nLab
universally closed morphism

Idea

A universally closed morphism is a closed morphism? all whose pullbacks are also closed.

Definition

Let $C$ be a category with pullbacks and with a notion of closed morphism? which is stable under composition and contains all the isomorphisms.

A morphism $f:X\to Y$ in $C$ is universally closed if for every $h:Z\to Y$ the pullback $h^{*}(f):Z{\times }_{Y}X\to Z$ is a closed morphism?.

In particular, for $h={\mathrm{id}}_Y$ we see that a universally closed morphism is itself closed.

Examples

• The assumptions are satisfied in the category of schemes. See valuative criterion of properness for a characterization of universally closed morphisms of schemes via valuation rings.