By C. Menini, F. van Oystaeyen
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Let D : ∆op → E be the diagonal of Z, defined by Dk = Zk,k ; thus hocolim∆op D ≈ f g hocolim∆op ×∆op Z ≈ 1. We define maps U − →D− → U as follows. The map f is given by the h h diagonal map fp : Up → Up ×Vp · · · ×Vp Up = Dp . The map g : D → U is adjoint to the map h : D|∆op → U |∆op defined as follows. For q ≤ n, Dq = Uq ×hVq · · · ×hVq Uq ≈ Uq since ≤n+1 ≤n+1 Uq = Vq in these degrees. For q = n+1 we define hn+1 : Dn+1 = Un+1 ×hVn+1 · · ·×hVn+1 Un+1 → Un+1 by projection to the first factor. One checks that this is indeed well-defined.
Let τ0 be the collection of sieves s : S → π0 yC in π0 C with the property that their lift s¯ : S¯ → yC is in T¯, and let τ denote the set of such lifts. It is easy to show that τ0 is a topology on C, and that τ¯ ⊆ T¯. I am going to show that elements of T¯ are actually ∞-connected maps in s PSh(C)τ and thus become weak equivalences after t-completion. This shows that E ≈ t(s PSh(C)τ ) if E is t-complete. Let f : X → Y ∈ T¯; we want to show that f is k-connected in s PSh(C)τ for all k. Let g : τkY f → Y be the relative truncation of f in s PSh(C).
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