The small étale site of a scheme

Let $X$ be a scheme in a fixed Grothendieck universe containing every scheme considered here. The small étale category of $X$ has as objects étale morphisms $U\to X$ and as arrows morphisms of $X$-schemes. A declared covering family of an object $U\to X$ is a set-indexed family of étale arrows $U_i\to U$ whose image union is $U$; the empty family covers the empty scheme. A family of this form is a Grothendieck pretopology when identity families cover, pullbacks of covers along arrows cover, and composites of covers cover. A sieve on $U$ is a collection of arrows with target $U$ closed under precomposition; the sieve generated by a family consists of arrows factoring through one of its members. The Grothendieck topology generated by the declared families is the smallest assignment of covering sieves containing every maximal sieve and every sieve generated by a declared family, and closed under pullback and local transitivity. The separate site-axiom lemma proves the declared families form a pretopology; the resulting site is denoted $X_{\acute{e}t}$.

Étale morphism of schemes

Jointly surjective étale families form a Grothendieck pretopology

The Stacks Project, Definitions 59.27.1 and 59.27.3

Milne, Lectures on Étale Cohomology, Chapter I, §§5–6

The pretopology certification invoked in the Definition assumes full AC, as recorded in the site-axiom lemma and in The Axiom of Choice; the category and the declared families can be specified without it.

For size bookkeeping, write $\mathcal{U}$ for the fixed universe in the Definition: schemes and covering index sets are $\mathcal{U}$-small. Work with these collections in a larger ambient Grothendieck universe $\mathcal{V}$ with $\mathcal{U}\in \mathcal{V}$, and take presheaf values to be $\mathcal{V}$-small sets. The collection of $\mathcal{U}$-small schemes, their arrows, and their $\mathcal{U}$-indexed covering families is a set in $\mathcal{V}$; it need not be a set belonging to $\mathcal{U}$. This is the ambient set-sized interpretation used in forming matching-family quotients for sheafification. The definition does not assert that an arbitrary sheaf of sets on this site is representable by a scheme, or that a torsor sheaf is representable.