In algebraic geometry, what kind of theory can only be described by topos but not a site?

Question

A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books about Etale cohomology don't use the notion of topos. It seems study of sites is enough for Etale cohomology. But topos was invented in algebraic geometry. I wonder in what area of algebraic geometry, study of topoi can not be replaced by study of sites?

Answer 1

Indeed, all the topoi I personally know to appear in algebraic geometry are Grothendieck toposes, hence toposes of sheaves over some site.

However, there is NO one-to-one correspondence between morphisms of sites and morphisms of the toposes over them. In particular the morphisms between crystalline toposes can not be realized by morphisms between the underlying sites. This is one reason why we're very much interested in toposes.

There are also other reasons, for instance that they provide a "base-independent" (site-independent) notion; exploiting that the same topos can be presented by vastly different sites is the heart of Olivia Caramello's program of toposes as bridges. We can also exploit the internal language of a topos allows to reduce some notions and theorems of algebraic geometry to notions and theorems of linear algebra and to develop a synthetic account of scheme theory (see these notes).