Six Functor Formalisms and Fibered Multiderivators

Abstract

We develop the theory of (op)fibrations of 2-multicategories and use it to define abstract six-functor-formalisms. We also give axioms for Wirthmüller and Grothendieck formalisms (where either $f^{!}=f^{\ast }$ or $f_{!}=f_{\ast }$) or intermediate formalisms where we have e.g. a natural morphism $f_{!}\to f_{\ast }$. Finally, it is shown that a fibered multiderivator (in particular, a closed monoidal derivator) can be interpreted as a six-functor-formalism on diagrams (small categories). This gives, among other things, a considerable simplification of the axioms and of the proofs of basic properties, and clarifies the relation between the internal and external monoidal products in a (closed) monoidal derivator. Our main motivation is the development of a theory of derivator versions of six-functor-formalisms.