Lax comma categories of ordered sets

TL;DR

This paper investigates the properties of lax comma categories of ordered sets, showing conditions under which they are topological, complete, and cartesian closed, and analyzing descent theory within these categories.

Contribution

It characterizes when the lax comma category of ordered sets is topological, complete, and cartesian closed, extending understanding of their categorical properties and descent behavior.

Findings

The forgetful functor is topological iff X is complete.

The category is complete and cartesian closed iff X is.

Pointwise effectiveness for descent implies effectiveness in the lax comma category.

Abstract

Let $\mathsf{Ord} $ be the category of (pre)ordered sets. Unlike $\mathsf{Ord}/X$, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category $\mathsf{Ord} //X$. In this paper we show that the forgetful functor $\mathsf{Ord} //X\to \mathsf{Ord} $ is topological if and only if $X$ is complete. Moreover, under suitable hypothesis, $\mathsf{Ord} // X$ 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 $\mathsf{Ord} //X$, being pointwise effective for descent in $\mathsf{Ord} $ is sufficient, while being effective for descent in $\mathsf{Ord} $ is necessary, to be effective for descent in $\mathsf{Ord} //X$.