universal epimorphism

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

In particular, setting $u={\mathrm{id}}_{Y}:Y\to Y$, we see that $f$ itself is an epimorphism.

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

Revised on May 17, 2011 03:01:02
by Mike Shulman
(71.136.238.9)