The functors Wbar and Diag o Nerve are simplicially homotopy equivalent

THE FUNCTORS W AND Diag ◦Nerve ARE SIMPLICIALLY HOMOTOPY EQUIVALENT

SEBASTIAN THOMAS

(communicated by Antonio Cegarra) Abstract

Given a simplicial groupG, there are two known classifying simplicial set constructions, the Kan classifying simplicial set WGand 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, WGand Diag NGare simplicially ho- motopy equivalent.

1. Introduction

We suppose given a simplicial group G. Kan introduced in [10] the Kan classi- fying simplicial set WG. The functor W from simplicial groups to simplicial sets is the right adjoint, and 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 sim- plicial set Diag NG. The latter construction is used for example by Quillen [7, appendix Q.3] andJardine[9, p. 41].

It is well-known that these two variants WGand Diag NGfor the classifying sim- plicial set of G are weakly homotopy equivalent. 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 and Mazur [1] (¹); and Ce- garraandRemedios[3] showed that already the total simplicial set functor and the diagonal functor, applied to a bisimplicial set, yield weakly homotopy equivalent results. 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 WGis a strong simplicial deforma- tion retract of Diag NG. In particular, WGand Diag NGare simplicially homotopy equivalent.

Received January 7, 2008, revised April 17, 2008; published on October 19, 2008.

2000 Mathematics Subject Classification: 18G30, 55U10

Key words and phrases: simplicial groups, Kan classifying functor, nerve, total simplicial set

°c 2008, Sebastian Thomas. Permission to copy for private use granted.

1This is not the total simplicial set as used byBousfieldandFriedlander[2, appendix B, p.

118].

This commutativity up to simplicial homotopy equivalence fits into the following diagram.

¡_simplicial

sets

¢ ^associatedcomplex //¡ _complexes

of abelian groups

¢ homology //¡_abelian

groups

¢

¡_simplicial

groups

¢

classifyingKan functor

W

99s

ss ss ss ss s

nerve NKKKKKKKK%%

KK

¡bisimplicial sets

¢

diagonal simplicial

set Diag

OO

total simplicial

set Tot

OO

associated double

complex //¡ _double

complexes

¢

total complex

OO

associated spectral

sequence //¡_spectral

sequences

¢

approximation

OO

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 toDold, PuppeandCartier[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, Lemma 4.1.3]

of G, which has E_p,n−p¹ ∼= Hn−p(Gp), and which converges to the homology of G.

Similarly for cohomology.

Conventions and notations

We use the following conventions and notations.

• The composite of morphismsX −→^f Y and Y −→^g Z is denoted byX −→^{f g} Z.

The composite of functorsC−→ D^F andD−→ E^G is denoted by C−−−→ E^G◦F .

• If C is a category and X, Y ∈ ObC are objects in C, we write C(X, Y) = MorC(X, Y) for the set of morphisms betweenX andY. Moreover, we denote by (((C,D))) the functor category that has functors between C andDas objects and natural transformations between these functors as morphisms.

• Given a functorI−→ C, we sometimes denote the image of a morphism^X i−→^θ j inI byXi Xθ

−−→Xj. This applies in particular ifI=∆^oporI=∆^op×∆^op.

• We use the notationsN={1,2,3, . . .} andN0=N∪ {0}.

• Given integers a, b ∈ Z, we write [a, b] := {z ∈ Z | a 6 z 6 b} for the set of integers lying betweena and b. Moreover, we write da, be:= (z ∈Z|a6 z 6b) for the ascending interval and ba, bc = (z ∈ Z | a >z > b) for the descending interval. Whereas we formally deal with tuples, we use the element notation, for example we writeQ

i∈d1,3eg_i=g₁g₂g₃ andQ

i∈b3,1cg_i =g₃g₂g₁ or (gi)_i∈b3,1c = (g3, g2, g1) for group elementsg1, g2, g3.

• Given an index setI, families of groups (Gi)i∈I and (Hi)i∈I and a family of group homomorphisms (ϕi)i∈I, whereϕi:Gi→Hifor alli∈I, we denote the direct product of the groups by

×

×

×

×

2. Simplicial preliminaries

We recall some standard definitions, cf. for example [5], [8] or [11].

Simplicial objects

Forn∈N₀, we let [n] denote the category induced by the totally ordered set [0, n]

with the natural order, and we let ∆ be the full subcategory in Cat defined by Ob∆:={[n]|n∈N0}.

The category of simplicial objects sC in a given category C is defined to be the functor category (((∆^op,C))). Moreover, thecategory of bisimplicial objects s²Cin Cis defined to be (((∆^op×∆^op,C))). The dual notion is that of thecategory csC:= (((∆,C))) of cosimplicial objects inC.

