The functors W and Diag Nerve are simplicially homotopy equivalent

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1 The functors W Diag Nerve are simplicially homotopy equivalent Sebastian Thomas January 7, 2008 Abstract Given a simplicial group G, there are two known classifying simplicial set constructions, the Kan classifying simplicial set WG Diag NG, where N denotes the dimensionwise nerve. They are known to be weakly homotopy equivalent. We will show that WG is a strong simplicial deformation retract of Diag NG. In particular, WG Diag NG are simplicially homotopy equivalent. 1 Introduction We suppose given a simplicial group G. Kan introduced in [10] the Kan classifying simplicial set WG. The functor W from simplicial groups to simplicial sets is the right adjoint, actually the homotopy inverse, to the Kan loop group functor, which is a combinatorial analogue to the topological loop space functor. Alternatively, dimensionwise application of the nerve functor for groups yields a bisimplicial set NG, to which we can apply the diagonal functor to obtain a simplicial set Diag NG. The latter construction is used for example by Quillen [12] Jardine [9, p. 41]. It is well-known that these two variants WG Diag NG for the classifying simplicial set of G are weakly homotopy equivalent ( 1. Better still, the Kan classifying functor W can be obtained as the composite of the nerve functor with the total simplicial set functor Tot as introduced by Artin Mazur [1] ( 2 ; Cegarra Remedios [3] showed that already the total simplicial set functor the diagonal functor, applied to a bisimplicial set, yield weakly homotopy equivalent results ( 3. Moreover, the model structures on the category of bisimplicial sets induced by Tot resp. by Diag are related [4]. The aim of this article is to prove the following Theorem. The Kan classifying simplicial set WG is a strong simplicial deformation retract of Diag NG. In particular, WG Diag NG are simplicially homotopy equivalent. This commutativity up to simplicial homotopy equivalence fits into the following diagram. ( simplicial groups Kan classifying functor W nerve N total simplicial set Tot ( simplicial sets ( bisimplicial sets diagonal simplicial set Diag associated complex associated double complex ( complexes homology of abelian groups total complex ( double complexes associated spectral sequence ( abelian groups ( spectral sequences approximation Mathematics Subject Classification 2000: 18G30, 55U10. This article has been published in Journal of Homotopy Related Structures 3(1 (2008, pp This is a slightly revised version from April 16, Addendum (December 19, 2011: The fact that WG Diag NG are weakly homotopy equivalent has been shown by Zisman [14, sec , cf. sec , rem. 1]. He shows that a morphism Diag NG WG, which is essentially the same as the morphism D G we consider in section 3, induces an isomorphism on the fundamental groups as well as isomorphisms on the homology groups of their universal coverings. 2 This is not the total simplicial set as used by Bousfield Friedler [2, app. B, p. 118]. 3 Addendum (December 19, 2011: To this end, Cegarra Remedios consider a morphism Diag X Tot X, which is essentially the same as the morphism φ X we consider in section 2. 1

