Where can I read a proof that the infinity-category of finite spectra is the stable infinity-category freely generated by an object?

@AlexanderCampbell SAG.C.1.1.7

@DylanWilson Excellent, thank you.

What's a simple example of a non-proper map of simplicial sets that still has the property that the pullback of any final map along it is final?

The fact that the Thom spectrum of the J-homomorphism $BU\times Z\to Pic(\mathbb{S})$ can be computed as the Thom spectrum of the map $Z\to Pic(MU)$ that takes $n$ to $\Sigma^{2n}MU$ follows from work of @JonathanBeardsley on relative Thom spectra. Is there a simpler explanation of this case?

@BrunoStonek if you care about the multiplicative structure beyond E_1 then I don't know of another way. If you only care about the E_1-ring structure, then $BU\times \mathbb{Z}$ is the tensor product of $BU$ and $\mathbb{Z}$ in $\mathsf{Alg}_{\mathbb{E}_1}(\mathsf{Spaces}_{/\mathrm{Pic}})$ so you could just use that the Thom spectrum functor is symmetric monoidal to get the formula $MU \wedge S^0[t]$ as $\mathbb{E}_1$-rings (and hence $MU$-modules)

I guess that splitting is probably as E_2 algebras too, but not more than that

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