/sci/ - Science & Math

And, how to apply the Yoneda lemma? There's this thing sometimes called the Yoneda principle:

If [math]\textbf{C}[/math] is a locally small category, [math]y \colon \textbf{C} \to \textbf{Sets} ^{\textbf{C} ^{op}}[/math] is the Yoneda embedding, and [math]A, B[/math] are any objects of the category, then [math]yA \cong yB[/math] implies [math]A \cong B[/math].

Now, to show the firepower of this idea, assume our locally small category is cartesian closed and has coproducts. The claim is that [math](A \times B) + (A \times C) \cong A \times (B+C)[/math]. Using the Yoneda principle, [math]y(A \times (B+C))=\text{Hom}(A \times (B+C), X) \cong \text{Hom}(B+C, X^A) \cong \text{Hom}(B, X^A) \times \text{Hom}(C, X^A) \cong \text{Hom}((A\times B)+(A \times C), X)=y((A\times B)+(A \times C))[/math].