2 By definition, the homology of a simplicial group is obtained by composition of the functors in the upper row. The generalised Eilenberg-Zilber theorem (due to Dold, Puppe Cartier [6, Satz 2.9] states that the quadrangle in the middle of the diagram commutes up to homotopy equivalence of complexes. The composition of the functors in the lower row yields the Jardine spectral sequence [9, lem ] of G, which has E 1 p,n p H n p (G p, which converges to the homology of G. Similarly for cohomology. Conventions notations We use the following conventions notations. The composite of morphisms X C F D D f Y Y G E is denoted by C G F E. g Z is denoted by X fg Z. The composite of functors If C is a category X, Y Ob C are objects in C, we write C (X, Y Mor C (X, Y for the set of morphisms between X Y. Moreover, we denote by (C, D the functor category that has functors between C D as objects natural transformations between these functors as morphisms. X Given a functor I C, we sometimes denote the image of a morphism i applies in particular if I op or I op op. We use the notations N {1, 2, 3,... } N 0 N {0}. θ j in I by X i X θ Xj. This Given integers a, b Z, we write [a, b] : {z Z a z b} for the set of integers lying between a b. Moreover, we write a, b : (z Z a z b for the ascending interval a, b (z Z a z b for the descending interval. Whereas we formally deal with tuples, we use the element notation, for example we write i 1,3 g i g 1 g 2 g 3 i 3,1 g i g 3 g 2 g 1 or (g i i 3,1 (g 3, g 2, g 1 for group elements g 1, g 2, g 3. Given an index set I, families of groups (G i i I (H i i I a family of group homomorphisms (ϕ i i I, where ϕ i : G i H i for all i I, we denote the direct product of the groups by i I G i the direct product of the group homomorphisms by i I ϕ i : i I G i i I H i, (g i i I (g i ϕ i i I. 2 Simplicial preliminaries We recall some stard definitions, cf. for example [5], [8] or [11]. Simplicial objects For n N 0, we let [n] denote the category induced by the totally ordered set [0, n] with the natural order, we let be the full subcategory in Cat defined by Ob : {[n] n N 0 }. The category of simplicial objects sc in a given category C is defined to be the functor category ( op, C. Moreover, the category of bisimplicial objects s 2 C in C is defined to be ( op op, C. The dual notion is that of the category csc : (, C of cosimplicial objects in C. δ For n N, k [0, n], we let [n 1] k [n] be the injection that omits k [0, n], for n N 0, k [0, n], we σ let [n + 1] k [n] be the surjection that repeats k [0, n]. The images of the morphisms δ k resp. σ k under a simplicial object X in a given category C are denoted by d k : X δ k, called the k-th face, for k [0, n], n N, resp. s k : X σ k, called the k-th degeneracy, for k [0, n], n N 0. Similarly, in a bisimplicial object X one defines horizontal vertical faces resp. degeneracies, d h k : X δ,id, d v k k : X id,δ k, sh k : X σ,id, s v k k : X id,σ k. Moreover, we use the ascending descending interval notation as introduced above for composites of faces resp. degeneracies, that is, we write d j,i : d j d j 1... d i resp. s i,j : s i s i+1... s j. The nerve We suppose given a group G. The nerve of G is the simplicial set NG given by N n G G n for all n N 0 by (g j j n 1,0 (N θ G ( g j i m 1,0 j (i+1θ 1,iθ 2

3 for (g j j n 1,0 N n G θ ([m], [n], where m, n N 0. Since the nerve construction is a functor Grp N sset, it can be applied dimensionwise to a simplicial group. This yields a functor sgrp N s 2 Set. From bisimplicial sets to simplicial sets We suppose given a bisimplicial set X. There are two known ways to construct a simplicial set from X, namely the diagonal simplicial set Diag X the total simplicial set Tot X, see [1, 3]. We recall their definitions. The diagonal simplicial set Diag X has entries Diag n X : X n,n for n N 0, while Diag θ X : X θ,θ for θ ([m], [n], where m, n N 0. θ To introduce the total simplicial set of X, we define the splitting at p [0, m] of a morphism [m] [n] in by Spl p (θ : (Spl p (θ, Spl p (θ, where [p] Spl p (θ [pθ] [m p] Spl p (θ [n pθ] are given by i Spl p (θ : iθ for i [0, p] i Spl p (θ : (i + pθ pθ for i [0, m p]. The total simplicial set Tot X is defined by Tot n X : { (x q q n,0 X q,n q xq d h q x q 1 d v 0 for all q n, 1 } for n N 0 q n,0 by (x q q n,0 (Tot θ X (x pθ X Splp (θ p m,0 for (x q q n,0 Tot n X θ ([m], [n], where m, n N 0. There is a natural transformation Diag φ Tot, where φ X is given by x n (φ X n (x n d h n,q+1 dv q 1,0 q n,0 for x n Diag n X, n N 0, X Ob s 2 Set; cf. [3, formula (1]. The Kan classifying simplicial set We let G be a simplicial group. For a morphism θ ([m], [n] non-negative integers i [0, m], j [iθ, n], we let θ [j] [i] ([i], [j] be defined by kθ [j] [i] : kθ for k [i]. Kan constructed a reduced simplicial set WG by W n G : j n 1,0 G j for every n N 0 (g j j n 1,0 W θ G : ( j (i+1θ 1,iθ g j G [j] θ i m 1,0 [i] for (g j j n 1,0 W n G θ ([m], [n], see [10, def. 10.3]. The simplicial set WG will be called the Kan classifying simplicial set of G. Notions from simplicial homotopy theory For n N, the stard n-simplex n in the category sset is defined to be the functor op Set represented by [n], that is, n : (, [n]. These simplicial sets yield a cosimplicial object cs(sset. We set d l : δl sset ( 0, 1 for l [0, 1]. For a simplicial set X we define ins 0 resp. ins 1 to be the composite morphisms X X 0 id d 1 X 1 resp. X X 0 id d 0 X 1, 3

