Yoneda embedding

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Named after the Japanese mathematician Nobuo Yoneda.


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] .

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