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My naïve cartoon picture of the construction of étale cohomology is this:

  1. start with a scheme, associate to it a Grothendieck topology (making a site).
  2. A functor from the Grothendieck topology to abelian groups (say) has all the relevant properties of a presheaf (by the definition of a Grothendieck topology) and so one gets cohomology by sheafifying and taking (as it were) sheaf cohomology.

My question is: is there a “minimal” reference describing the second step above without caring about schemes or étale cohomology (the first step)? Of course, I don't mind if the reference covers étale cohomology, as long as steps 1 and 2 are separated.

Having formulated this question and anticipating the answer let me ask a second question: are there, among the first six exposés of SGA4, parts that I can skip while trying to learn about step 2?

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    $\begingroup$ The words used are the same, and indeed one is a generalisation of the other, but a sheaf on a Grothendieck site is a different thing from a sheaf on a topological space so you need different tools to compute cohomology. (For example, there is no analogue of the Godement resolution.) $\endgroup$
    – Zhen Lin
    Jun 17, 2021 at 11:32
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    $\begingroup$ One takes the topos of sheaves on the site, then considers abelian group objects in it, then maybe there are some assumptions that ensure you have an abelian category with the right properties to then use Grothendieck's Tohoku paper? $\endgroup$ Jun 17, 2021 at 12:59
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    $\begingroup$ Johnstone’s book on topos theory from way back has a section on topos cohomology. Not the elephant book but his original book. The book is not exactly reader friendly for non category theorists but it is shorter than SGA. I think he users Barr's covering theorem to get that there are enough injectives but once you have that you are good to go. There are no schemes in the book $\endgroup$ Jun 17, 2021 at 13:55
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    $\begingroup$ The category of abelian group objects in a Grothendieck topos is a Grothendieck abelian category – you get local presentability by the usual yoga and you inherit the required exactness properties because sheafification is exact – so the Tôhoku paper is indeed applicable. It would be nice if the OP could indicate whether this is the level of answer sought. $\endgroup$
    – Zhen Lin
    Jun 18, 2021 at 15:10
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    $\begingroup$ @ZhenLin Maybe I'm not understanding what you mean, but I think there is indeed a Godement resolution for Grothendieck sites: link.springer.com/article/10.1007/s13348-014-0123-x and link.springer.com/article/10.1007/s13348-016-0171-5 $\endgroup$ Jun 19, 2021 at 6:31

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Artin, M. Grothendieck topologies. (English) Zbl 0208.48701 Cambridge, Mass.: Harvard University. 133 p. (1962). (pdf copy)

These notes seem to fit your description precisely. They are concise, start from first principles, assuming basically only knowledge of Grothendieck's Tôhoku paper. The focus is specifically on how to define cohomology in a topos, as opposed to many references on topos theory aimed more in the direction of algebraic stacks, logic, motivic homotopy...

(This text was recommended in a now deleted answer by another user. In my opinion Artin's notes are quite nice and I thought the recommendation was worth preserving.)

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