Lexic.us

Definition of Yoneda embedding

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

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