The question is how to classify the possible P's up to isomorphism. Let's pick another G bundle EG over BG. It will have the property that every G bundle is determined by the homotopy class of a map p from M to BG. Then the bundle over M becomes the pullback of this model bundle.
Why should this special bundle even exist? We have a contravariant functor from the homotopy category to set which takes my space X to all G-bundles on it up to isomorphism. The fact that we pull back bundles not push them indicates the contravariance. The question then becomes is this functor representable by some object in the homotopy category. This is the object that we will call BG. Or at least it is one of the possible choices all of which are equally good.
For a moment let us specialize to U(1) gauge theories. We will then be looking at line bundles. What is BU(1)? Let's use the fact that the infinite sphere is contractible. To see this think about the sphere as the normalized wavefunctions on the interval then push everything to the end so that eventually get the constant function. Now mod out by the U(1) of phases. This gives BU(1)=normalized pure states and EU(1)=all normalized states. Since EU(1) is contractible, this is actually a good classifying space that has the universal property/represents the functor. This is because contractibility of the total space means that the loop space of U(1) and BU(1) are homotopy equivalent via the long exact sequence having a whole bunch of zeroes. This allows us to see isomorphisms between each homotopy group at all n. Now we need to figure out how to classify maps from M to BU(1)=pure states. This is where we need to know about spectra for generalized cohomology.
Define K(Z,0)=Z and K(Z,n)=BK(Z,n-1). This gives K(Z,1)=BZ a space which has Z as its space of loops. Well that's U(1) because we can homotope any loop to a standard form z^n by gradually getting rid of those lower terms. Taking the next step is K(Z,2)=BU(1)=pure states. So to classify U(1) bundles we need the classes of maps from X to K(Z,2) but K(Z,2) is in the spectrum for plain old cohomology and so [X to K(Z,2)] is the second integral cohomology. This is clearly the easiest so far to calculate and is equivalent. We have our way to classify all U(1) bundles. Just compute the second integral cohomology. There will be a bundle associated with each class. This is the chern class of the line bundle. We have lot's of ways of seeing that chern classes classify line bundles. We could also use the long exact sequence in sheaf cohomology associated to transition maps or connections and curvature.
All line bundles have some chern class and the universal one EU(1) over BU(1) is no different so we have to go from the class c in the cohomology of BU(1) to the class in the cohomology of M. But remember we have that map p from X to BU(1), and this is co so we can just pull back c to M and changing p to another homotopic one will not make a bit of difference.
Let's give some more examples of the classifying spaces for some other gauge groups.
For U(n) start with the total space of orthonormal n frames sitting inside countable Hilbert space.
Modding out by U(n) makes all the n frames in the same n plane the same. Therefore BU(n) is the set of n-planes sitting inside Hilbert space. This checks out with n=1 too because we can associate to each pure state the line in Hilbert space that when properly normalized and phased out would give this state.
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