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The classifying space BG of a group G classifies principal G-bundles, in that homotopy classes of maps [X, BG] are naturally identified with isomorphism classes of principal G-bundles P ⭢ X.
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Bijection homotopy class of maps and homomorphisms of fundamental group
I was told there was a bijection between $[X;BG]$, the set of homotopy types of maps from a topological space $X$ to the classifying space $BG$, and the set of group homomorphisms $Hom(\pi_1 (X), G)$. …