Effective Descent Morphisms of Filtered Preorders

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 \(X\rightarrow A\) $X\to A$ and, equivalently, as a set A equipped with a family \((A_x)_{x\in X}\) ${(A_x)}_{x\in X}$ of upclosed subsets of A with \(x'\leqslant x\Rightarrow A_x\subseteq A_{x'}\) $x^{\prime }⩽x\Rightarrow A_x\subseteq A_{x^{\prime }}$ .