Forn∈N,k∈[0, n], we let [n−1]−→^δk [n] be the injection that omitsk∈[0, n], and forn∈N0,k∈[0, n], we let [n+ 1]−→^σk [n] be the surjection that repeatsk∈[0, n].

The images of the morphisms δ^k resp. σ^k under a simplicial objectX in a given category C are denoted by dk :=X_δk, called the k-th face, for k ∈ [0, n], n ∈ N, resp. sk :=X_σ^k, called thek-th degeneracy, for k∈ [0, n], n∈ N0. Similarly, in a bisimplicial object X one defines horizontal and vertical faces resp.degeneracies, d^h_k :=X_δ^k_,id, d^v_k:=X_id,δ^k, s^h_k :=X_σ^k_,id, s^v_k:=X_id,σ^k.

We use the ascending and descending interval notation as introduced above for composites of faces resp. degeneracies, that is, we write d_bj,ic:= djdj−1. . .di resp.

s_di,je:= sisi+1. . .sj. The nerve

We suppose given a group G. The nerve of G is the simplicial set NG given by NnG=G^×n for alln∈N0 and by

(gj)j∈bn−1,0c(NθG) = ( Y

j∈b(i+1)θ−1,iθc

gj)i∈bm−1,0c

for (gj)j∈bn−1,0c ∈NnGandθ∈∆([m],[n]), where m, n∈N0.

Since the nerve construction is a functorGrp−→^N sSet, it can be applied dimen- sionwise to a simplicial group. This yields a functorsGrp−→^N s²Set.

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 DiagX and the total simplicial set TotX, see [1,§3]. We recall their definitions.

Thediagonal simplicial set DiagX has entries Diag_nX :=Xn,n for n∈N0, while Diag_θX :=Xθ,θ forθ∈∆([m],[n]), wherem, n∈N0.

To introduce the total simplicial set ofX, we define thesplitting atp∈[0, m] of a morphism [m]−→^θ [n] in∆ by Spl_p(θ) := (Spl_6p(θ),Spl_>p(θ)), where

[p]−−−−−→^Spl6p(θ) [pθ] and [m−p]−−−−−→^Spl>p(θ) [n−pθ]

are given by iSpl_6p(θ) := iθ for i ∈ [0, p] and iSpl_>p(θ) := (i+p)θ −pθ for i∈[0, m−p]. Thetotal simplicial set TotX is defined by

TotnX :=©

(xq)_q∈bn,0c∈

×

q∈bn,0c

Xq,n−q

¯¯xqd^h_q =xq−1d^v₀ for allq∈ bn,1cª

forn∈N0 and by

(xq)_q∈bn,0c(TotθX) = (xpθX_Splp(θ))_p∈bm,0c

for (xq)_q∈bn,0c∈TotnX andθ∈∆([m],[n]), wherem, n∈N0. There is a natural transformation

Diag−→^φ Tot,

where φX is given by xn(φX)n = (xnd^h_bn,q+1cd^v_bq−1,0c)_q∈bn,0c for xn ∈ Diag_nX, n∈N0,X ∈Obs²Set; cf. [3, formula (1)].

The Kan classifying simplicial set

We let Gbe a simplicial group. For a morphismθ ∈∆([m],[n]) and non-negative integers i∈[0, m], j ∈[iθ, n], we letθ|^[j]_[i] ∈∆([i],[j]) be defined by kθ|^[j]_[i] :=kθ for k∈[i].Kanconstructed a reduced simplicial set WGby

WnG:=

×

j∈bn−1,0c

Gj for everyn∈N0

and

(gj)j∈bn−1,0cWθG:= ( Y

j∈b(i+1)θ−1,iθc

gjG_θ|[j]

[i])i∈bm−1,0c

for (gj)j∈bn−1,0c∈WnGandθ∈∆([m],[n]), see [10, Definition 10.3]. The simplicial set WGwill be called theKan classifying simplicial set ofG.

Notions from simplicial homotopy theory

For n ∈ N, the standardn-simplex ∆ⁿ in the category sSet is defined to be the functor∆^op−→Setrepresented 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 setX we define ins0 resp. ins1 to be the composite morphisms X −→^∼= X×∆^{0 id}−−−−→^×d1 X×∆¹resp.X −→^∼= X×∆^{0 id}−−−−→^×d0 X×∆¹,

where the cartesian product is defined dimensionwise and the isomorphisms are canonical.

