Topological hypercovers and A^1-realizations
Abstract:
Let {U_a} be an open cover of a topological space X. From this data one may build the Cech complex, which is the simplicial space that in dimension n consists of all the (n+1)-fold intersections of the U_a's. The first thing we show in this paper is that the homotopy colimit of this Cech complex is weakly equivalence to X. The main subject of the paper is generalizing this statement to apply to hypercovers: these are simplicial spaces which in each dimension are a disjoint union of open subsets of X, with the property that in dimension n one has a cover of the (n+1)-fold intersections. In other words, hypercovers are similar to Cech complexes except that at each level we may refine the intersections by choosing a smaller cover. The main result of this paper is a proof that the homotopy colimit of a hypercover is again weakly equivalent to the space X we started with. Various corollaries are deduced concerning homotopical decompositions. We use this result to construct a topological realization functor relating the Morel-Voevodsky model category for A^1-homtopy theory over the complex numbers to the usal model structure on topological spaces.