  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(Omega), with Omega a locale, the
category of ordered objects in the topos of sheaves on Omega. In general
Q-orders can equivalently be described as Cauchy complete categories enriched
in the split-idempotent completion of Q. Applied to a locale Omega this
generalizes and unifies previous treatments of (ordered) sheaves on Omega in
terms of Omega-enriched structures.
