This is really more deserving of being a comment(but I don't have enough points to make them), check out this MO post http://mathoverflow.net/questions/92206/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.