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?
They're not particularly related in any way that I know of.
They're both just two examples of the generic notion of "lifting": given some spaces $X,Y$ with a "projection" map $p : Y \to X$ (typically very far from injective), for each $x \in X$, identify, in a "canonical" way, some $y \in Y$ with $p(y) = x$.
In the video, $X,Y$ are topological spaces and $p$ is a covering map.
In rough paths theory, $X$ could be the space of ordinary paths, $Y$ the space of "enhanced" rough paths (where an element consists of a path together with some higher-order objects that can be seen as iterated integrals of the path), and the map $p$ is "forget the higher-order stuff". So "lifting" just means making a "good" choice for what the higher-order stuff ought to be, in a particular context.