§4 The Kan classifying functor

Constπ0G

Constπ₀ϕ

H ^εH //Constπ₀H

Thus we have a natural transformationidsGrp

−→ε Const ◦π0. Finally, we have to show thatεandη are unit and counit. Indeed,

(εConstP)n(ConstηP)n = idPνConstPηP =νConstPηP = idP = (idConstP)n

holds for alln∈N0, P∈ObGrp, as well as

(g₀B₀MG)(π₀ε_G)(η_π0G) = ((g₀(ε_G)₀)B₀M(Constπ₀G))η_π0G= (g₀ν_G{1})ηπ0G=g₀ν_G=g₀B₀MG for allG∈ObsGrp,g₀∈G₀.

(4.16) Definition(0th coskeleton). For every groupP we define thecoskeletonofP to be the simplicial group CoskP:= ConstP. The functor

Grp−−−→^Cosk sGrp is called the (0th)coskeleton.

§4 The Kan classifying functor

We have already seen that we can define the homology of a group as the homology of the classifying simplicial set. In this section, we want to generalise this procedure to simplicial groups, introducing Kan’s classifying functor

sGrp−→^W sSet,

which generalises the classifying simplicial set functor Grp−→^B sSet

in the sense thatB = W◦Cosk. After this, we constructKan’s loop group functorGas a left adjoint toW.

The reader is refered to [20].

(4.17) Definition (nerve of a simplicial group). We define the nerve NG of a simplicial group G to be the bisimplicial set which corresponds to the nerve of the group object insSetarising fromG. Analogously for the morphisms insGrp.

sGrp ^N //

OO

∼=

s²SetOO

∼=

Grp(sSet) ^N //s(sSet)

(4.18) Remark(the nerve of a simplicial group is build componentwise). The nerve of a simplicial groupGis given byN_m,−G= NGmfor allm∈N0 andN_θ,−G= NGθ for all morphismsθ∈Mor∆.

Proof. Follows from example (1.27)(c) and definition (1.32).

(4.19) Remark. There is a functor sGrp−→^W sSet

isomorphic toTot◦Nthat is given on objects byWnG:=

×

j∈bn−1,0cGj and (gj)_{j∈bn−1,0c}WθG:= ( Y

j∈b(i+1)θ−1,iθc

gjG

θ|^[j]_[i])_{i∈bm−1,0c}

for (g_j)_{j∈bn−1,0c} ∈ W_nG, where θ ∈ ∆([m],[n]), m, n ∈ N0, G ∈ ObsGrp, and on morphisms by W_nϕ :=

×

j∈bn−1,0cϕ_j forn∈N0, ϕ∈sGrp(G, H),G, H∈ObsGrp.

Proof. We letGbe a simplicial group and computeTot NG. The set ofn-simplices is given by Tot_nNG={(xq)_q∈bn,0c∈

×

q∈bn,0c

N_q,n−qG|x_qd^h_q =x_q−1d^v₀ forq∈ bn,1c}

={((gq,j)j∈bn−q−1,0c)_q∈bn,0c∈

×

q∈bn,0c

G^×(n−q)_q |(g_q,j)j∈bn−q−1,0cd^h_q = (g_q−1,j)_{j∈bn−q,0c}d^v₀ forq∈ bn,1c}

={((gq,j)j∈bn−q−1,0c)_q∈bn,0c∈

×

q∈bn,0c

G^×(n−q)_q |(gq,jdq)j∈bn−q−1,0c= (gq−1,j+1)j∈bn−q−1,0c forq∈ bn,1c}

={((gq,j)j∈bn−q−1,0c)_q∈bn,0c∈

×

q∈bn,0c

G^×(n−q)_q |gq,jdq =gq−1,j+1 forj∈ bn−q−1,0c, q∈ bn,1c}

={((gq,j)j∈bn−q−1,0c)_q∈bn,0c∈

×

q∈bn,0c

G^×(n−q)_q |gq,j =gq+1,j−1dq+1 forj∈ bn−q−1,1c, q∈ bn−1,0c}

={((gq,j)j∈bn−q−1,0c)_q∈bn,0c∈

×

q∈bn,0c

G^×(n−q)_q |gq,j =gq+j,0d_bq+j,q+1c forj∈ bn−q−1,1c, q∈ bn−1,0c}

={((gq,j)j∈bn−q−1,0c)_q∈bn,0c∈

×

q∈bn,0c

G^×(n−q)_q |gq,j =gq+j,0d_bq+j,q+1c forj∈ bn−q−1,0c, q∈ bn,0c}

={((gq+j,0d_bq+j,q+1c)j∈bn−q−1,0c)_q∈bn,0c|gq+j,0∈Gq+j forj∈ bn−q−1,0c, q∈ bn,0c}

