Elementary characterisation of small quantaloids of closed cribles

TL;DR

This paper characterizes small quantaloids of closed cribles via properties of Grothendieck topologies and categorical conditions, providing a precise equivalence criterion for their structure.

Contribution

It establishes necessary and sufficient conditions for a small quantaloid to be equivalent to a quantaloid of closed cribles from a small site, linking categorical properties with topological structures.

Findings

A small quantaloid is equivalent to R(C,J) iff certain categorical conditions hold.

The topology J on Map(Q) is canonical if coreflexives split in Q.

Provides a characterization connecting quantaloids, Grothendieck topologies, and categorical properties.

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.