Fork∈[0, n+ 1],n∈N0, we letτ^k ∈∆¹_n=∆([n],[1]) be the morphism given by [0, n−k]τ^k ={0}and [n−k+ 1, n]τ^k ={1}. Note that (xn)(ins0)n= (xn,τ⁰) and (xn)(ins1)n= (xn,τⁿ⁺¹) forxn ∈Xn.

In the following, we assume given simplicial setsX andY.

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 ins0H = f and ins1H =g. In this case,H is called asimplicial homotopy fromf tog.

The simplicial setsX andY are said to be simplicially homotopy equivalent if there are simplicial mapsX −→^f Y and Y −→^g X such thatf g∼idX and gf ∼idY. In this case we writeX'Y and we callf andgmutually inversesimplicial homotopy equivalences.

Finally, we suppose given a dimensionwise injective simplicial map Y −→ⁱ X, that is,in is assumed to be injective for alln∈N0. We callY a simplicial deformation retract of X if there exists a simplicial map X −→^r Y such that ir = idY and ri ∼idX. In this case,r is said to be a simplicial deformation retraction. If there exists a homotopy ri−→^H idX which is constant along i, that is, if (ynin,τ^k)Hn = yninfn=yningnforyn ∈Yn,k∈[0, n+1],n∈N0, then we callY astrongsimplicial deformation retract ofX andra strong simplicial deformation retraction.

3. Comparing WWbar and Diag ◦NDiag 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 (DG)n =

×

Proposition. The natural transformationDis a retraction. A corresponding core- traction is given by

W−→^S Diag◦N, where

(SG)n: WnG→Diag_nNG,(gi)_{i∈bn−1,0c}7→(yi)_{i∈bn−1,0c} with, defined by descending recursion,

yi:= Y

j∈di+1,n−1e

(y⁻¹_j dbj,i+1csdi,j−1e) Y

j∈bn−1,ic

(gjdbj,i+1csdi,n−1e)∈Gn

for eachi∈ bn−1,0c,n∈N0,G∈ObsGrp.

Proof. We suppose given a simplicial groupG. Then we have to show that the maps (SG)n forn∈N0 commute with the faces and degeneracies ofG.

First, we consider the faces. We letn∈Nandk∈[0, n]. For (gi)_{i∈bn−1,0c} ∈WnG we compute

(gi)i∈bn−1,0cdk(SG)n−1= (fi)i∈bn−2,0c(SG)n−1= (xi)i∈bn−2,0c,

where fi:=







gi+1dk fori∈ bn−2, kc, (gkdk)gk−1 fori=k−1, gi fori∈ bk−2,0c and

xi := Y

j∈di+1,n−2e

(x⁻¹_j d_bj,i+1cs_di,j−1e) Y

j∈bn−2,ic

(fjd_bj,i+1cs_di,n−2e) fori∈ bn−2,0c. On the other hand, we get

(gi)_{i∈bn−1,0c}(SG)ndk = (yi)_{i∈bn−1,0c}dk= (x⁰_i)_{i∈bn−2,0c} with

yi:= Y

j∈di+1,n−1e

(y⁻¹_j dbj,i+1csdi,j−1e) Y

j∈bn−1,ic

(gjdbj,i+1csdi,n−1e) fori∈ bn−1,0cand

x⁰_i :=







yi+1dk fori∈ bn−2, kc, (ykdk)(yk−1dk) fori=k−1, yidk fori∈ bk−2,0c.

We have to show that xi = x⁰_i for alli ∈ bn−2,0c. To this end, we proceed by induction oni.

Fori∈ bn−2, kc, we calculate

xi = Y

j∈di+1,n−2e

(x⁻¹_j dbj,i+1csdi,j−1e) Y

j∈bn−2,ic

(fjdbj,i+1csdi,n−2e)

= Y

j∈di+1,n−2e

(x⁰_j⁻¹d_bj,i+1cs_di,j−1e) Y

j∈bn−2,ic

(fjd_bj,i+1cs_di,n−2e)

= Y

j∈di+1,n−2e

(y⁻¹_j+1dkd_bj,i+1cs_di,j−1e) Y

j∈bn−2,ic

(gj+1dkd_bj,i+1cs_di,n−2e)

=³ Y

j∈di+2,n−1e

(y_j⁻¹d_bj,i+2cs_di+1,j−1e) Y

j∈bn−1,i+1c

(gjd_bj,i+2cs_di+1,n−1e)´ dk

=yi+1dk. Fori=k−1, we have

xk−1= Y

j∈dk,n−2e

(x⁻¹_j d_bj,kcs_{dk−1,j−1e}) Y

j∈bn−2,k−1c

(fjd_bj,kcs_{dk−1,n−2e})

