Definition:Yoneda Functor/Yoneda Embedding

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

• Yoneda Embedding Theorem
• Definition:Contravariant Yoneda Functor

Source of Name

This entry was named for Nobuo Yoneda.

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.