I have a series of vague questions, related to localization of symmetric monoidal categories.

Here is the context. Say we are working over a field of characteristic zero. Then the "one category level higher" version of (DG) commutative ring is a (DG) symmetric monoidal category. It is well-known that for $X$ a scheme (or even, IIUC, a Noetherian stack with affine diagonal) $X$ can be recovered from the DG symmetric monoidal category of its quasicoherent sheaves. (Moreover, the functor $\text{Schemes}\to\text{SymMonCat}$ is fully faithful, including in an $\infty$-categorical context.

Now if $R$ is a commutative ring, we say that $S$ is a localization of $R$ if it can be obtained from $R$ by inverting some set of elements. If $R, S$ are both Noetherian, then there is a very nice alternative way of characterizing localizations:

(*) A map $R\to S$ is a localization if and only if the product map $S\otimes_R S\to S$ (derived tensor product) is an equivalence.

Now there are (at least) three interesting notions of localization for a DG symmetric monoidal category $\mathcal{C}$ (note all make sense also for just monoidal categories, and that when taking universal objects in the category of categories I'm going to be vague about what I'm requiring from categories: I'm willing to assume compactly generated, idempotent complete, etc.).

- Localization along a morphism $f:X\to Y.$
- Localization along an object $X$ (defined as the universal symmetric monoidal DG category admitting a functor from $\mathcal{C}$ where $X$ is $\otimes$-invertible, maybe satisfying some additional conditions).
- "Parallel" localization along a morphism: if $f:X\to Y$ is a morphism, I'm defining this to be the initial category in which $X,Y$ are invertible and there exists a map $f':X^{-1}\to Y^{-1}$ with $f\otimes f' = \text{id}:\mathbb{I}\to \mathbb{I},$ where the equality is understood via an appropriate system of coherences.

(Of course 2. is a special case of 3.)

Here are some questions.

- Defining $\otimes$ in terms of colimit in the category of symmetric monoidal categories, is there a context where (*) holds for symmetric monoidal categories (i.e., localizations can be characterized by a tensor-idempotence condition)?
- Are there "interesting" localizations of this type of the category $\mathcal{C}$ of DG quasicoherent coherent sheaves on a scheme $X$ which do not come from geometric localizations? If no, are there any examples in more general contexts that have been studied or computed in some sense? (For example, what happens if you $\otimes$-localize the category of vector spaces along a two-dimensional vector space?)
- By functoriality and universality, localizations of any of the types 1., 2., 3. can be "combined" (in a commutative way), and (by formal nonsense), the result of applying two localizations $\mathcal{C}\to \mathcal{C}_1$ and $\mathcal{C}\to \mathcal{C}_2$ is the colimit $\mathcal{C}_1\otimes_{\mathcal{C}}\mathcal{C}_2$. **Definition**. For $\mathcal{C}_1\leftarrow\mathcal{C}\to \mathcal{C}_2$ a pair of localizations of $\mathcal{C}$ as above, set $$\mathcal{C}_{12}: = \mathcal{C}_1\otimes_{\mathcal{C}}\mathcal{C}_2$$ for the "combined" localization. Say that $\mathcal{C}_1$ and $\mathcal{C}_2$ *cover* $\mathcal{C}$ if $\mathcal{C}$ is the limit of the diagram $\mathcal{C}_1\to \mathcal{C}_{12}\leftarrow \mathcal{C}_2.$ My question now is: are there interesting examples of covers of DG symmetric monoidal categories in this sense other than Zariski covers in geometry?