= Y

j∈dk,n−2e

(x⁰_j⁻¹d_bj,kcs_{dk−1,j−1e}) Y

j∈bn−2,k−1c

(fjd_bj,kcs_{dk−1,n−2e})

= Y

j∈dk,n−2e

(y⁻¹_j+1dkdbj,kcsdk−1,j−1e) Y

j∈bn−2,kc

(gj+1dkdbj,kcsdk−1,n−2e)

·((gkdk)gk−1)s_{dk−1,n−2e}

= Y

j∈dk+1,n−1e

(y⁻¹_j d_bj,kcs_{dk−1,j−2e}) Y

j∈bn−1,k−1c

(gjd_bj,kcs_{dk−1,n−2e})

= (ykdk) Y

j∈dk,n−1e

(y_j⁻¹d_bj,kcs_{dk−1,j−2e}) Y

j∈bn−1,k−1c

(gjd_bj,kcs_{dk−1,n−2e})

= (ykdk)

·³ Y

j∈dk,n−1e

(y⁻¹_j d_bj,kcs_{dk−1,j−1e}) Y

j∈bn−1,k−1c

(gjd_bj,kcs_{dk−1,n−1e})´ dk

= (ykdk)(yk−1dk).

Fori∈ bk−2,0c, we finally get

xi = Y

j∈di+1,n−2e

(x⁻¹_j dbj,i+1csdi,j−1e) Y

j∈bn−2,ic

(fjdbj,i+1csdi,n−2e)

= Y

j∈di+1,n−2e

(x⁰_j⁻¹d_bj,i+1cs_di,j−1e) Y

j∈bn−2,ic

(fjd_bj,i+1cs_di,n−2e)

= ( Y

j∈di+1,k−2e

(y⁻¹_j d_kd_bj,i+1cs_di,j−1e))((y_ky_k−1)⁻¹d_kd_bk−1,i+1cs_di,k−2e)

·( Y

j∈dk,n−2e

(y_j+1⁻¹ dkdbj,i+1csdi,j−1e))( Y

j∈bn−2,kc

(yj+1dkdbj,i+1csdi,n−2e))

·(((gkdk)gk−1)d_bk−1,i+1cs_di,n−2e)( Y

j∈bk−2,ic

(gjd_bj,i+1cs_di,n−2e))

= Y

j∈di+1,k−1e

(y⁻¹_j d_kd_bj,i+1cs_di,j−1e) Y

j∈dk,n−1e

(y⁻¹_j d_bj,i+1cs_di,j−2e)

· Y

j∈bn−1,kc

(gjdbj,i+1csdi,n−2e) Y

j∈bk−1,ic

(gjdbj,i+1csdi,n−2e)

=

³ Y

j∈di+1,n−1e

(y_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bn−1,ic

(gjdbj,i+1csdi,n−1e)

´

dk =yidk. Next, we come to the degeneracies. We let n∈ N0, k ∈ [0, n] and (gi)i∈bn−1,0c ∈ WnG. Then we have

(gi)i∈bn−1,0csk(SG)n+1= (hi)i∈bn,0c(SG)n+1= (zi)i∈bn,0c, where

hi :=







gi−1sk fori∈ bn, k+ 1c, 1 fori=k,

gi fori∈ bk−1,0c and

zi:= Y

j∈di+1,ne

(z_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bn,ic

(hjdbj,i+1csdi,ne)

fori∈ bn,0c. Further, we get

(gi)_{i∈bn−1,0c}(SG)nsk= (yi)_{i∈bn−1,0c}sk= (z_i⁰)_i∈bn,0c with

yi:= Y

j∈di+1,n−1e

(y⁻¹_j d_bj,i+1cs_di,j−1e) Y

j∈bn−1,ic

(gjd_bj,i+1cs_di,n−1e) fori∈ bn−1,0cand

z⁰_i:=







yi−1sk fori∈ bn, k+ 1c, 1 fori=k,

yisk fori∈ bk−1,0c.

Thus we have to show thatzi=z_i⁰ for everyi∈ bn,0c. To this end, we perform an induction oni∈ bn,0c.

