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Six-functor-formalisms and fibered multiderivators

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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 \(f^{!}=f^{*}\) 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.

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  1. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany

    Fritz Hörmann

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  1. Fritz Hörmann
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Hörmann, F. Six-functor-formalisms and fibered multiderivators. Sel. Math. New Ser. 24, 2841–2925 (2018). https://doi.org/10.1007/s00029-018-0424-0

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