Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid
摘要
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.
引用
@article{arxiv.math/0409477,
title = {Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid},
author = {Isar Stubbe},
journal= {arXiv preprint arXiv:math/0409477},
year = {2007}
}
备注
21 pages