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Like what the title said, I want to use Yoneda's lemma to "guess" the definition of exponential object in $SET$. So basically I want to say that given any two sets $A, B$ in $SET$, the exponential object "$A^B$" exists in the category of sets. So let's pretend for a moment, that a person has no notions of set theory, how would one give an good guess on how we should define "$A^B$". I have seen similar things being done using Yoneda's lemma on the category of presheafs, but I want to take a step back even further, just to see how to use Yoneda's lemma in this manner.

Cheers

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This depends on how you define the exponential object at all. Without any knowledge of set theory, how do you motivate the isomorphism $\mathcal{C}(A, B^C) \simeq \mathcal{C}(A \times C, B)$?

If you accept that, though, it becomes fairly clear. $A^B \simeq \mathcal{Set}(1, A^B) \simeq \mathcal{Set}(1 \times B, A) \simeq \mathcal{Set}(B, A)$. So $A^B$ is the set of functions from $B$ to $A$ (if it exists at all: you'll still need to check that $\mathcal{Set}(A, B^C) \simeq \mathcal{Set}(A \times C, B)$ in general).

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  • $\begingroup$ ah thanks. So I don't really need yoneda to for the "guess", I just need to accept (check) that the natural isomorphism you stated. $\endgroup$ – user16319 Nov 20 '19 at 21:38
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    $\begingroup$ @user16319 Yes, though in a sense the Yoneda lemma is needed to show that $\mathcal{C}(A, B^C) \simeq \mathcal{C}(A \times C, B)$ uniquely defines the exponential object. That is, if $\mathcal{C}(A, E) \simeq \mathcal{C}(A \times C, B) \simeq \mathcal{C}(A, E')$, then $E \simeq E'$. $\endgroup$ – SCappella Nov 21 '19 at 4:24

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