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There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions are discussed in this paper by Hinich.

I am looking for a definition of derived functors that answers all of the following criteria.

  1. It's reasonably conceptual (i.e defined by universal properties) and does not require too many constructions (so nothing like injective resolutions).
  2. It admits a reasonably quick path to proving the classical comparison theorems for sheaf cohomology.
  3. No mention of triangulated categories or their structure.

I haven't been able to find a reference that gives a good definition of derived functors and also uses it to prove the comparison theorem, so if you know one, a reference would be great.

I asked about the relation between the $\delta$-functor approach and Kan extension along localizations here, but ideally, I'd like to avoid $\delta$-functors entirely. Is there such a path?

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  • $\begingroup$ Does your desire to avoid triangulated categories extend to stable ∞-categories? If so I cannot see how you could avoid $\delta$-functors. $\endgroup$
    – Denis Nardin
    Commented Apr 3, 2020 at 8:50
  • $\begingroup$ I don't know what stable $\infty$-categories are but I have nothing against them. My fear is that they don't admit a "reasonably quick path" to concrete comparison theorems, but I'd be happy to discover otherwise. I should emphasize that I want to be able to write down all the details. $\endgroup$
    – Arrow
    Commented Apr 3, 2020 at 9:01
  • $\begingroup$ See my answer here for a quick rundown of stable ∞-categories: mathoverflow.net/questions/344219/… . Admittedly you need to develop some theory before working with them. And what do you mean by "write down all details"? Starting from ZFC set theory (I assume not)? What kind of things are you willing to take for granted? $\endgroup$
    – Denis Nardin
    Commented Apr 3, 2020 at 9:04
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    $\begingroup$ @DenisNardin: If the OP already consider injective resolutions to have "too many constructions", then suggesting stable ∞-categories is a bit weird. Recall that Lurie's definition of the derived ∞-category (Definition 1.3.2.7 in Higher Algebra) uses projective resolutions, for example. So you suggested approach definitely violates OP's requirement of not using resolutions. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 3, 2020 at 17:30
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    $\begingroup$ Theoretically, some time in the future, homotopy type theory may be able to provide enough tools to state and prove comparison theorems without referring to injective resolutions, triangulated categories, or similiar tools. But right now, in the current state of the field, there is hardly anything simpler than injective resolutions, model categories, triangulated categories, stable ∞-categories, etc., all of which inevitably pass through some form of injective or projective resolutions. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 3, 2020 at 17:41

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