Grothendieck topology

Let $\mathcal C$ be a category. A sieve $S$ on an object $U$ is a collection of morphisms with codomain $U$ that is closed under precomposition: if $V\to U$ belongs to $S$, then every composite $W\to V\to U$ belongs to $S$.

A Grothendieck topology $J$ on $\mathcal C$ assigns to every object $U$ a collection $J(U)$ of sieves, called covering sieves, such that:

1. the maximal sieve of all morphisms into $U$ belongs to $J(U)$;
2. if $S\in J(U)$ and $f:V\to U$, then the pullback sieve $f^*S$ belongs to $J(V)$;
3. if $S$ is a sieve on $U$, $R\in J(U)$, and $f^*S\in J(V)$ for every $f:V\to U$ in $R$, then $S\in J(U)$.

Thus a Grothendieck topology specifies what it means to work locally on the objects of a category; it need not arise from a topology on a set of points.