= Solution
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 $f^*R$. The maximal <sieve on a category> clearly has this property. If $R$ is persistently nonempty, then $u^*R$ is too: for any $v$ into the domain of $u$, apply density to $uv$ to find an extension $uvh\in R$, giving $vh\in u^*R$.
For transitivity, suppose $R$ is dense on $n$ and $S$ is a <sieve on a category> such that $r^*S$ is dense for each $r\in R$. Given $f:m\to n$, choose $h$ with $fh\in R$. Density of $(fh)^*S$, 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 \b[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.
Back to article page
