higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Fix some scheme $S$.
The fpqc-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;
for every affine open $W \hookrightarrow X$ there exists $n \geq 0$, a function $a : \{1, \cdots, n\} \to I$ and affine opens $V_j \hookrightarrow T_{a(j)}$ with
This appears as (Stacks Project, def. 27.8.1). (DR: this reference is now incorrect! It should be tag number, but I can’t find it)
The last condition does imply that $\cup_i f_i(U_i) = X$.
The abbreviation “fpqc” is for fidèlement plat quasi-compacte : faithfully flat and quasi-compact.
Because the collection of fpqc covers of a scheme does not have a small collection of refinements (Stacks project, Tag 0BBK), working with the fpqc topology can be set-theoretically tricky. Indeed, in 1975, Waterhouse gave an example of a functor on schemes that admits no fpqc sheafification. This contradicts many claims in the literature that fpqc sheafification and stackification is functorial (and such claims continue to be made).
fpqc-site $\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site
Chaper 27.8 in