Suppose $\mathcal{C}$ is a dg category (over some base) with all colimits. We say that $X\in \mathcal{C}$ is a generator if $\mathcal{C}$ is equivalent to $\operatorname{End}\_\mathcal{C}X$-modules (via the Yoneda functor). My question: are there some "niceness" conditions we can impose on the pair $(\mathcal{C}, X)$ which guarantee that all deformations of $X$ (i.e. all fibers of a suitably perfect sheaf of objects of $\mathcal{C}$ over a curve with special fiber $X$, say) are still generators?



Is the property of being a dg generator open?
Let me give an example (which you are probably aware of) that shows how a generator may become a non-generator after a deformation. 


Let $\mathcal C$ be the derived category of $\mathbb{P}^1$ and $X = \mathcal{O} \oplus \mathcal{O}(-1)[1]$. This is clearly a generator. On the other hand, we can deform the sum to an extension
$$
0 \to \mathcal{O}(-1) \to \mathcal{O} \to \mathcal{O}\_P \to 0,
$$
where $P \in \mathbb{P}^1$ is a point, and the sheaf $\mathcal{O}\_P$ is definitely not a generator. 


I think a similar trick can be used in a much more general situation, so I do not believe there is a reasonable condition (besides the rigidity of $X$) ensuring that the property "being a generator" is open.


