When two varieties have equivalent derived categories of coherent sheaves are the stable $\infty$-categories of coherent sheaves also equivalent?
Are the stable $\infty$-categories of varieties "discrete" in some sense?
When two varieties have equivalent derived categories of coherent sheaves are the stable $\infty$-categories of coherent sheaves also equivalent?
Are the stable $\infty$-categories of varieties "discrete" in some sense?
Orlov proved that any derived equivalence (of smooth projective varieties) is realized by a Fourier-Mukai functor; this should imply equivalence on $\infty$-level.
The answer of this question in the language of $(\infty,1)$-categories can be found in the following paper by Benjamin Antieau, On the uniqueness of infinity-categorical enhancements of triangulated categories. Where it is shown that derived categories admit unique $\infty$-categorical enhancements ( by stable $(\infty,1)$-categories ). See Theorem 6.4 and Corollary 6.5.