Homotopy lemma

Proof of nerve lemma. Let T> be the nerve diagram of tl, and let D be the trivial diagram over BdA/"(W), i.e., all associated spaces are points. Consider the unique diagram map X -.V V. Since all maps X cr) for a Af(U) are homotopy equivalences, we can conclude by the homotopy lemma that also the map hocolimX hocolimP — hocolimH is a homotopy equivalence. 

Now consider the general case for a. As the functors in question on F changes coproducts to products, the map in question is a quasi-isomorphism if F is a direct sum of complexes bounded above with equivariant cohomology groups. Indeed, a direct product of quasi-isomorphisms of complexes of PM(A i) is again quasi-isomorphic. In particular, the lemma holds if F is a homotopy colimit of objects of As any object F of Dem X,) is the... 

We prove the case c. By Lemma 13.6, we may assume that G is a homotopy limit of if-injective complexes with locally quasi-coherent bounded below cohomology groups. As the functors on G in consideration commute with homotopy limits, the problem is reduced to the case a. ... 

Lemma 3.7. — For any 36 the canonical morphism 36 HomSf,.% ) is a strict -homotopy equivalence, and thus an 1-weak equivalence. 

Note that the realization functor — Iaj A SfwJT) A Shv(T) takes values in the fuU subcategory of simplicial sheaves of simplicial dimension zero, i.e. factors through a functor — a A ShifT) Sfw(Tj which is left adjoint to the restriction of C, to Shv(T). Together with Lemma 3.12 this fact can be used to obtain an alternative description of the homotopy category, 9(0 (T, I) as follows. 

We consider the functoriality of homotopy categories of sites with intervals only in the case of reasonable continuous maps of sites cf 1.55). We have the following obvious lemma. 

We will also use the obvious notations Z", Z", and By Lemma 2.13 these suspension functors define functors on, , (5) and by Lemma 2.15 on the level of A -homotopy categories we have a canonical isomorphism of functor Zr = Z, oZ,. 

Our definition of a good G-space together wdth Lemma 1.16 implies that any simplicial sheaf on (G — Tk)j is simplicially weakly equivalent to a simplicial sheaf whose terms arc direct sums of sheaves represented by G-spaces Ua such that Ua —> 7t()(Ucx) is a G-homotopy equivalence. Applying the functor tiq to termwise we get a new simplicial sheaf which clearly belongs to the image of tc. On... 

Proof — By definition BrtG = R7i,(BG) where n Sm/Sf, (5m/S) ij is the obvious morphism of sites. Since the third condition of Lemma 3.15 clearly holds for 7t so does the first and therefore it is sufficient to show that BG is A -local in A° ShVet Sm/S). Let, i/9G be a simplicially fibrant model for BG. Using Lemma 2.8(2) wc see that it is sufficient to show that for any striedy henselian local scheme S and a finite etale group scheme G over S of order prime to char S) the map of simplicial sets. G(S). G(As) is a weak equivalence. Since S is strictly henselian G is just a finite group. In particular we obviously have G(S) = G(A ). We also have H],(S,G) = and H],(As, G) = where the second equality holds because of the homotopy invariance of the completion of outside of characteristic ([13]) and therefore our map is a weak equivalence by Proposition 1.16. 

In Euclidean space, it can be seen that the weighted Voronoi regions and all their intersections are contractible therefore it follows by the nerve lemma (Theorem 15.21) that the dual complex of a ball collection is homotopy equivalent to the union of these balls. 

Furthermore, by the projection lemma, the map pj hocolimP —> colimP is a homotopy equivalence. Since coIimP = X and hocolimP = N IA), the nerve lemma follows. ... 

Let Si colimPi —> hocolimbe the Segal map defined in the proof of the projection lemma, where it was also shown that it is a homotopy equivalence. Finally, the projection mappj hocolim X>2 —cohmP2 is a homotopy equivalence by the projection lemma, and therefore we conclude that the composition... 

WZZ99] V. Welker, G.M. Ziegler, R. Zivaljevic, Homotopy colimits - comparison lemmas for combinatorial applications, J. Reine Angew. Math. 509 (1999), pp. 117-149. 

It follows from the Lemma below that homotopy equivalent... 

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