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Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid | | Isar Stubbe
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24 Sep 2004 | | Subject: | Category Theory | math.CT | | Abstract: | 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. | | Source: | arXiv, math.CT/0409477 | | Services: | Forum | Review | PDF | Favorites | |
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