Lift (mathematics)

We say that f factors through h. 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.

[1] 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].

Thus in the category of topological spaces with continuous maps as morphisms, we have The notations of first-order predicate logic are streamlined when quantifiers are relegated to established domains and ranges of binary relations.

Gunther Schmidt and Michael Winter have illustrated the method of lifting traditional logical expressions of topology to calculus of relations in their book Relational Topology.

[2] They aim "to lift concepts to a relational level making them point free as well as quantifier free, thus liberating them from the style of first order predicate logic and approaching the clarity of algebraic reasoning."

denotes the identity relation on the range of M. "The notation for quantification is hidden and stays deeply incorporated in the typing of the relational operations (here transposition and composition) and their rules."

For maps of a circle, the definition of a lift to the real line is slightly different (a common application is the calculation of rotation number).