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Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime ideal 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 $\mathcal{T}$?

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?

Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime ideal 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 its $R$-points?

Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime ideal 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 $\mathcal{T}$?