Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid

Isar Stubbe (1, 2)

1 ULCO - Université du Littoral Côte d'Opale
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

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.

Domaines: Mathématiques [math]; Informatique [cs]; Physique [physics]

Isar Stubbe. Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid. Applied Categorical Structures, 2005, 13 (3), pp.235-255. ⟨10.1007/s10485-004-7421-5⟩. ⟨hal-03606840⟩