Fori∈ bn, k+ 1c, we have zi= Y

j∈di+1,ne

(z_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bn,ic

(hjdbj,i+1csdi,ne)

= Y

j∈di+1,ne

(z_j⁰⁻¹d_bj,i+1cs_di,j−1e) Y

j∈bn,ic

(hjd_bj,i+1cs_di,ne)

= Y

j∈di+1,ne

(y⁻¹_j−1skd_bj,i+1cs_di,j−1e) Y

j∈bn,ic

(gj−1skd_bj,i+1cs_di,ne)

=³ Y

j∈di,n−1e

(y_j⁻¹d_bj,ics_{di−1,j−1e}) Y

j∈bn−1,i−1c

(gjd_bj,ics_{di−1,n−1e})´ sk

=yi−1sk. Fori=k, we compute

zk= Y

j∈dk+1,ne

(z_j⁻¹dbj,k+1csdk,j−1e) Y

j∈bn,kc

(hjdbj,k+1csdk,ne)

= Y

j∈dk+1,ne

(z_j⁰⁻¹d_bj,k+1cs_dk,j−1e) Y

j∈bn,kc

(hjd_bj,k+1cs_dk,ne)

= Y

j∈dk+1,ne

(y_j−1⁻¹ skd_bj,k+1cs_dk,j−1e) Y

j∈bn,k+1c

(gj−1skd_bj,k+1cs_dk,ne)

= Y

j∈dk+1,ne

(y_j−1⁻¹ dbj−1,k+1csdk,j−1e) Y

j∈bn,k+1c

(gj−1dbj−1,k+1csdk,ne)

= Y

j∈dk+1,ne

(y_j−1⁻¹ skd_bj,k+2cs_dk+1,j−1e) Y

j∈bn,k+1c

(gj−1skd_bj,k+2cs_dk+1,ne)

= Y

j∈dk+1,ne

(z_j⁰⁻¹d_bj,k+2cs_dk+1,j−1e) Y

j∈bn,k+1c

(hjd_bj,k+2cs_dk+1,ne)

=z_k+1⁻¹ Y

j∈dk+2,ne

(z_j⁻¹dbj,k+2csdk+1,j−1e) Y

j∈bn,k+1c

(hjdbj,k+2csdk+1,ne)

=z_k+1⁻¹ zk+1= 1.

Fori∈ bk−1,0c, we get zi= Y

j∈di+1,ne

(z_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bn,ic

(hjdbj,i+1csdi,ne)

= Y

j∈di+1,ne

(z_j⁰⁻¹dbj,i+1csdi,j−1e) Y

j∈bn,ic

(hjdbj,i+1csdi,ne)

= Y

j∈di+1,k−1e

(y_j⁻¹skd_bj,i+1cs_di,j−1e) Y

j∈dk+1,ne

(y⁻¹_j−1skd_bj,i+1cs_di,j−1e)

· Y

j∈bn,k+1c

(gj−1skdbj,i+1csdi,ne) Y

j∈bk−1,ic

(gjdbj,i+1csdi,ne)

= Y

j∈di+1,k−1e

(y_j⁻¹skdbj,i+1csdi,j−1e) Y

j∈dk,n−1e

(y⁻¹_j skdbj+1,i+1csdi,je)

· Y

j∈bn−1,kc

(gjskd_bj+1,i+1cs_di,ne) Y

j∈bk−1,ic

(gjd_bj,i+1cs_di,ne)

=³ Y

j∈di+1,n−1e

(y_j⁻¹d_bj,i+1cs_di,j−1e) Y

j∈bn−1,ic

(gjd_bj,i+1cs_di,n−1e)´

sk =yisk. Thus (SG)n∈Nyields a simplicial map

WG−−→^SG Diag NG.

Finally, we have to prove thatDG is a retraction with coretractionSG, that is, (SG)n(DG)n= id_W

nG for alln∈N0.

Again, we let (yi)_{i∈bn−1,0c} denote the image of an element (gi)_{i∈bn−1,0c} ∈ WnG under (SG)n. Then we have

(gi)i∈bn−1,0c(SG)n(DG)n= (yi)i∈bn−1,0c(DG)n= (yidbn,i+1c)i∈bn−1,0c. Induction oni∈ bn−1,0cshows that

yid_bn,i+1c = Y

j∈di+1,n−1e

(y⁻¹_j d_bj,i+1cs_di,j−1ed_bn,i+1c)

· Y

j∈bn−1,ic

(gjdbj,i+1csdi,n−1edbn,i+1c)

= Y

j∈di+1,n−1e

(y⁻¹_j dbn,i+1c) Y

j∈bn−1,ic

(gjdbj,i+1c)

= Y

j∈di+1,n−1e

(g_j⁻¹d_bj,i+1c) Y

j∈bn−1,ic

(gjd_bj,i+1c) =gi. This implies that (SG)n(DG)n = id_WnG for alln∈N0.

Theorem. We suppose given a simplicial groupG. The Kan classifying simplicial set WGis a strong simplicial deformation retract of Diag NGwith a strong simplicial deformation retraction given by

Diag NG−−→^DG WG.

Proof. We consider the coretraction W−→^S Diag N as in the preceding proposition.

