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 \(f^{!}=f^{*}\) 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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Correspondence to 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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Keywords

  • Derivators
  • Fibered derivators
  • Multiderivators
  • (Op)fibered 2-multicategories
  • Six-functor-formalisms
  • Grothendieck contexts
  • Wirthmüller contexts

Mathematics Subject Classification

  • 55U35
  • 14F05
  • 18D10
  • 18D30
  • 18E30
  • 18G99