }$ which sends

an object $X \in C$ to the covariant hom-functor $h^X = \map {\operatorname {Hom} } {X, -}$
a morphism $f : X \to Y$ to the precomposition natural transformation $h^f : \map {\operatorname {Hom} } {Y, -} \to \map {\operatorname {Hom} } {X, -} : g \mapsto g \circ f$


Also see


Source of Name

This entry was named for Nobuo Yoneda.