nLab lift

Redirected from "lifting problems".
Contents

Contents

Idea

A lift or lifting of a morphism f:XYf \colon X \to Y (in some category) through an epimorphism (or more general map) p:Y^Yp \colon \widehat{Y} \to Y, is a morphism f˜:XY^\tilde{f} \colon X\to \widehat{Y} such that f=pf˜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 X^\widehat{X} is an initial object. (Whereas, when YY is a terminal object then it reduces to the situation of an extension).

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

Last revised on October 15, 2024 at 18:03:56. See the history of this page for a list of all contributions to it.