Question

Another classical example of "looking at all choices instead of one" is the idea of the fundamental groupoid of a topological space. Instead of choosing one base point and letting all loops begin and end in this point one considers all paths between all points (modulo homotpy). This notion makes theorems like the Seifert-Van-Kampen-theorem much more natural. One does no longer have to add technical conditions that certain intersections contain the base point and are path connected.