Realization functors via Grothendieck derivators and applications

Realization functors are some of the main tools in the study of equivalences between derived categories. Having an abelian category A and a t-structure on D(A) with heart H, one wishes to construct a triangle functor, denoted by “real", from D(H) to D(A) which is potentially an equivalence under certain assumptions on the subcategory H of D(A). Originally, such a functor could be constructed only from the bounded derived category D^b(H), by A. A. Beilinson et al. in their paper Faisceaux Pervers. In a more recent article, S. Virili constructed both bounded and unbounded realization morphisms of Grothendieck derivators (strict 2-functors from some 2-category of small categories to the 2-category of all categories satisfying some additional axioms) and showed their use in Tilting Theory. Precisely and briefly, one considers a t-structure on a strong and stable derivator D with heart H and obtains a pseudonatural transformation from the derivator D_H associated to H to D. These morphisms generalize the former realization functors, and their properties follow by formal arguments with little room for choice in the construction. This thesis aims to explain said construction in a mostly self-contained way, fill in additional details of the necessary arguments, look into more possible applications of realization morphisms in Representation Theory and hopefully illustrate in a compact way how realization functors from the 2-categorical approach are useful.