Example of Yoneda’s Lemma


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Example of Yoneda's Lemma

For each element A on a category $ \mathscr{C}$ we define a presheaf y(A) such that:
  • Given an object D of $ \mathscr{C}$ we have
    $ y(A)D=Hom_{\mathscr{C}}(D,A)$


  • Given a morphism $ \alpha :B\rightarrow D$ and $ \theta:D\rightarrow A$ we obtain: $ y(A)(\alpha):Hom_{\mathscr{C}}(D,A)\rightarrow Hom_{\mathscr{C}}(B,A)$
    $ y(A)(\alpha)(\theta)=\theta o \alpha$
Given the above construction it follows that, for any morphism on $ \mathscr{C}$ of the form $ f:A\rightarrow A_1$ there exists a natural transformation 3.5 $ y(A)\rightarrow y(A_1)$.
We can therefore deduce that y is actually a functor from the category C to the set of presheaves defined on $ \mathscr{C}$, i.e $ y:\mathscr{C}\rightarrow Sets^{\mathscr{C}^{op}}$, such that to each object A in $ \mathscr{C}$, y assigns the set of Homeomorphisms which have as codomain A, $ A\rightarrow y(A)=Hom_{\mathscr{C}}(-,A)$
where Hom(-,A) corresponds to a Presheaf on $ \mathscr{C}$ defined on A.
A very simple graphical example of the above is the following:

$\displaystyle \xymatrix{ &&E&&\\ A\ar[rru]^h&&&&B\ar[llu]_g\\ &&D\ar[rru]_{\alpha}\ar[llu]^{\theta}&&\\ &&\Downarrow^y(A)&&\\ &&Hom_{\mathscr{C}}(E,A)\ar[rrd]^{y(A)(g)}\ar[lld]_{y(A)(h)}&&\\ Hom_{\mathscr{C}}(A,A)\ar[rrd]_{y(A)(\theta)}&&&&Hom_{\mathscr{C}}(B,A)\ar[lld]^{y(A)(\alpha)}\\ &&Hom_{\mathscr{C}}(D,A)&&\\ }$


Cecilia Flori 2007-01-02