*[van Kampen’s theorem can handle the situation in which the intersection is not simply connected (the free product becomes an amalgamated free product over the fundamental group of the intersection. – T.]*

1. Given the universal cover of M, a map is the same as a covering space of M that is a principal G-bundle (with G considered as a discrete group).

2. (Proposition 2) The pushout of two covering spaces over an open covering of M is a covering space of M, and if they are both principal G-bundles, then so is the pushout.

3. The restriction of a $G$-bundle to an open subspace U of M corresponds to the restriction of to .

The proof is then: by Point 1 we can do the construction entirely with G-bundles; by Point 2 we can create the map to G, which by Point 3 has the desired restriction properties and is unique because pushouts are unique.

This is, I guess, the Grothendieck-Galois-theoretic proof of the theorem. I think you could summarize even further by using the word “stack”, but I won’t :)

]]>That’s a funny one! ]]>