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



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> 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.
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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.



Tohoku and cohomology of toposes
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).


