Lax comma categories of ordered sets

Abstract

Let Ord be the category of (pre)ordered sets. Unlike Ord/X , whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category Ord//X . In this paper we show that the forgetful functor Ord//X → Ord is topological if and only if X is complete. Moreover, under suitable hypothesis, Ord//X is complete and cartesian closed if and only if X is. We end by analysing descent in this category. Namely, when X is complete, we show that, for a morphism in Ord//X , being pointwise effective for descent in Ord is sufficient, while being effective for descent in Ord is necessary, to be effective for descent in Ord//X .

Mathematics Subject Classification (2020):

• 06A07
• 18A25
• 18A30
• 18N10
• 18D20
• 18E50

Key words:

• Effective descent morphisms
• lax comma 2-categories
• comma categories
• exponentiability
• cartesian closed categories
• topological functors
• enriched categories
• Ord-enriched categories