Yoneda's lemma and $K$-theory.

Question

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The (reduced) suspension/loopspace adjunction implies that $\langle \Sigma^2 X, BU\times \mathbb{Z}\rangle \cong \langle X,\Omega^{2} (BU\times \mathbb{Z}) \rangle$. Therefore, $\langle X, BU\times \mathbb{Z} \rangle \cong \langle X, \Omega^{2} (BU\times \mathbb{Z}) \rangle$ for all compact Hausdorff spaces $X$. How can I use Yoneda's lemma to prove that if $$\langle X, BU\times \mathbb{Z} \rangle \cong \langle X, \Omega^{2} (BU\times \mathbb{Z}) \rangle$$ then $$ BU\times \mathbb{Z} \cong \Omega^{2} (BU\times \mathbb{Z}) ?$$

Answer 1

Given natural bijections $F_X:Hom(X,A) \cong Hom(X,B)$, the Yoneda lemma says there is a unique morphism $f:A \rightarrow B$ that provides the action of this natural function. That is, you get from $Hom(X,A)$ to $Hom(X,B)$ by applying $f$. Note that we didn't need to use that the things were bijections to define this map, only naturality. To see where this map comes from explicitly, apply $F_X$ to $1_A \in Hom(A,A)$.

Now, apply all this machinery to the inverses of the natural bijections, producing a morphism $g:B \rightarrow A$. The uniqueness part of Yoneda then says that $fg$ and $gf$ are the identities on the corresponding objects, since their action on the Homsets applies $F_X$ and then $F_X^{-1}$, or the reverse, both of which are the identity, which obviously arises by composition with the identity map.

Edit: I'd like to add that I provided an answer for a general category, under the assumption that $\langle X,Y\rangle \cong \langle X,Z \rangle$ denoted a natural isomorphism between homsets in some category. If this is not the case, let me know and I'll delete this answer!

I also don't know the specifics of what you're doing, but the fact that your $X$ can only take values in compact Hausdorff spaces (or things homotopic to them) causes an issue, since the whatever $BU \times \mathbb{Z}$ is, I wouldn't imagine it's homotopic to something compact. You want your isomorphism to be a homotopy, so the category you must work in is some subcategory of spaces up to homotopy, so to use this machinery you have to find some way of allowing $X$ to be any homotopy class of some subcatgory of spaces that both of the objects of interest lie in.