Lax comma categories of ordered sets

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 $\mathsf{Ord}//X$.

12 pages

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• Topological lax comma categories