Lax comma categories of ordered sets

Abstract

AbstractLet 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): 06A0718A2518A3018N1018D2018E50Key words: Effective descent morphismslax comma 2-categoriescomma categoriesexponentiabilitycartesian closed categoriestopological functorsenriched categoriesOrd-enriched categories