4 where the cartesian product is defined dimensionwise the isomorphisms are canonical. For k [0, n + 1], n N 0, we let τ k 1 n ([n], [1] be the morphism given by [0, n k]τ k {0} [n k + 1, n]τ k {1}. Note that (x n (ins 0 n (x n, τ 0 (x n (ins 1 n (x n, τ n+1 for x n X n. In the following, we assume given simplicial sets X Y. Simplicial maps f, g sset (X, Y are said to be simplicially homotopic, written f g, if there exists a simplicial map X 1 H Y such that ins0 H f ins 1 H g. In this case, H is called a simplicial homotopy from f to g. The simplicial sets X Y are said to be simplicially homotopy equivalent if there are simplicial maps X Y Y g X such that fg id X gf id Y. In this case we write X Y we call f g mutually inverse simplicial homotopy equivalences. i Finally, we suppose given a dimensionwise injective simplicial map Y X, that is, i n is assumed to be injective for all n N 0. We call Y a simplicial deformation retract of X if there exists a simplicial map X r Y such that ir id Y ri id X. In this case, r is said to be a simplicial deformation retraction. If there exists H a homotopy ri id X which is constant along i, that is, if (y n i n, τ k H n y n i n f n y n i n g n for y n Y n, k [0, n + 1], n N 0, then we call Y a strong simplicial deformation retract of X r a strong simplicial deformation retraction. 3 Comparing W Diag N We have W Tot N. The natural transformation Diag φ Tot composed with the nerve functor N yields a natural transformation Diag N D W, given by (D G n i n 1,0 d n,i+1 : Diag n NG W n G for n N 0 G Ob sgrp. Proposition. The natural transformation D is a retraction. A corresponding coretraction is given by where W S Diag N, (S G n : W n G Diag n NG, (g i i n 1,0 (y i i n 1,0 with, defined by descending recursion, y i : for each i n 1, 0, n N 0, G Ob sgrp. (g j d j,i+1 s i,n 1 G n Proof. We suppose given a simplicial group G. Then we have to show that the maps (S G n for n N 0 commute with the faces degeneracies of G. First, we consider the faces. We let n N k [0, n]. For an n-tuple (g i i n 1,0 W n G we compute (g i i n 1,0 d k (S G n 1 (f i i n 2,0 (S G n 1 (x i i n 2,0, where g i+1 d k for i n 2, k, f i : (g k d k g k 1 for i k 1, for i k 2, 0 g i f x i : (x 1 j i+1,n 2 j n 2,i (f j d j,i+1 s i,n 2 for each i n 2, 0. 4

5 On the other h, we get (g i i n 1,0 (S G n d k (y i i n 1,0 d k (x i i n 2,0 with y i : y i+1 d k for i n 2, k, x i : (y k d k (y k 1 d k for i k 1, y i d k for i k 2, 0. (g j d j,i+1 s i,n 1 for i n 1, 0 We have to show that x i x i for all i n 2, 0. To this end, we proceed by induction on i. For i n 2, k, we calculate x i (x 1 (f j d j,i+1 s i,n 2 j i+1,n 2 j i+1,n 2 j i+1,n 2 ( j i+2,n 1 For i k 1, we have x k 1 j k,n 2 j k,n 2 j k,n 2 (x j 1 d j,i+1 s i,j 1 j+1 d kd j,i+1 s i,j 1 (x 1 j d j,i+2 s i+1,j 1 j d j,k s k 1,j 1 (x j 1 d j,k s k 1,j 1 j k+1,n 1 (y k d k j+1 d kd j,k s k 1,j 1 ( (y k d k (y k d k (y k 1 d k. j d j,k s k 1,j 2 j n 2,i j n 2,i j n 2,i j n 1,i+1 j n 2,k 1 j d j,k s k 1,j 2 j n 2,k 1 j n 2,k j d j,k s k 1,j 1 For i k 2, 0, we finally get x i (x 1 ( j i+1,n 2 j i+1,n 2 ( j k,n 2 (x j 1 d j,i+1 s i,j 1 j n 1,k 1 j n 2,i j n 2,i (f j d j,i+1 s i,n 2 (g j+1 d k d j,i+1 s i,n 2 (g j d j,i+2 s i+1,n 1 d k y i+1 d k. (f j d j,k s k 1,n 2 (f j d j,k s k 1,n 2 (g j+1 d k d j,k s k 1,n 2 ((g k d k g k 1 s k 1,n 2 j n 1,k 1 (g j d j,k s k 1,n 2 j n 1,k 1 (f j d j,i+1 s i,n 2 (f j d j,i+1 s i,n 2 (g j d j,k s k 1,n 2 (g j d j,k s k 1,n 1 d k j d k d j,i+1 s i,j 1 ((y k y k 1 1 d k d k 1,i+1 s i,k 2 j+1 d kd j,i+1 s i,j 1 ( j n 2,k (y j+1 d k d j,i+1 s i,n 2 5

