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 and cartesian closed, 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.