Six-functor-formalisms and fibered multiderivators

  • Fritz Hörmann


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.


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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  1. 1.
    Ayoub, J.: Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I. Astérisque 314, x+466 pp (2007a)Google Scholar
  2. 2.
    Ayoub, J.: Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II. Astérisque 315, vi+364 pp (2007b)Google Scholar
  3. 3.
    Bakovic, I.L: Fibrations of bicategories. Preprint (2009)
  4. 4.
    Buckley, M.: Fibred 2-categories and bicategories. J. Pure Appl. Algebra 218(6), 1034–1074 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cisinski, D.C.: Images directes cohomologiques dans les catégories de modèles. Ann. Math. Blaise Pascal 10(2), 195–244 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gordon, R., Power, A.J., Street, R.: Coherence for tricategories. Mem. Am. Math. Soc. 117(558), vi+81 (1995)Google Scholar
  7. 7.
    Gurski, N.: An Algebraic Theory of Tricategories. Ph.D. Thesis, University of Chicago, (2006)
  8. 8.
    Hermida, C.: Some properties of \(\text{ Fib }\) as a fibred 2-category. J. Pure Appl. Algebra 134(1), 83–109 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hermida, C.: Representable multicategories. Adv. Math. 151(2), 164–225 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hermida, C.: Fibrations for abstract multicategories. Galois Theory Hopf Algebras and Semiabelian Categ. 43, 281–293 (2004)MathSciNetMATHGoogle Scholar
  11. 11.
    Hörmann, F.: Fibered multiderivators and (co)homological descent. Theory Appl. Categ. 32(38), 1258–1362 (2017a)MathSciNetMATHGoogle Scholar
  12. 12.
    Hörmann, F.: Derivator six-functor-formalisms—definition and construction I. arXiv:1701.02152 (2017b)
  13. 13.
    Kapranov, M.M., Voevodsky, V.A.: 2-Categories and Zamolodchikov tetrahedra equations. In: Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Volume 56 of Proceedings of Symposium Pure Mathematics, pp. 177–259. American Mathematical Society, Providence, RI (1994)Google Scholar
  14. 14.
    Lipman, J., Hashimoto, M.: Foundations of Grothendieck duality for diagrams of schemes. Lecture Notes in Mathematics, vol. 1960. Springer, Berlin (2009)Google Scholar
  15. 15.
    Shulman, M.A.: Constructing symmetric monoidal bicategories. arXiv:1004.0993 (2010)

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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