Yoneda embedding

Named after the Japanese mathematician Nobuo Yoneda.

Yoneda embedding (plural Yoneda embeddings)

1. (category theory) Given category ${\displaystyle {\mathcal {C}}}$ , a Yoneda embedding for this category is a functor ${\displaystyle \phi }$ such that for any object A in ${\displaystyle {\mathcal {C}}}$ , ${\displaystyle \phi :A\mapsto h^{A}}$ and for any morphism ${\displaystyle f:B\rightarrow A}$ in ${\displaystyle {\mathcal {C}}}$ , ${\displaystyle \phi :f\mapsto \eta :h^{A}\rightarrow h^{B}}$ where the natural transformation η has components ${\displaystyle \eta _{X}:s\mapsto s\circ f}$ . Then ${\displaystyle \phi :{\mathcal {C}}^{op}\rightarrow [{\mathcal {C}},{\mathcal {S}}ets]}$ . Otherwise, it is a functor ${\displaystyle \phi }$ such that ${\displaystyle \phi :A\mapsto h_{A}}$ and for any ${\displaystyle f:A\rightarrow B}$ in ${\displaystyle {\mathcal {C}}}$ , ${\displaystyle \phi :f\mapsto \eta :h_{A}\rightarrow h_{B}}$ where η has components ${\displaystyle \eta _{X}:s\mapsto f\circ s}$ . Then ${\displaystyle \phi :{\mathcal {C}}\rightarrow [{\mathcal {C}}^{op},{\mathcal {S}}ets]}$ .

• representable functor

• TheCatsters – Representables and Yoneda 1