Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 22 / 7 / i

A dense sieve, also called a persistently nonempty sieve, is one whose every pullback sieve on a category is nonempty: a member factoring through $f$ has the form $fh$, corresponding to a member $h$ of $f^{\ast }R$. The maximal sieve on a category clearly has this property. If $R$ is persistently nonempty, then $u^{\ast }R$ is too: for any $v$ into the domain of $u$, apply density to $uv$ to find an extension $uvh\in R$, giving $vh\in u^{\ast }R$.

For transitivity, suppose $R$ is dense on $n$ and $S$ is a sieve on a category such that $r^{\ast }S$ is dense for each $r\in R$. Given $f:m\to n$, choose $h$ with $fh\in R$. Density of $(fh{)}^{\ast }S$, tested on its identity arrow, supplies $k$ with $fhk\in S$. Thus $S$ is dense. These are the maximality, pullback and transitivity axioms of a Grothendieck topology. Therefore the specified sieves on a category form the coverage $J$, namely the double-negation topology. The empty sieve on a category is never covering, because its identity pullback is empty.