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yoneda_embedding [2015/03/14 01:10] nikolaj |
yoneda_embedding [2015/03/14 01:13] nikolaj |
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| Let $g: X\to Y$ and $f: A\to B$. | Let $g: X\to Y$ and $f: A\to B$. | ||
| - | We make $f$ into a natural transformation from $\mathrm{Hom}(-,A)$ to $\mathrm{Hom}(-B)$ as follows: | + | We make $f$ into a natural transformation from $\mathrm{Hom}(-,A)$ to $\mathrm{Hom}(-B)$ via the following square: |
| $h: \mathrm{Hom}(Y,A)$ ... $f\circ h : \mathrm{Hom}(Y,B)$ | $h: \mathrm{Hom}(Y,A)$ ... $f\circ h : \mathrm{Hom}(Y,B)$ | ||
| $h\circ g : \mathrm{Hom}(X,A)$ ... $f\circ h\circ g : \mathrm{Hom}(X,B)$ | $h\circ g : \mathrm{Hom}(X,A)$ ... $f\circ h\circ g : \mathrm{Hom}(X,B)$ | ||
| + | |||
| + | Note how the $\mathrm{Hom}$-functors themselves are contravariant, as seen by the direction of the vertical arrows: $g: X\to Y$ but e.g. the associated fmap goes from $\mathrm{Hom}(Y,A)$ to $\mathrm{Hom}(X,A)$. | ||
| === Yoneda lemma === | === Yoneda lemma === | ||
