category theory

# Contents

## 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 $h=id_Y$ we see that a universally closed morphism is itself closed.

## Examples

Revised on May 1, 2011 08:43:10 by Zoran Škoda (109.227.47.152)