This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Lift_(mathematics) 00:00:36 See also Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago. Learning by listening is a great way to: - increases imagination and understanding - improves your listening skills - improves your own spoken accent - learn while on the move - reduce eye strain Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone. Listen on Google Assistant through Extra Audio: https://assistant.google.com/services/invoke/uid/0000001a130b3f91 Other Wikipedia audio articles at: https://www.youtube.com/results?search_query=wikipedia+tts Upload your own Wikipedia articles through: https://github.com/nodef/wikipedia-tts Speaking Rate: 0.794069559500817 Voice name: en-US-Wavenet-E "I cannot teach anybody anything, I can only make them think." - Socrates SUMMARY ======= 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 f = g ∘ h. We then say that f factors through h. A basic example in topology is lifting a path in one space to a path in a covering space. Consider, for instance, 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 f : [ 0 , 1 ] → R P 2 , (projective plane path) g : S 2 → R P 2 , (covering map) h : [ 0 , 1 ] → S 2 . (sphere path) {\displaystyle {\begin{aligned}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{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.

• published: 07 Oct 2019
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