nLab
universal epimorphism

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

In particular, setting u=id Y:YYu = id_Y\colon Y \to Y, we see that ff itself is an epimorphism.

A morphism g:XYg\colon X\to Y is a universal monomorphism if its opposite g :Y X g^\circ\colon 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)