6 (((g k d k g k 1 d k 1,i+1 s i,n 2 ( j n 1,k ( j d k d j,i+1 s i,j 1 (g j d j,i+1 s i,n 2 j k 2,i j k 1,i (g j d j,i+1 s i,n 2 j d j,i+1 s i,j 2 (g j d j,i+1 s i,n 2 (g j d j,i+1 s i,n 1 d k y i d k. Next, we come to the degeneracies. We let n N 0, k [0, n] (g i i n 1,0 W n G. Then we have (g i i n 1,0 s k (S G n+1 (h i i n,0 (S G n+1 (z i i n,0, where g i 1 s k for i n, k + 1, h i : 1 for i k, for i k 1, 0 g i z i : (z 1 j i+1,n j n,i (h j d j,i+1 s i,n for each i n, 0. Further, we get (g i i n 1,0 (S G n s k (y i i n 1,0 s k (z i i n,0 with y i : y i 1 s k for i n, k + 1, z i : 1 for i k, y i s k for i k 1, 0. (g j d j,i+1 s i,n 1 for i n 1, 0 Thus we have to show that z i z i for every i n, 0. To this end, we perform an induction on i n, 0. For i n, k + 1, we have z i (z 1 (h j d j,i+1 s i,n j i+1,n j i+1,n j i+1,n ( j i,n 1 (z j 1 d j,i+1 s i,j 1 j 1 s kd j,i+1 s i,j 1 j d j,i s i 1,j 1 For i k, we compute z k (z 1 j d j,k+1 s k,j 1 j k+1,n j n,i j n,i j n,i (h j d j,i+1 s i,n j n 1,i 1 j n,k (g j 1 s k d j,i+1 s i,n (g j d j,i s i 1,n 1 s k y i 1 s k. (h j d j,k+1 s k,n 6

7 j k+1,n j k+1,n j k+1,n j k+1,n j k+1,n z 1 (z j 1 d j,k+1 s k,j 1 j 1 s kd j,k+1 s k,j 1 j 1 d j 1,k+1 s k,j 1 j n,k j 1 s kd j,k+2 s k+1,j 1 (z j 1 d j,k+2 s k+1,j 1 k+1 (z 1 j k+2,n j d j,k+2 s k+1,j 1 For i k 1, 0, we get z i (z 1 j i+1,n j i+1,n j n,k+1 j n 1,k ( (z j 1 d j,i+1 s i,j 1 j n,i j n,i j s k d j,i+1 s i,j 1 (g j 1 s k d j,i+1 s i,n j s k d j,i+1 s i,j 1 (g j s k d j+1,i+1 s i,n Thus (S G n N yields a simplicial map WG S G Diag NG. j n,k+1 (h j d j,k+1 s k,n j n,k+1 j n,k+1 j n,k+1 j n,k+1 (g j 1 s k d j,k+1 s k,n (g j 1 d j 1,k+1 s k,n (g j 1 s k d j,k+2 s k+1,n (h j d j,k+2 s k+1,n (h j d j,i+1 s i,n (h j d j,i+1 s i,n j k+1,n j k 1,i j k 1,i (h j d j,k+2 s k+1,n z 1 k+1 z k+1 1. j 1 s kd j,i+1 s i,j 1 (g j d j,i+1 s i,n j s k d j+1,i+1 s i,j (g j d j,i+1 s i,n (g j d j,i+1 s i,n 1 s k y i s k. Finally, we have to prove that D G is a retraction with coretraction S G, that is, (S G n (D G n id WnG for all n N 0. Again, we let (y i i n 1,0 denote the image of an element (g i i n 1,0 W n G under (S G n. Then we have (g i i n 1,0 (S G n (D G n (y i i n 1,0 (D G n (y i d n,i+1 i n 1,0. Induction on i n 1, 0 shows that y i d n,i+1 j d j,i+1 s i,j 1 d n,i+1 j d n,i+1 (g 1 j d j,i+1 This implies that (S G n (D G n id WnG for all n N 0. (g j d j,i+1 (g j d j,i+1 g i. (g j d j,i+1 s i,n 1 d n,i+1 7

