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Topological hypercovers and 1-realizations

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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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Correspondence to Daniel Dugger.

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Mathematics Subject Classification (2000):55U35, 14F20, 14F42

The second author was supported by an NSF Postdoctoral Research Fellowship

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Dugger, D., Isaksen, D. Topological hypercovers and 1-realizations. Math. Z. 246, 667–689 (2004). https://doi.org/10.1007/s00209-003-0607-y

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