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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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Classifying space of large category?
Whenever I've seen the definition of the classifying space of a category, the category is always specified to be small. I understand the definition well enough for my purposes (I think), but it occurr …