Topological hypercovers and 1-realizations

Abstract.We show that if U* is a hypercover of a topological space X then the natural map hocolim U* → X is a weak equivalence. This fact is used to construct topological realization functors for the 1-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.

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