8 Theorem. We suppose given a simplicial group G. The Kan classifying simplicial set WG is a strong simplicial deformation retract of Diag NG with a strong simplicial deformation retraction given by Diag NG D G WG. Proof. We consider the coretraction W S Diag N as in the preceding proposition. Now, we shall show that D G S G id Diag NG via a simplicial homotopy constant along S G. A simplicial homotopy H from D G S G to id Diag NG is given by H n : Diag n NG 1 n Diag n NG, ((g n,i i n 1,0, τ n+1 k (y (n+1 k i i n 1,0 for all n N 0, where k [0, n + 1], defined by descending recursion, g n,i for i n 1, k 1 N 0, y (n+1 k i : ((y(n+1 k j 1 d j,i+1 s i,j 1 j k 2,i (g n,jd k 1,i+1 s i,k 2 for i k 2, 0. To facilitate the following calculations, we abbreviate ỹ i : y (n+1 k i for the respective index k [0, n] under consideration, if no confusion can arise. We have to verify that the maps H n for n N 0 yield a simplicial map. First, we show the compatibility with the faces. For k [0, n], l [0, n + 1], n N 0, (g n,i i n 1,0 Diag n NG, we have ((g n,i i n 1,0, τ n+1 l d k H n 1 ((g n,i i n 1,0 d k, τ n+1 l d k H n 1 ((f i i n 2,0, δ k τ n+1 l H n 1 { } ((f i i n 2,0, τ n l H n 1 for k l, ((f i i n 2,0, τ n+1 l ( x i i n 2,0, H n 1 for k < l where g n,i+1 d k for i n 2, k, f i : (g n,k d k (g n,k 1 d k for i k 1, g n,i d k for i k 2, 0 for all i n 2, 0 f i for i n 2, l 1, ( x 1 x i : (f jd l 1,i+1 s i,l 2 for i l 2, 0 f i for i n 2, l 2, j i+1,l 3 ( x 1 j l 3,i (f jd l 2,i+1 s i,l 3 for i l 3, 0 if k l, if k < l for all i n 2, 0. On the other h, we have ((g n,i i n 1,0, τ n+1 l H n d k (ỹ i i n 1,0 d k ( x i i n 2,0 with ỹ i : { g n,i for i n 1, l 1, (ỹ 1 (g n,jd l 1,i+1 s i,l 2 for i l 2, 0 for i n 1, 0 ỹ i+1 d k for i n 2, k, x i : (ỹ k d k (ỹ k 1 d k for i k 1, ỹ i d k for i k 2, 0 8

9 for i n 2, 0. We have to show that x i x i for all i n 2, 0. To this end, we consider three cases we hle each one by induction on i n 2, 0. We suppose that k n, l. For i n 2, k, we have x i f i g n,i+1 d k ỹ i+1 d k x i. For i k 1, we get x k 1 f k 1 (g n,k d k (g n,k 1 d k (ỹ k d k (ỹ k 1 d k x k 1. For i k 2, l 1, we get x i f i g n,i d k ỹ i d k x i Finally, for i l 2, 0, we calculate x i ( x 1 (f j d l 1,i+1 s i,l 2 ( ( x j 1 d j,i+1 s i,j 1 j d k d j,i+1 s i,j 1 (f j d l 1,i+1 s i,l 2 (g n,j d k d l 1,i+1 s i,l 2 (g n,j d l 1,i+1 s i,l 2 d k ỹ i d k x i. Next, we suppose that k l 1. For i n 2, k, we have x i f i g n,i+1 d k ỹ i+1 d k x i. For i k 1, we compute x k 1 f k 1 (g n,k d k (g n,k 1 d k (g n,k d k (g n,k 1 d k s k 1 d k (ỹ k d k (ỹ k 1 d k x k 1. For i k 2, 0, we get x i ( x 1 (f j d k 1,i+1 s i,k 2 ( ( x j 1 d j,i+1 s i,j 1 j d k d j,i+1 s i,j 1 j k 2,i j k 2,i j k 2,i j k 1,i (f j d k 1,i+1 s i,k 2 (g n,j d k d k 1,i+1 s i,k 2 (g n,j d k,i+1 s i,k 1 d k ỹ i d k x i. Finally, we suppose that k l 2, 0. For i n 2, l 2, we see that x i f i g n,i+1 d k ỹ i+1 d k x i. For i l 3, k, we have x i ( x 1 (f j d l 2,i+1 s i,l 3 j i+1,l 3 j i+1,l 3 ( x j 1 d j,i+1 s i,j 1 j+1 d kd j,i+1 s i,j 1 j l 3,i j l 3,i j i+1,l 3 j l 3,i (f j d l 2,i+1 s i,l 3 (g n,j+1 d k d l 2,i+1 s i,l 3 9

