higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Fix some scheme $S$.
the fppf-site (over $S$) is the site
whose underlying category is the category $Aff/S$ of affine schemes over $S$;
whose coverage has as covering families $\{f : U_i \to X\}$ those families of morphisms that are such that
each $f_i$ is a flat morphism;
each $f_i$ is locally of finite presentation;
we have that $X = \cup_i f_i(U_i)$.
This appears as (de Jong, def. 27.7.1, def 27.7.6, lemma 27.7.11).
The abbreviation “fppf” is for fidèlement plat de présentation finie : faithfully flat and of finite presentation.
But notice that it is common practice, as in the above definition, to require only local finite presentability.
fpqc-site $\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site
Chaper 27.7 in