nLab lift

Idea

A lift or lifting of a morphism $f \colon X \to Y$ (in some category) through an epimorphism (or more general map) $p \colon \widehat{Y} \to Y$, is a morphism $\tilde{f} \colon X\to \widehat{Y}$ such that $f = p\circ\tilde{f}$:

This is the concept formally dual to extension.

More generally, given a square commuting diagram, then one says a lift in the diagram is a dashed morphism from the bottom left to the top right, making both resulting triangles commute:

This reduces to the previous situation in the case that $\widehat{X}$ is an initial object. (Whereas, when $Y$ is a terminal object then it reduces to the situation of an extension).

If such a lift exists at all, one also says that $h$ has the left lifting property against $p$, and equivalently that $p$ has the right lifting property against $h$.

Related concepts

• lifting property

• Kan lift

• lifts and extensions

• section

• projective object

• orthogonality