Let $E\rightarrow B$ be a fibration. It is well-known that this ensures that the map $PE\rightarrow PB×_BE$ has a section. What condition would ensure that this section can be choosen to lift constant paths to constant paths?
2026-04-13 10:09:43.1776074983
Path lifting function associated to a fibration
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This happens if and only if the fiber is totally path disconnected. For instance, the projection from the product with of the base space with the pseudoarc has your property whereas the projection from the product with the interval does not.