Spherical DG-functors

Understand

For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A -> B.

• We construct its associated autoequivalences: the twist T of D(B) and the co-twist F of D(A).
• We give powerful sufficiency criteria for a quasi-functor to be spherical and for the twists associated to a collection of spherical quasi-functors to braid.
• Using the framework of DG-enhanced triangulated categories, we translate all of the above to Fourier-Mukai transforms between the derived categories of algebraic varieties.