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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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Why isn't the orbifold cohomology of $pt/G$ equal to the cohomology of $BG$?

Orbifold cohomology is not a model for the cohomology of the orbifold (or more generally stack) itself, but for that of its inertia stack (a.k.a. derived loop space), which parametrizes points of the …
David Ben-Zvi's user avatar