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ADITTYA CHAUDHURI (Nov 08 2021 at 05:59):Is there an analogue of homotopy lifting property for "thin homotopy" of smooth paths?
Morgan Rogers (he/him) (Nov 08 2021 at 10:53):More details please! What's a "thin homotopy"?
Reid Barton (Nov 08 2021 at 12:24):[[thin homotopy]], basically a homotopy that "does not cover any area"
Morgan Rogers (he/him) (Nov 08 2021 at 13:11):So the intuition that you're hoping is valid is "if I lift a thin homotopy it stays thin"? That seems reasonable.
But wait, the question was about the homotopy lifting property; certainly you can give a definition of "thin homotopy lifting property" by refining the categorical definition, but is there a further question that you have about this @ADITTYA CHAUDHURI ?
ADITTYA CHAUDHURI (Nov 08 2021 at 18:56):Reid Barton said:
[[thin homotopy]], basically a homotopy that "does not cover any area"
Yes.
ADITTYA CHAUDHURI (Nov 08 2021 at 18:57):Morgan Rogers (he/him) said:
So the intuition that you're hoping is valid is "if I lift a thin homotopy it stays thin"? That seems reasonable.
But wait, the question was about the homotopy lifting property; certainly you can give a definition of "thin homotopy lifting property" by refining the categorical definition, but is there a further question that you have about this ADITTYA CHAUDHURI ?
Yes, my question is "under what condition thin homotopy remains thin?"!!
Morgan Rogers (he/him) (Nov 09 2021 at 10:51):Assuming that you're lifting along a nice enough map (a proper covering map, say), the lift of a homotopy locally looks like the original homotopy. In particular, in a setting where the spaces are structured enough to compute areas, the lifted homotopy should have the same area as the original homotopy. You'll need to impose the relevant niceness conditions to ensure this, but these should be "the obvious ones" and the proof that thinness lifts will proceed by lifting the computation of the area of the homotopy.
Morgan Rogers (he/him) (Nov 09 2021 at 10:51):(Hard to say anything more precise unless you give us the setting you're working in)
ADITTYA CHAUDHURI (Nov 18 2021 at 07:02):@Morgan Rogers (he/him) Thanks, the question is "what extra conditions have to be imposed on a usual topological Serre/ Hurewicz fibration, so that the usual homotopy lifting property can be restricted to a thin homotopy lifting property? Is being a submersion enough?