10 ( j i+2,l 2 j d j,i+2 s i+1,j 1 For i k 1, we have x k 1 ( x 1 j d j,k s k 1,j 1 j k,l 3 ( j k,l 3 j k,l 3 ( x j 1 d j,k s k 1,j 1 j l 2,i+1 j l 3,k 1 (ỹ j+1 d k d j,k s k 1,j 1 ( j l 3,k 1 j l 3,k (g n,j d l 1,i+2 s i+1,l 2 d k ỹ i+1 d k x i. (f j d l 2,k s k 1,l 3 (f j d l 2,k s k 1,l 3 (g n,j+1 d k d l 2,k s k 1,l 3 (g n,k d k d l 2,k s k 1,l 3 (g n,k 1 d k d l 2,k s k 1,l 3 j d j,k s k 1,j 2 (g n,j d l 1,k s k 1,l 3 j k+1,l 2 ( (ỹ k d k x k 1. j k,l 2 j l 2,k 1 j d j,k s k 1,j 1 For i k 2, 0, we get x i ( x 1 ( j i+1,l 3 j i+1,l 3 j ( j k,l 3 ( x j 1 d j,i+1 s i,j 1 j l 3,i j l 3,i j l 2,k 1 (f j d l 2,i+1 s i,l 3 (f j d l 2,i+1 s i,l 3 (g n,j d l 1,k s k 1,l 2 d k (ỹ k d k (ỹ k 1 d k d k d j,i+1 s i,j 1 k 1 d kd k 1,i+1 s i,k 2 k d kd k 1,i+1 s i,k 2 j+1 d kd j,i+1 s i,j 1 ( (g n,k 1 d k d l 2,i+1 s i,l 3 ( ( j d k d j,i+1 s i,j 1 j k 2,i j l 3,k j k,l 2 (g n,j+1 d k d l 2,i+1 s i,l 3 (g n,k d k d l 2,i+1 s i,l 3 (g n,j d k d l 2,i+1 s i,l 3 j d k d j 1,i+1 s i,j 2 (g n,j d l 1,i+1 s i,l 2 d k ỹ i d k x i. (g n,j d k d l 2,i+1 s i,l 3 Now we consider the degeneracies. We let n N 0, k [0, n], l [0, n + 1], (g n,i i n 1,0 Diag n NG. We compute ((g n,i i n 1,0, τ n+1 l s k H n+1 ((g n,i i n 1,0 s k, τ n+1 l s k H n+1 ((h i i n 1,0, σ k τ n+1 l H n+1 { } ((h i i n 1,0, τ n+2 l H n+1 for k l, ((h i i n 1,0, τ n+1 l ( z i i n,0, H n+1 for k < l where g n,i 1 s k for i n, k + 1, h i : 1 for i k, g n,i s k for i k 1, 0 { } h i for i n, l 1, z i : ( z 1 (h jd l 1,i+1 s i,l 2 for i l 2, 0 { } h i for i n, l, j i+1,l 1 ( z 1 j l 1,i (h jd l,i+1 s i,l 1 for i l 1, 0 10 if k l, if k < l.

11 Furthermore, we have where ((g n,i i n 1,0, τ n+1 l H n s k (ỹ i i n 1,0 s k ( z i i n,0, ỹ i : { g n,i for i n 1, l 1, (ỹ 1 (g n,jd l 1,i+1 s i,l 2 for i l 2, 0 ỹ i 1 s k for i n, k + 1, z i : 1 for i k, ỹ i s k for i k 1, 0. Thus we have to show that z i z i for every i n, 0. Again, we distinguish three cases, in each one, we perform an induction on i n, 0. We suppose that k n, l. For i n, k + 1, we calculate z i h i g n,i 1 s k ỹ i 1 s k z i. For i k, we get z k h k 1 z k. For i k 1, l 1, we have z i h i g n,i s k ỹ i s k z i. For i l 2, 0, we get z i ( z 1 ( ( z j 1 d j,i+1 s i,j 1 j s k d j,i+1 s i,j 1 (h j d l 1,i+1 s i,l 2 (h j d l 1,i+1 s i,l 2 (g n,j s k d l 1,i+1 s i,l 2 Now we suppose that k l 1. For i n, k + 1, we calculate z i h i g n,i 1 s k ỹ i 1 s k z i. For i k, we get z k h k d k+1 s k 1 z k. For i k 1, 0, we get z i ( z 1 j i+1,k j i+1,k ( z j 1 d j,i+1 s i,j 1 j k,i j k,i j s k d j,i+1 s i,j 1 (g n,j d l 1,i+1 s i,l 2 s k ỹ i s k. (h j d k+1,i+1 s i,k (h j d k+1,i+1 s i,k j k 1,i (g n,j s k d k+1,i+1 s i,k 11

