Lift (mathematics)

In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such that f = g∘h. We say that f factors through h.

Lift of f (commutative diagram)

A basic example in topology is lifting a path in one topological space to a path in a covering space. For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval [0,1]. We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the category of topological spaces with continuous maps as morphisms, we have

${\displaystyle {\begin{aligned}f\colon \,&[0,1]\to \mathbb {RP} ^{2}&\quad &{\text{(projective plane path)}}\\g\colon \,&S^{2}\to \mathbb {RP} ^{2}&\quad &{\text{(covering map)}}\\h\colon \,&[0,1]\to S^{2}&\quad &{\text{(sphere path)}}\end{aligned}}}$

Lifts are ubiquitous; for example, the definition of fibrations (see homotopy lifting property) and the valuative criteria of separated and proper maps of schemes are formulated in terms of existence and (in the last case) uniqueness of certain lifts.

In algebraic topology and homological algebra, tensor product and the Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Ext functor and the Tor functor.