I am trying to learn rough path theory as described in https://en.wikipedia.org/wiki/Rough_path. In this wikipedia page "rough path lift" is mentioned. I have gone though the following link which talks about path lifting property https://www.youtube.com/watch?v=snHIkGVlmkM .
Is there a connection between path lifting property mentioned in the youtube video and "rough path lift" mentioned in the wikipedia page.
An explanation how they are related or unrelated would be highly appreciated.
My intuition is that can we assume the d-dimensional rough path space as X as mentioned in the video. Further, is Y of video is the space of iterated integrals or something like that? In such case what are $\phi$ of the video or $p$ of the video?