Tuesday, December 17, 15:15, 7.527

Simone Virili, Morita theory for stable derivators I and II.

Abstract:
The talk will be divided in two parts. In the first part we will review some of the basic definitions and examples in the theory of stable derivators. After that, we will recall the definition of a t-structure and show that t-structures on the base D(1) of a strong and stable derivator D are in bijection with (co)localizations of D. If time allows, we will mention some applications of this result as, for example, the fact that the heart of a compactly generated t-structure on the homotopy category of a stable model category (e.g., any algebraic or topological triangulated category), is a Grothendieck category. This is based on a joint work with Manuel Saorin and Jan Stovicek.

In the second part of the talk, we will start with a strong and stable derivator D, and a t-structure on the base D(1), whose heart we denote by A. We will then concentrate on the problem of comparing Der(A) (the unbounded derived category of the heart), with D(1). We will see that, under very mild assumptions one can always construct a morphism of prederivators, called realization morphism,
           real : Der_A → D
where Der_A is the natural prederivator enhancing the derived category of A, and we will give a set of sufficient conditions for this morphism to be an equivalence. In particular, if the t-structure t is induced by a "suitably bounded" co/tilting object, real is always an equivalence. This construction unifies and extends most of the derived co/tilting equivalences appeared in the literature in recent years.