A site in the stacks project is different from what is called a site in SGA4. What we call a site is what is called a category endowed with a pretopology (see Exposee II, Definition 1.3 of SGA4). In other words a site is category C endowed with a set Cov of families of morphisms with fixed target called coverings such that

  1. If V —> U is an isomorphism then {V —> U} is a covering,
  2. if {U_i —> U} is a covering and {V_{ij} —> U_i} is a covering for each i, then {V_{ij} —> U} is a covering,
  3. if {U_i \to U} is a covering and V —> U is a morphism of C then U_i \times_U V exists and {U_i \times_U V —> V} is a covering.

A sheaf on C is then a presheaf which satisfies the sheaf axiom for all the coverings. Note that in general there are many choices of Cov which give rise to the same category of sheaves. For example on (Sch), see previous post for notation, the etale coverings and the smooth coverings give rise to the same category of sheaves. For this reason you will sometimes hear people say that the etale and smooth topology are the same. But for us the etale site and the smooth site are different.

In this post I wanted to mention that working with sites as above is useful in that the types of coverings you allow can be used to express properties of the site which cannot be expressed in terms of the topology alone. For example, we can say that a property P of objects of C is local on the site if given a covering {U_i —> U} we have P(U) <=> P(U_i) for all i. Then it is clear that the property P(X) =”dim(X) < 17" is local on the etale site (Sch)_{etale} but not local on the smooth site (Sch)_{smooth}. Similarly for properties of morphisms, e.g., P(f)="f is locally quasi-finite" is local on the target on the etale site, but not local on the target on the smooth site. For a previous discussion of what it means for a property of morphisms to be "etale local on source and target", see this post.