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→ A X → A and, equivalently, as a set A equipped with a family (Aₓ)x∈ X ( A x ) x ∈ X of upclosed subsets of A with x' x⇒ Aₓ⊆ Ax' x ′ ⩽ x ⇒ A x ⊆ A x ′ .