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

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

2004-09-24

arxiv.org/abs/math/0409477v1

Mathematics

Category Theory

21 pages

Scientific paper

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.

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