# Lift (mathematics)

In the branch of mathematics called category theory, given a morphism f from an object X to an object Y, and a morphism g from an object Z to Y, a lift (or lifting) of f to Z is a morphism h from X to Z such that g $\circ$ h = f.
\begin{align} f\colon& [0,1] \to \mathbb{RP}^2 , &\qquad&\text{(projective plane path)} \\ g\colon& S^2 \to \mathbb{RP}^2 , &\qquad&\text{(covering map)} \\ h\colon& [0,1] \to S^2 . &\qquad&\text{(sphere path)} \end{align}