nLab universal epimorphism

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