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Selecta Mathematica

, Volume 24, Issue 4, pp 2841–2925 | Cite as

Six-functor-formalisms and fibered multiderivators

  • Fritz Hörmann
Article

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.

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 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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