 Coverage on a category

A coverage specifies covering families on a small category, stable under refinement along arrows: for a covering family into $C$ and $f:B\to C$, there is a covering family into $B$ whose composites with $f$ factor through the original family. Passing to generated sieves and imposing maximality and local transitivity gives its associated Grothendieck topology, with the same sheaves on a site. One can equivalently specify the saturated covering sieves directly. Rigidity refers to these covering sieves, so it is unchanged by replacing a family presentation by its generated topology.