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In McLarty's The Rising Sea: Grothendieck on simplicity and generality I found the following quote:

The same, Grothendieck knew, would work for cases yet unimagined. He notes that Tohoku [Grothendieck 1957] already gave foundations for the cohomology of any topos [Grothendieck 1985–1987, p. P41n.]. That context was hardly foreseen as he wrote Tohoku in 1955. This is one more proof that it was the right idea of cohomology.

In which sense gave Tohoku a foundation for the cohomology of any topos? In particular, which theorem in Tohoku proves or constructs the cohomology of toposes?

[Grothendieck 1985–1987, p. P41n.] is Récoltes et Semailles. (I can spot the passage in which Grothendieck refers to Tohoku, but this doesn't answer my question.)

I could swear I heard the claim before that Tohoku is the only place in the literature which shows that toposes have cohomology (of course without mentioning the word "topos"), although I can't recall at the moment where I heard that.

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    $\begingroup$ Maybe this is referring to the fact that Tohoku establishes that what we now call Grothendieck abelian categories have enough injectives. Sheaves of abelian groups on a (Grothendieck) topos form a Grothendieck abelian category, hence have enough injectives, so one can take derived functors of global sections $\endgroup$
    – Dylan Wilson
    Commented Nov 22, 2021 at 14:06
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    $\begingroup$ I suspect Dylan Wilson's comment is the answer. However, the claim that Tohoku is the only place... seems pretty dubious, cf SGA4 exp. V. $\endgroup$
    – Donu Arapura
    Commented Nov 22, 2021 at 14:20
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    $\begingroup$ I would also not say that Tohoku was the only place... but I think your question is the topic of this talk, (or here). $\endgroup$
    – user234212323
    Commented Nov 22, 2021 at 14:31
  • $\begingroup$ @scritoriano I can't speak French. Can you translate some points which are relevant to my question? $\endgroup$
    – user469007
    Commented Nov 22, 2021 at 15:11
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    $\begingroup$ I agree too with @DylanWilson that the comment refers to the existence of injectives on a Grothendieck category. I suggest to post its as an answer. $\endgroup$
    – Leo Alonso
    Commented Nov 22, 2021 at 16:22

1 Answer 1

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As requested:

By Theorem 1.10.1 in Tohoku, an Grothendieck abelian category has enough injectives. Sheaves of abelian groups on a Grothendieck topos form a Grothendieck abelian category. By Theorem 2.2.2 in Tohoku, one may then take derived functors of global sections.

As mentioned in the comments, though, sheaf cohomology on toposes is developed quite a bit elsewhere (e.g. in SGA 4).

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    $\begingroup$ Thanks! Are derived functors due to Grothendieck? $\endgroup$
    – user469007
    Commented Nov 22, 2021 at 17:38
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    $\begingroup$ @user469007 I think they are due to Cartan-Eilenberg $\endgroup$
    – user234212323
    Commented Nov 22, 2021 at 18:35
  • $\begingroup$ @user469007 I think Cartan-Eilenberg are responsible for derived functors in the category of modules over a ring. Grothendieck for the more general version. $\endgroup$
    – Descartes Before the Horse
    Commented Nov 23, 2021 at 20:25

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