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Topological hypercovers and \(\mathbb{A}^1\)-realizations. (English) Zbl 1055.55016
The authors define a hypercover of a topological space X as an augmented simplicial space \(U_*\to X\) such that the maps \(U_n\to M^X_nU\) are open covering maps for all \(n\geq 0\). Here \(M^X_nU\) denotes the \(n\)th matching object of \(U_*\) computed in the category of simplicial spaces over \(X\). The main result of this paper asserts that if \(U_*\to X\) is a hypercover then the maps \(\operatorname{hocolim} U_*\to | U_*| \to X\) are both weak equivalences, where \(| U_*| \) denotes geometric realization. As a consequence, topological realization functors for the \(\mathbb{A}^1\)-homotopy theory of schemes over real and complex fields [J. F. Jardine, Doc. Math., J. DMV 5, 445–553 (2000; Zbl 0969.19004)] are constructed.
An interesting fact, proved and used in the paper, is the following. For topological spaces, the homotopy colimit of a simplicial space \(U_*\) is weakly equivalent to the realization of \(U_*\), even if \(U_n\) are not cofibrant.

55U35 Abstract and axiomatic homotopy theory in algebraic topology
14F20 √Čtale and other Grothendieck topologies and (co)homologies
14F42 Motivic cohomology; motivic homotopy theory
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