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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.

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