Elementary characterisation of small quantaloids of closed cribles

Abstract

Each small site (C,J) determines a small quantaloid of closed cribles R(C,J). We prove that a small quantaloid Q is equivalent to R(C,J) for some small site (C,J) if and only if there exists a (necessarily subcanonical) Grothendieck topology J on the category Map(Q) of left adjoints in Q such that Q=R(Map(Q),J), if and only if Q is locally localic, map- discrete, weakly tabular and weakly modular. If moreover coreflexives split in Q, then the topology J on Map(Q) is the canonical topology.