202602201955

Tags : Category Theory

Applications of Yoneda Lemma

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In Cartesian Closed Categories

To show that

$$(A^B{)}^C\cong A^{B\times C}$$

We now only need to show:

$$h((A^B{)}^C)\cong h(A^{B\times C})$$

Which is now pretty straigthforward:

$$\begin{array}{rl}\text{Hom}(X,(A^B{)}^C)& \cong \text{Hom}(X\times C,A^B)\\ & \cong \text{Hom}((X\times C)\times B,A)\\ & \cong \text{Hom}(X\times (B\times C),A)\\ & \cong \text{Hom}(X,A^{B\times C})\end{array}$$

And all of this is natural in $X$. Given any function $f:Y\to X$ we simply pre-compose to get the natural transformation.

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References

• Yoneda Lemma
• Yoneda Embedding