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added to Kan lift the def of absolute Kan lifts, and some more examples.
Btw, in all of the sources I’ve read about this, Kan lifts are simply called (left, right) liftings. What do you think about renaming this to liftings? will it conflict with some other kind of “lifting”?
Btw, in all of the sources I’ve read about this, Kan lifts are simply called (left, right) liftings. What do you think about renaming this to liftings? will it conflict with some other kind of “lifting”?
I wouldn’t quite want to rename the entry, as “Kan lifts” crucially differ from the usual default meaning of “lift” (in homotopy thory) by allowing the 2-cell to be non-invertible.
But what I do very much agree with is that it would be good amplify the relation to other notions of lifts. One could create a table that lists various notions of “lift” and the contexts in which they apply.
I think this is one of those things where once one is known to be working in a particular context, the adjective is unnecessary. So papers that are about, say, Yoneda structures, can say at the beginning “by a lifting we mean …” and then it is unambiguous for that paper. But there are enough other types of “liftings” in mathematics as a whole that the nLab page about this concept needs to distinguish itself from those other meanings somehow. But the page could helpfully remark that often the adjective is omitted.
Which is basically to say that I agree with Urs, I guess.
Now that I think about it, you’re right and guess I have to disagree with myself :)
I’ve added a remark about the Kan-omitted terminology, and a redirect from Kan lifting.
Urs,
It’d certainly be nice to have “lift disambiguation” page, but I don’t feel qualified to that (my knowledge of homotopy theory is pretty shallow). Out of curiosity, what you were referring to with “allowing the 2-cell to be non-invertible” ??
what you were referring to with “allowing the 2-cell to be non-invertible” ??
Precisely what it says in the entry: the 2-cell called $\epsilon$ there is in general not invertible.
For a lift in the sense of homotopy theory it would be invertible.
Precisely what it says in the entry: the 2-cell called ϵ there is in general not invertible.
For a lift in the sense of homotopy theory it would be invertible.
Ok I see, thanks.
Maybe that would be expressible as a Kan lift in some locally groupoidal 2-category.
Yes. In a (2,1)-category we “have homotopy theory”.
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