This question is related to this question from Dinakar, which I found interesting but don't yet have the background to understand at that level.

Unless I'm mistaken, the rough statement is that H^{n}(X;G) (the n-dimensional cohomology of X with coefficients in G) should somehow correspond to (free?) homotopy classes of maps X --> K(G,n). I want to understand this better, in relatively elementary terms. Here are some questions which (I hope) will point me in the right direction.

- What category are we working in? My guess is that X should just be a topological space, the cohomology is singular cohomology, and our maps X --> K(G,n) just need to be continuous.
- Does this carry over if we give X a smooth structure, take de Rham cohomology, and require our maps X --> K(G,n) to be smooth?
- How does addition in H
^{n}(X;G) carry over? - How does the ring structure on H*(X;G) carry over? (This has probably been adequately answered to Dinakar already.)