Talk:Lift (mathematics)

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I would propose to move the example given in the text to covering space, because it is quite a specific example - the lifting not only exists, but it is also unique (after choosing one point) which is a peculiar feature of covering spaces. A generic diagram might be enough. (?) Jakob.scholbach 15:35, 7 April 2007 (UTC)

Any example will have special features; that is the nature of examples, and why they do not substitute for definitions, but supplement them. I would object to removing this example, which is seminal; I would support appending another one, which is more generic, if you wish.
In fact, I believe many readers will encounter "lifting" in topology before they encounter category theory, so I wonder if this article has not gotten off to the wrong start. I claim that the concept of lifting is too broadly used to have the definition hijacked by this category theory definition.
And, I'm not sure this one definition is so helpful in other contexts.
  • Consider a fiber bundle, F → E → B, and let the identity on B be f, and let the bundle projection be g; then h in the categorical definition is a section. In that context we prefer to distinguish lifts from sections, because lifts always exist, but (global) sections usually do not.
  • Or consider a short exact sequence of groups, 0 → K → G → H → 0, and let the identity on H be f, and let the quotient projection be g; then h would be a split. This, too, is a very special case (though a rather important idea), and not representative.
Usual practice would be to start with something like an orienting discussion and perhaps a topology definition, and work up to all the different uses, including this category theory definition. Even when lifting is discussed in the presence of categories, this definition may not be the most helpful one. (Example: Introduction to algebraic stacks by Voight.)
I realize this is a stub that doesn't know what it will be when it grows up. Michael Hardy asked for input, and I'm also curious as to what others think. --KSmrqT 20:34, 7 April 2007 (UTC)