12 ( j k 1,i At last, we suppose that k l 2, 0. For i n, l, we have z i h i g n,i 1 s k ỹ i 1 s k z i. For i l 1, k + 1, we get z i ( z 1 ( j i+1,l 1 j i+1,l 1 j i+1,l 1 j i,l 2 For i k, we have z k j k+1,l 1 j k+1,l 1 j k+1,l 1 k s k( ( z j 1 d j,i+1 s i,j 1 j 1 s kd j,i+1 s i,j 1 j d j,i s i 1,j 1 ( z 1 j d j,k+1 s k,j 1 ( z j 1 d j,k+1 s k,j 1 j l 1,i j l 1,i j l 1,i j l 2,i 1 j 1 s kd j,k+1 s k,j 1 j k+1,l 2 (( k s k z k. j k+1,l 2 j l 1,k j l 1,k j d j,k+1 s k,j ( For i k 1, 0, we get z i ( z 1 j i+1,l 1 j i+1,l 1 ( z j 1 d j,i+1 s i,j 1 j l 1,k+1 ( j s k d j,i+1 s i,j 1 (g n,j d k,i+1 s i,k 1 s k ỹ i s k. (h j d l,i+1 s i,l 1 (h j d l,i+1 s i,l 1 (g n,j 1 s k d l,i+1 s i,l 1 (g n,j d l 1,i s i 1,l 2 s k ỹ i 1 s k z i. j l 1,k+1 j d j,k+1 s k,j 1 j l 1,i j l 1,i (g n,j 1 s k d l,i+1 s i,l 1 j s k d j,i+1 s i,j 1 Altogether, we obtain a simplicial map Diag NG 1 H Diag NG. (h j d l,k+1 s k,l 1 (h j d l,k+1 s k,l 1 j l 2,k j k+1,l 1 j k,l 2 (g n,j 1 s k d l,k+1 s k,l 1 j l 2,k (h j d l,i+1 s i,l 1 (g n,j d l 1,k+1 s k,l 1 (h j d l,i+1 s i,l 1 j k 1,i (g n,j d l 1,k+1 s k,l 2 s k k s k(ỹ k s k 1 j 1 s kd j,i+1 s i,j 1 (g n,j s k d l,i+1 s i,l 1 j s k d j+1,i+1 s i,j (g n,j d l 1,i+1 s i,l 2 s k ỹ i s k z i. (g n,j s k d l,i+1 s i,l 1 To prove that H is a simplicial homotopy from D G S G to id Diag NG, it remains to show that ins 0 H D G S G ins 1 H id Diag NG. For n N 0, k [0, n + 1], (g n,i i n 1,0 Diag n NG, n N 0, we have (g n,i i n 1,0 (D G n (S G n (g n,i d n,i+1 i n 1,0 (S G n (y i i n 1,0 12

13 with y i for i n 1, 0, ((g n,i i n 1,0, τ n+1 k H n (y (n+1 k i i n 1,0 (g n,j d n,j+1 d j,i+1 s i,n 1 (g n,j d n,i+1 s i,n 1 with g n,i for i n 1, k 1 N 0, y (n+1 k i : ((y(n+1 k j 1 d j,i+1 s i,j 1 j k 2,i (g n,jd k 1,i+1 s i,k 2 for i k 2, 0. But by descending induction on i n 1, 0, we get y i (g n,j d n,i+1 s i,n 1 ((y (0 j 1 d j,i+1 s i,j 1 Hence the simplicial map H fulfills (g n,j d n,i+1 s i,n 1 y (0 (g n,i i n 1,0 (ins 0 n H n ((g n,i i n 1,0, τ 0 H n (y (0 i i n 1,0 (y i i n 1,0 (g n,i i n 1,0 (D G n (S G n i. (g n,i i n 1,0 (ins 1 n H n ((g n,i i n 1,0, τ n+1 H n (g n,i i n 1,0 for each (g n,i i n 1,0 Diag n NG, n N 0. In order to prove that WG is a strong deformation retract of Diag NG, it remains to show that H is constant along S G. Concretely, this means the following. For (g i i n 1,0 W n G, we have where ((g i i n 1,0 (S G n, τ n+1 k H n ((y i i n 1,0, τ n+1 k H n (y (n+1 k i i n 1,0, y i : (g j d j,i+1 s i,n 1 y i for i n 1, k 1 N 0, y (n+1 k i : ((y(n+1 k j 1 d j,i+1 s i,j 1 j k 2,i (y jd k 1,i+1 s i,k 2 for i k 2, 0. Now, we have to show that y (n+1 k i y i for all i n 1, 0, k [0, n + 1]. For k {n + 1, 0}, this follows since H is a simplicial homotopy from D G S G to id Diag NG since S G D G S G S G. So we may assume that k n, 1 have to show that y (n+1 k i y i for every i k 2, 0. But we have ( y i d k 1,i+1 s i,k 2 (g j d j,i+1 s i,n 1 d k 1,i+1 s i,k 2 13

