Categorical Structures Enriched in a Quantaloid: Orders and Ideals over a Base Quantaloid

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call ‘Q-order’. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a locally ordered category Ord(Q) of Q-orders and monotone maps; actually, Ord(Q) = Map(Idl(Q)). In particular is Ord(Ω), with Ω a locale, the category of ordered objects in the topos of sheaves on Ω. In general Q-orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of Q. Applied to a locale Ω this generalizes and unifies previous treatments of (ordered) sheaves on Ω in terms of Ω-enriched structures.