Natural Transformations


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Natural Transformations

Definition 3.6  

A natural transformation from $ Y:\mathscr{C}\rightarrow set$ to $ X:\mathscr{C}\rightarrow set$ is an assignment of an arrow $ N:Y\rightarrow X$ that associates to each object A in $ \mathscr{C}$ an arrow $ N_A:Y(A)\rightarrow X(A)$ in Set such that, for any $ \mathscr{C}$-arrow $ f:A\rightarrow B$ the following diagram commutes

$\displaystyle \xymatrix{
 A\ar[dd]^f&&Y(B)\ar[rr]^{N_B}\ar[dd]_{Y(f)}&&X(B)\ar[dd]^{X(f)}\\
 &&&&\\
 B&&Y(A)\ar[rr]^{N_A}&&X(A)\\
 }$

i.e.

$\displaystyle N_A o Y(f)=X(f) o N_B$    

where $ N_A:Y(A)\rightarrow X(A)$ are the components on N, while N is the natural transformation.
From this diagram it is clear that the two arrows $ N_A$ and $ N_B$ turn the Y-picture of $ f:A\rightarrow B$ into the respective X-picture.



Cecilia Flori 2007-01-02