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?



Cohomology of Grothendieck topology
Artin, M. Grothendieck topologies. (English) [Zbl 0208.48701](https://www.zbmath.org/?q=an%3A0208.48701) Cambridge, Mass.: Harvard University. 133 p. (1962). ([pdf copy](https://www.math.nagoya-u.ac.jp/%7Elarsh/teaching/S2013_AG/grothendiecktopologies.pdf))


These notes seem to fit your description precisely. They are concise, start from first principles, assuming basically only knowledge of Grothendieck's [Tôhoku paper](https://ncatlab.org/nlab/show/Tohoku). 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.)


