Definition 7.10.
Let \((C,C^{\ad})\) be a category with an admissibility structure. A topology \(\tau\) is said to be compatible with \(C^{\ad}\) if every \(\tau\)-covering sieve contains a \(\tau\)-covering sieve generated by admissible maps. We refer to such triples
\[\Gg = (C,C^{\ad},\tau)\]
as geometric sites. Covering families generated by admissible morphisms will be called admissible covers. If \(C\) is small and idempotent complete, we will also call such a triple a geometry.