This is really more deserving of being a comment(but I don't have enough points to make them), check out this MO post What properties make $[0,1]$ a good candidate for defining fundamental groups?

Perhaps you will find some inspiration in terms of other ways to think about the role of $[0,1]$ and what its analogue should be in scheme-land.

This is really more deserving of being a comment(but I don't have enough points to make them), check out this MO post What properties make $[0,1]$ a good candidate for defining fundamental groups?

Perhaps you will find some inspiration in terms of other ways to think about the role of $[0,1]$ and what its analogue should be in scheme-land.