Now, we shall show that DGSG ∼ idDiag NG via a simplicial homotopy constant alongSG.

A simplicial homotopyH fromDGSG to idDiag NG is given by Hn: Diag_nNG×∆¹_n→Diag_nNG,

((gn,i)_{i∈bn−1,0c},τ^n+1−k)7→(y_i^(n+1−k))_{i∈bn−1,0c}

for alln∈N0, where k∈[0, n+ 1] and, defined by descending recursion, y_i^(n+1−k):=







gn,i fori∈ bn−1, k−1c ∩N0,

Q

j∈di+1,k−2e((y^(n+1−k)_j )⁻¹dbj,i+1csdi,j−1e)

·Q

j∈bk−2,ic(gn,jdbk−1,i+1csdi,k−2e) fori∈ bk−2,0c.

To facilitate the following calculations, we abbreviate ˜yi:=y^(n+1−k)_i for the respec- tive indexk∈[0, n] under consideration, if no confusion can arise.

We have to verify that the mapsHn forn∈N0yield a simplicial map.

First, we show the compatibility with the faces. Fork∈[0, n],l∈[0, n+ 1],n∈N0, (gn,i)_{i∈bn−1,0c}∈Diag_nNG, we have

((g_n,i)_{i∈bn−1,0c},τ^n+1−l)d_kH_n−1= ((g_n,i)_{i∈bn−1,0c}d_k,τ^n+1−ld_k)H_n−1

= ((fi)_{i∈bn−2,0c},δ^kτ^n+1−l)Hn−1

= (

((fi)i∈bn−2,0c,τ^n−l)Hn−1 fork>l, ((fi)_{i∈bn−2,0c},τ^n+1−l)Hn−1 fork < l

)

= (˜xi)_{i∈bn−2,0c}, where

fi:=







gn,i+1dk fori∈ bn−2, kc,

(gn,kdk)(gn,k−1dk) fori=k−1, gn,idk fori∈ bk−2,0c for alli∈ bn−2,0cand

˜ xi :=



























fi fori∈ bn−2, l−1c,

Q

j∈di+1,l−2e(˜x⁻¹_j dbj,i+1csdi,j−1e)

·Q

j∈bl−2,ic(fjd_bl−1,i+1cs_di,l−2e) fori∈ bl−2,0c





 ifk>l,







fi fori∈ bn−2, l−2c,

Q

j∈di+1,l−3e(˜x⁻¹_j dbj,i+1csdi,j−1e)

·Q

j∈bl−3,ic(fjdbl−2,i+1csdi,l−3e) fori∈ bl−3,0c





 ifk < l for alli∈ bn−2,0c. On the other hand, we have

((gn,i)_{i∈bn−1,0c},τ^n+1−l)Hndk= (˜yi)_{i∈bn−1,0c}dk= (˜x⁰_i)_{i∈bn−2,0c} with

˜ yi:=







gn,i fori∈ bn−1, l−1c,

Q

j∈di+1,l−2e(˜y_j⁻¹dbj,i+1csdi,j−1e)

·Q

j∈bl−2,ic(gn,jdbl−1,i+1csdi,l−2e) fori∈ bl−2,0c fori∈ bn−1,0cand

˜ x⁰_i :=







˜

yi+1dk fori∈ bn−2, kc, (˜ykdk)(˜yk−1dk) fori=k−1,

˜

yidk fori∈ bk−2,0c

fori∈ bn−2,0c. We have to show that ˜xi= ˜x⁰_i for alli∈ bn−2,0c. To this end, we consider three cases and we handle each one by induction oni∈ bn−2,0c.

We suppose thatk∈ bn, lc. Fori∈ bn−2, kc, we have

˜

xi =fi=gn,i+1dk= ˜yi+1dk = ˜x⁰_i. Fori=k−1, we get

˜

xk−1=fk−1= (gn,kdk)(gn,k−1dk) = (˜ykdk)(˜yk−1dk) = ˜x⁰_k−1. Fori∈ bk−2, l−1c, we get

˜

xi =fi=gn,idk= ˜yidk= ˜x⁰_i Finally, fori∈ bl−2,0c, we calculate

˜

xi = Y

j∈di+1,l−2e

(˜x⁻¹_j d_bj,i+1cs_di,j−1e) Y

j∈bl−2,ic

(fjd_bl−1,i+1cs_di,l−2e)

= Y

j∈di+1,l−2e

(˜x⁰_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bl−2,ic

(fjdbl−1,i+1csdi,l−2e)

= Y

j∈di+1,l−2e

(˜y_j⁻¹dkdbj,i+1csdi,j−1e) Y

j∈bl−2,ic

(gn,jdkdbl−1,i+1csdi,l−2e)

=

³ Y

j∈di+1,l−2e

(˜y_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bl−2,ic

(gn,jdbl−1,i+1csdi,l−2e)

´ dk

= ˜yidk= ˜x⁰_i.

