Here's an entirely unrelated question:
Given a stable $\infty$-category with a t-structure $\mathcal{C}$ let $\mathcal{A}$ denote its heart. Then there's a canonical functor $F: D(\mathcal{A}) \to \mathcal{C}$ (maybe not in this generality, though at least I hope so). Let $\mathsf{A}$ be the full subcategory of $\mathcal{C}$ spanned by shifts of the objects in the heart. What exactly should this category satisfy in order for the $F$ to be fully faithful? I think something like "generated by cup products from Ext^1" should be enough but I'm not sure how to make this precise... does anyone kn…