Applied Categorical Structures

, Volume 13, Issue 3, pp 235–255 | Cite as

Categorical Structures Enriched in a Quantaloid: Orders and Ideals over a Base Quantaloid

Article

Abstract

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid \(\mathcal{Q}\) , which we call ‘ \(\mathcal{Q}\) -order’. This requires a theory of semicategories enriched in the quantaloid \(\mathcal{Q}\) , that admit a suitable Cauchy completion. There is a quantaloid \(\mathsf{Idl}(\mathcal{Q})\) of \(\mathcal{Q}\) -orders and ideal relations, and a locally ordered category \(\mathsf{Ord}(\mathcal{Q})\) of \(\mathcal{Q}\) -orders and monotone maps; actually, \(\mathsf{Ord}(\mathcal{Q})=\mathsf{Map}(\mathsf{Idl}(\mathcal{Q}))\) . In particular is \(\mathsf{Ord}(\Omega)\) , with Ω a locale, the category of ordered objects in the topos of sheaves on Ω. In general \(\mathcal{Q}\) -orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of \(\mathcal{Q}\) . Applied to a locale Ω this generalizes and unifies previous treatments of (ordered) sheaves on Ω in terms of Ω-enriched structures.

Keywords

quantaloid quantale locale ordered sheaf enriched categorical structure Cauchycompletion 

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité de LouvainLouvain-la-NeuveBelgique

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