Derivator Six Functor Formalisms -- Definition and Construction I

A theory of a derivator version of six-functor-formalisms is developed, using an extension of the notion of fibered multiderivator due to the author. Using the language of (op)fibrations of 2-multicategories this has (like a usual fibered multiderivator) a very neat definition. This definition not only encodes all compatibilities among the six functors but also their interplay with homotopy Kan extensions. One could say: a nine-functor-formalism. This is essential for dealing with (co)descent questions. Finally, it is shown that every fibered multiderivator (for example encoding any kind of derived four-functor formalism $(f_*, f^*, \otimes, \mathcal{HOM})$ occurring in nature) satisfying base-change and projection formula formally gives rise to such derivator six-functor-formalism in which $f_!=f_*$, i.e. a derivator Grothendieck context.