Six Functor Formalisms and Fibered Multiderivators

TL;DR

This paper develops a comprehensive framework for six-functor formalisms using fibered multiderivators, simplifying axioms and clarifying relations between internal and external monoidal products, with applications to derivator theories.

Contribution

It introduces a new approach to six-functor formalisms via fibered multiderivators, unifying various formalisms and simplifying foundational axioms.

Findings

Axioms for Wirthmüller and Grothendieck formalisms established.

Fibered multiderivators can be interpreted as six-functor formalisms.

Simplification of axioms and proofs in six-functor theories.

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 Wirthm\"uller and Grothendieck formalisms (where either $f^!=f^*$ or $f_!=f_*$) or intermediate formalisms where we have e.g. a natural morphism $f_! \rightarrow f_*$. 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.