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Yoneda embedding
Functor
| context | ${\bf C}$ … small category |
| definiendum | $\mathcal y$ |
| inclusion | $\mathcal y$ in ${\bf C}\longrightarrow{\bf Set}^{{\bf C}^\mathrm{op}}$ |
| definition | $\mathcal y\,A:={\bf C}[-,A]$ |
| definition | $\mathcal y(\,f):=g\mapsto f\circ g$ |
Discussion
To elaborate on the action of $\mathcal y$ on arrows:
Consider a general arrow $f:{\bf C}[A,B]$.
In the presheaf category ${\bf Set}^{{\bf C}^\mathrm{op}}$, the image arrow
$\mathcal y(\,f)$ in $\mathrm{nat}\left({\bf C}[-,A],{\bf C}[-,C]\right)$
is the natural transformation, with components
$y(\,f)_X:{\bf C}[X,A]\to{\bf C}[X,B],$
mapping an arrow $g$ to an arrow $f\circ g$.
Reference
Wikipedia: Functor category
Parents
Element of
Requirements
