So, what is a derived functor?

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We rethink the notion of derived functor in terms of corre-spondences, that is, functors ε → [1]. While derived functors in our sense, when they exist, are given by Kan extensions, their existence is a strictly stronger property than the existence of Kan extensions. We show, however, that derived functors exist in the cases one expects them to exist. Our definition is espe-cially convenient for the description of a passage from an adjoint pair (F;G) of functors to a derived adjoint pair (LF;RG). In particular, canonicity of such a passage is immediate in our approach. Our approach makes perfect sense in the context of ∞-categories.

Original languageEnglish
Pages (from-to)279-293
Number of pages15
JournalHomology, Homotopy and Applications
Issue number2
StatePublished - 2020

Bibliographical note

Funding Information:
This work, being formally independent of George Maltsiniotis’ [Mal], stems from a similar dissatisfaction with the existing definition of derived functor. We are grateful to him for bringing our attention to this work. We are grateful to B. Keller who explained to us that Deligne’s definition leads to automatic adjunction of the derived functors. We are also grateful to D.-C. Cisinski who informed us about his book [C]. Discussions with Ilya Zakharevich were very useful. We are very grateful to the referee for his request to clarify some sloppy passages in the original version of the paper. The present work was partially supported by ISF grants 446/15 and 786/19.

Publisher Copyright:
© 2020, Vladimir Hinich.


  • Derived functor
  • ∞-category

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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