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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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Topos associated to a category

This is described in the paper Bunge and Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, 1995. which describes the sense in which the construction you call Spec i …
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