Next, we suppose thatk=l−1. For i∈ bn−2, kc, we have

˜

xi =fi=gn,i+1dk= ˜yi+1dk = ˜x⁰_i. Fori=k−1, we compute

˜

xk−1=fk−1= (gn,kdk)(gn,k−1dk) = (gn,kdk)(gn,k−1dksk−1dk)

= (˜ykdk)(˜yk−1dk) = ˜x⁰_k−1. Fori∈ bk−2,0c, we get

˜

x_i = Y

j∈di+1,k−2e

(˜x⁻¹_j d_bj,i+1cs_di,j−1e) Y

j∈bk−2,ic

(f_jd_bk−1,i+1cs_di,k−2e)

= Y

j∈di+1,k−2e

(˜x⁰_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bk−2,ic

(fjdbk−1,i+1csdi,k−2e)

= Y

j∈di+1,k−2e

(˜y⁻¹_j dkd_bj,i+1cs_di,j−1e) Y

j∈bk−2,ic

(gn,jdkd_bk−1,i+1cs_di,k−2e)

=

³ Y

j∈di+1,k−1e

(˜y_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bk−1,ic

(gn,jdbk,i+1csdi,k−1e)

´ dk

= ˜yidk= ˜x⁰_i.

Finally, we suppose thatk∈ bl−2,0c. Fori∈ bn−2, l−2c, we see that

˜

xi =fi=gn,i+1dk= ˜yi+1dk = ˜x⁰_i. Fori∈ bl−3, kc, we have

˜

xi = Y

j∈di+1,l−3e

(˜x⁻¹_j dbj,i+1csdi,j−1e) Y

j∈bl−3,ic

(fjdbl−2,i+1csdi,l−3e)

= Y

j∈di+1,l−3e

(˜x⁰_j⁻¹d_bj,i+1cs_di,j−1e) Y

j∈bl−3,ic

(fjd_bl−2,i+1cs_di,l−3e)

= Y

j∈di+1,l−3e

(˜y⁻¹_j+1dkdbj,i+1csdi,j−1e) Y

j∈bl−3,ic

(gn,j+1dkdbl−2,i+1csdi,l−3e)

=³ Y

j∈di+2,l−2e

(˜y⁻¹_j d_bj,i+2cs_di+1,j−1e) Y

j∈bl−2,i+1c

(gn,jd_bl−1,i+2cs_di+1,l−2e)´ dk

= ˜yi+1dk= ˜x⁰_i. Fori=k−1, we have

˜

xk−1= Y

j∈dk,l−3e

(˜x⁻¹_j d_bj,kcs_{dk−1,j−1e}) Y

j∈bl−3,k−1c

(fjd_bl−2,kcs_{dk−1,l−3e})

= Y

j∈dk,l−3e

(˜x⁰_j⁻¹d_bj,kcs_{dk−1,j−1e}) Y

j∈bl−3,k−1c

(fjd_bl−2,kcs_{dk−1,l−3e})

= ( Y

j∈dk,l−3e

(˜yj+1dkdbj,kcsdk−1,j−1e))( Y

j∈bl−3,kc

(gn,j+1dkdbl−2,kcsdk−1,l−3e))

·(gn,kdkd_bl−2,kcs_{dk−1,l−3e})(gn,k−1dkd_bl−2,kcs_{dk−1,l−3e})

= Y

j∈dk+1,l−2e

(˜y⁻¹_j d_bj,kcs_{dk−1,j−2e}) Y

j∈bl−2,k−1c

(gn,jd_bl−1,kcs_{dk−1,l−3e})

= (˜ykdk)

³ Y

j∈dk,l−2e

(˜y_j⁻¹dbj,kcsdk−1,j−1e) Y

j∈bl−2,k−1c

(gn,jdbl−1,kcsdk−1,l−2e)

´ dk

= (˜ykdk)(˜yk−1dk) = ˜x⁰_k−1. Fori∈ bk−2,0c, we get

˜

xi = Y

j∈di+1,l−3e

(˜x⁻¹_j dbj,i+1csdi,j−1e) Y

j∈bl−3,ic

(fjdbl−2,i+1csdi,l−3e)

= Y

j∈di+1,l−3e

(˜x⁰_j⁻¹d_bj,i+1cs_di,j−1e) Y

j∈bl−3,ic

(fjd_bl−2,i+1cs_di,l−3e)

