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.