Topological hypercovers and 1-realizations

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Motivic Eilenberg-MacLane spaces (2008, background): “93] (see also [4]), it is sufficient to show that for any point x ∈ X(C) there exists an open neighborhood U ′ of p such that for any open subset U of U ′ the map q(U) → U is a weak equivalence.”
Factorization homology of topological manifolds (2012, unknown): “6 of [13] gives that the map colim (V ↪→N)∈Diskk/N ConfJ(f−1V )ΣJ −→ ConfJ (M)ΣJ is an equivalence of spaces, which completes the proof.”
Hypercovers and simplicial presheaves (2002, unknown): “This functor has ˇCech descent, because X is weakly equivalent to the homotopy colimit of the ˇCech complex associated to any open cover; see [ DI1 , theorem 1 · 1].”