14 j d j,i+1 s i,j 1 d k 1,i+1 s i,k 2 j d j,i+1 s i,j 1 d k 1,j+1 d j,i+1 s i,k 2 j d j,i+1 s i,k 1 s k,j 1 d k 1,i+1 s i,k 2 (g j d j,i+1 s i,k 1 s k,n 1 d k 1,i+1 s i,k 2 j d j,i+1 d i+k 1 j,i+1 s i,j 1 d j,i+1 s i,k 2 j d j,i+1 s i,k 1 d k 1,i+1 s i+1,i+j k s i,k 2 (g j d j,i+1 s i,k 1 d k 1,i+1 s i+1,i+n k s i,k 2 j d k 1,i+1 s i,k 2 this implies, by induction on i k 2, 0, that y (n+1 k i ((y (n+1 k j 1 d j,i+1 s i,j 1 for all i k 2, 0. References j k 2,i j k 2,i+1 j d k 1,i+1 s i,k 2 j k 2,i (g j d j,i+1 s i,n 1 d k 1,i+1 s i,k 2 (y j d k 1,i+1 s i,k 2 (y j d k 1,i+1 s i,k 2 (y j d k 1,i+1 s i,k 2 (g j d j,i+1 s i,n 1 y i (g j d j,i+1 s i,n 1, (g j d j,i+1 s i,n 1 [1] Artin, Michael; Mazur, Barry. On the van Kampen theorem. Topology 5 (1966, pp [2] Bousfield, Aldridge K.; Friedler, Eric M. Homotopy theory of Γ-spaces, spectra, bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf. Evanston, Ill. 1977, II, pp , Lecture Notes in Mathematics, vol Springer, Berlin, [3] Cegarra, Antonio M.; Remedios, Josué. The relationship between the diagonal the bar constructions on a bisimplicial set. Topology its Applications 153(1 (2005, pp [4] Cegarra, Antonio M.; Remedios, Josué. The behaviour of the W -construction on the homotopy theory of bisimplicial sets. Manuscripta Mathematica 124(4 (2007, pp [5] Curtis, Edward B. Simplicial homotopy theory. Advances in Mathematics 6 (1971, pp [6] Dold, Albrecht; Puppe, Dieter S. Homologie nicht-additiver Funktoren. Anwendungen. Annales de l Institut Fourier 11 (1961, pp [7] Friedler, Eric M.; Mazur, Barry. Filtrations on the homology of algebraic varieties. Memoirs of the American Mathematical Society, vol. 110, no American Mathematical Society (

15 [8] Goerss, Paul G.; Jardine, John F. Simplicial Homotopy Theory. Progress in Mathematics, vol Birkhäuser Verlag, Basel, [9] Jardine, John F. Algebraic Homotopy Theory, Groups, K-Theory. Ph.D. thesis, The University of British Columbia, [10] Kan, Daniel M. On homotopy theory c.s.s. groups. Annals of Mathematics 68 (1958, pp [11] May, J. Peter. Simplicial objects in algebraic topology. Van Nostr Mathematical Studies, vol. 11. D. Van Nostr Co., Inc., Princeton, N.J.-Toronto, Ont.-London, [12] Quillen, Daniel G. On the group completion of a simplicial monoid. Appendix in [7]. [13] Thomas, Sebastian. (Cohomology of crossed modules. Diploma thesis, RWTH Aachen University, [14] Zisman, Michel. Suite spectrale d homotopie et ensembles bisimpliciaux. Manuscript, Université scientifique et médicale de Grenoble, Sebastian Thomas Lehrstuhl D für Mathematik RWTH Aachen University Templergraben Aachen Germany sebastian.thomas@math.rwth-aachen.de 15