Yoneda embedding

English

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Etymology

Named after the Japanese mathematician Nobuo Yoneda.

Noun

Yoneda embedding ‎(plural Yoneda embeddings)

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