A morphism $f\colon X \to Y$ is a **universal epimorphism** if for every morphism $u\colon V \to Y$ there is a pullback $X \times_Y V$ and its projection $u^*(f)\colon X \times_Y V \to V$ is an epimorphism.

In particular, setting $u = id_Y\colon Y \to Y$, we see that $f$ itself is an epimorphism.

A morphism $g\colon X\to Y$ is a **universal monomorphism** if its opposite $g^\circ\colon Y^\circ \to X^\circ$ is a universal epimorphism in the opposite category. In particular, it is a monomorphism.

Last revised on May 17, 2011 at 03:01:02. See the history of this page for a list of all contributions to it.