Spherical DG-functors

Abstract

For two DG-categories A and B we define the notion of a spherical Morita quasi-functor $\mathcal{A}\to \mathcal{B}$. We construct its associated autoequivalences: the twist $T\in \mathrm{Aut}\mathcal{D}(\mathcal{B})$ and the cotwist $F\in \mathrm{Aut}\mathcal{D}(\mathcal{A})$. We give 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. This is a broad generalisation of the results on spherical objects in [ST01] and on spherical functors in [Ann07]. In fact, this paper replaces [Ann07], which has a fatal gap in the proof of its main theorem. Though conceptually correct, the proof was impossible to fix within the framework of triangulated categories.