Definition:Yoneda Functor/Yoneda Embedding

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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.


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 $h_f : \map {\operatorname {Hom} } {-, X} \to \map {\operatorname {Hom} } {-, Y}$


Also denoted as

The Yoneda embedding is also denoted by $Y$.


Also known as

The Yoneda embedding can also be referred to as the (covariant) Yoneda functor.


Also see


Source of Name

This entry was named for Nobuo Yoneda.


Sources