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Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime spectra?

Given a tensor triangulated category $\mathcal{T}$ an $R$-point can be possibly thought about as a tensor triangulated functor $\mathcal{T} \to D^b(R)$.

What can be said about the faithfulness of the functor $$\otimes\Delta Cat \to PSh_{Sets}(CRings)$$ sending a $\mathcal{T}$ to the presheaf sending $R$ to the set of $R$-points of $\mathcal{T}$?

Can this point of view be used to talk about points of the stable homotopy category, similar to the classification of its thick prime ideals by Devinatz, Hopkins, Smith?

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    $\begingroup$ I doubt that that functor is fully faithful (since for example I don't expect the stable homotopy category to have enough points in an ordinary ring) but I don't know how to make it precise $\endgroup$
    – Denis Nardin
    Commented Mar 30, 2015 at 15:03
  • $\begingroup$ Also, my intuition says that the tensor triangulated functors should be considered as a groupoid and not a set if you want to get a reasonable theory $\endgroup$
    – Denis Nardin
    Commented Mar 30, 2015 at 15:09

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