Lemma 24: Fix basepoints as usual. There is a lift of such that if and only if
Furthermore, if the lift exists, it is unique.
Lemma 25: Given a choice of , there exists an elevation of at . Furthermore, is unique in the sense that if is another elevation of and then there is a homeomorphism and the diagram
Let be defined by . By Lemma 24, the composition
lifts at , call this lift . Suppose
Then . But by Lemma 24, . This implies .
Another, more categorical construction uses the fibre product.
The fibre product is defined by
There are obvious maps
given by forgetting factors.
Exercise: If is a covering map then is a covering map.
Lemma 26: Fix . Let and let for . Let , and let be the connected component containing . Then is an elevation of at , and every elevation of arises in this way.
Proof: To prove that is an elevation we just observe that . Now suppose
is an elevation. Then , with .
The covering map factors trhough , and so is a covering map. Because is an elevation, is a homeomorphism onto its image, a connected component of .
What has this got to do with graphs of spaces/groups?
Let be a vector space, and let be an edge map. Define to be the mapping cylinder
comes with a map such that and . This is an inclusion , and . Let be a covering map.
Let be the fibre product. There’s a map ; . Clearly, .
Therefore, is an injection. It’s easy to see that induces a bijection at the level .
Lemma 27: Any covering space arises as the fibre product of a covering map .
Proof: Let be a covering map
let be the fibre product of and .
Define by . As before, is a covering map.