= ( Y

j∈di+1,k−2e

(˜y⁻¹_j dkd_bj,i+1cs_di,j−1e))(˜y⁻¹_k−1dkd_bk−1,i+1cs_di,k−2e)

·(˜y_k⁻¹dkdbk−1,i+1csdi,k−2e)( Y

j∈dk,l−3e

(˜y_j+1⁻¹ dkdbj,i+1csdi,j−1e))

·( Y

j∈bl−3,kc

(gn,j+1dkd_bl−2,i+1cs_di,l−3e))(gn,kdkd_bl−2,i+1cs_di,l−3e)

·(gn,k−1dkd_bl−2,i+1cs_di,l−3e)( Y

j∈bk−2,ic

(gn,jdkd_bl−2,i+1cs_di,l−3e))

= Y

j∈di+1,k−1e

(˜y⁻¹_j dkdbj,i+1csdi,j−1e) Y

j∈dk,l−2e

(˜y_j⁻¹dkdbj−1,i+1csdi,j−2e)

· Y

j∈bl−2,ic

(gn,jdkd_bl−2,i+1cs_di,l−3e)

=

³ Y

j∈di+1,l−2e

(˜y_j⁻¹dbj,i+1csdi,j−1e) Y

j∈bl−2,ic

(gn,jdbl−1,i+1csdi,l−2e)

´ dk

= ˜yidk= ˜x⁰_i.

Now we consider the degeneracies. We let n ∈ N0, k ∈ [0, n], l ∈ [0, n+ 1], and (gn,i)i∈bn−1,0c∈Diag_nNG. We compute

((gn,i)_{i∈bn−1,0c},τ^n+1−l)skHn+1= ((gn,i)_{i∈bn−1,0c}sk,τ^n+1−lsk)Hn+1

= ((hi)i∈bn−1,0c,σ^kτ^n+1−l)Hn+1

=

(((hi)_{i∈bn−1,0c},τ^n+2−l)Hn+1 fork>l, ((hi)i∈bn−1,0c,τ^n+1−l)Hn+1 fork < l

)

= (˜zi)i∈bn,0c,

where hi :=







gn,i−1sk fori∈ bn, k+ 1c, 1 fori=k,

gn,isk fori∈ bk−1,0c and

˜ zi:=



























hi fori∈ bn, l−1c,

Q

j∈di+1,l−2e(˜z_j⁻¹d_bj,i+1cs_di,j−1e)

·Q

j∈bl−2,ic(hjd_bl−1,i+1cs_di,l−2e) fori∈ bl−2,0c





 ifk>l,







hi fori∈ bn, lc,

Q

j∈di+1,l−1e(˜z_j⁻¹dbj,i+1csdi,j−1e)

·Q

j∈bl−1,ic(hjd_bl,i+1cs_di,l−1e) fori∈ bl−1,0c





 ifk < l.

Furthermore, we have

((gn,i)_{i∈bn−1,0c},τ^n+1−l)Hnsk= (˜yi)_{i∈bn−1,0c}sk= (˜z_i⁰)_i∈bn,0c, where

˜ yi:=







gn,i fori∈ bn−1, l−1c,

Q

j∈di+1,l−2e(˜y_j⁻¹dbj,i+1csdi,j−1e)

·Q

j∈bl−2,ic(gn,jdbl−1,i+1csdi,l−2e) fori∈ bl−2,0c and

˜ z⁰_i:=







˜

yi−1sk fori∈ bn, k+ 1c, 1 fori=k,

˜

yisk fori∈ bk−1,0c.

Thus we have to show that ˜zi= ˜z_i⁰ for every i∈ bn,0c. Again, we distinguish three cases, and in each one, we perform an induction oni∈ bn,0c.

We suppose thatk∈ bn, lc. Fori∈ bn, k+ 1c, we calculate

˜

zi=hi=gn,i−1sk = ˜yi−1sk = ˜z_i⁰. Fori=k, we get

˜

zk=hk= 1 = ˜z_k⁰. Fori∈ bk−1, l−1c, we have

˜

zi=hi=gn,isk = ˜yisk= ˜z_i⁰. Fori∈ bl−2,0c, we get

˜

zi= Y

j∈di+1,l−2e

(˜z_j⁻¹d_bj,i+1cs_di,j−1e) Y

j∈bl−2,ic

(hjd_bl−1,i+1cs_di,l−2e)

= Y

j∈di+1,l−2e

(˜z_j⁰⁻¹dbj,i+1csdi,j−1e) Y

j∈bl−2,ic

(hjdbl−1,i+1csdi,l−2e)