Effective descent morphisms of filtered preorders

Abstract

We characterize effective descent morphisms of what we call filtered preorders, and apply these results to slightly improve a known result, due to the first author and F. Lucatelli Nunes, on the effective descent morphisms in lax comma categories of preorders. A filtered preorder, over a fixed preorder $X$, is defined as a preorder $A$ equipped with a profunctor $X\to A$ and, equivalently, as a set $A$ equipped with a family $(A_x{)}_{x\in X}$ of upclosed subsets of $A$ with $x^{\prime }⩽x\Rightarrow A_x\subseteq A_{x^{\prime }}$.