Yoneda embedding

Yoneda embedding

English

Etymology

Named after the Japanese mathematician Nobuo Yoneda.

Noun

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 ${\\displaystyle \\phi :{\\mathcal {C}}\\rightarrow [{\\mathcal {C}}^{op},{\\mathcal {S}}ets]}$.

See also

• representable functor

References

• TheCatsters – Representables and Yoneda 1