Site

Remarks

Example: for a topological space $X$, let $\mathcal{O}(X)$ be the category of open sets with inclusions. Let a sieve on $U$ cover when the union of its members is $U$. This rule satisfies the three axioms of a Grothendieck topology, so $(\mathcal{O}(X), J)$ is a small site. Non-example: the category $\mathcal{O}(X)$ alone, with no choice of covering sieves, is not a site, since it does not say which families cover. Some authors define a site with a Grothendieck pretopology, that is, with covering families of morphisms instead of covering sieves; each pretopology determines a topology with the same sheaves.