={((gjd_bj,q+1c)_{j∈bn−1,qc})_q∈bn,0c|gj ∈Gj forj∈ bn−1, qc, q∈ bn,0c}

forn∈N0. For an element((gjd_bj,q+1c)_{j∈bn−1,qc})_q∈bn,0c∈TotnNG, we compute

((gjd_bj,q+1c)_{j∈bn−1,qc})_q∈bn,0c(TotθNG) = ((gq+jd_bq+j,q+1c)j∈bn−q−1,0c)_q∈bn,0c(TotθNG)

= ((gpθ+jdbpθ+j,pθ+1c)j∈bn−pθ−1,0c(NG)_Spl

p(θ))_p∈bm,0c

= ((g_pθ+jdbpθ+j,pθ+1c)j∈bn−pθ−1,0c(NG)_Spl

≤p(θ),n−pθ(NG)_p,Spl

≥p(θ))_p∈bm,0c

= ((gpθ+jdbpθ+j,pθ+1cG_Spl≤p(θ))j∈bn−pθ−1,0c(NG)_{p,Spl≥p(θ)})_p∈bm,0c

= (( Y

j∈b(i+1)Spl_≥p(θ)−1,iSpl_≥p(θ)c

gpθ+jdbpθ+j,pθ+1cG_Spl≤p(θ))i∈bm−p−1,0c)_p∈bm,0c

= (( Y

Thus, by transport of structure,WGwithW_nG=

×

j∈bn−1,0cG_j for alln∈N0 becomes a reduced simplicial set isomorphic toTot NGvia the bijections

(f_G)_n: Tot_nNG→WG,((g_jd_bj,q+1c)_{j∈bn−1,qc})_q∈bn,0c 7→(g_j)_{j∈bn−1,0c}.

Furthermore, given another simplicial groupH and a simplicial group homomorphismG−→^ϕ H, we have, again by transport of structure,

Altogether, we have constructed a functorsGrp−→^W sSetand a natural isotransformationTot◦N−→^f W.

(4.20) Definition (Kan classifying simplicial set). We letGbe a simplicial group. The reduced simplicial set WGgiven as in remark (4.19) by

WnG=

×

theKan classifying simplicial set ofG.

(4.21) Proposition. We letGbe a simplicial group. The faces and degeneracies of its Kan classifying simplicial setWGare given by

for(gj)_{j∈bn−1,0c} ∈WnG,k∈[0, n],n∈N, and

resp.

Hⁿ(W CoskG, M;R) = Hⁿ(BG, M;R)∼= Hⁿ(G, M;R) forn∈N0 and a moduleM over a commutative ringR.

Now we want to construct a left adjoint forW.

(4.23) Remark. We letX be a reduced simplicial set and we letX −→X be a simplicial bijection given by Xn →Xn, xn7→xn for alln∈N0. For everyn∈N0, we letGnX be the free group

GnX :=hxn+1|xn+1∈Xn+1, xnsn= 1for allxn∈XniGrp∼=hxn+1|xn+1∈Xn+1\(Im sn)iGrp. Further, forθ∈_∆([m],[n]),m, n∈N0, we define a group homomorphismG_θX: G_nX →G_mXon the generating setX_n+1 ofG_nX by setting

xn+1(GθX) :=xn+1(PθX)(xn+1dn+1sn(PθX))⁻¹ for allxn+1∈Xn+1. ThenGX is a simplicial group.

Proof. First of all, we note that ifxn+1=xnsn for somexn∈Xn, then

x_n+1(P_θX)(x_n+1d_n+1s_n(P_θX))⁻¹=x_ns_n(P_θX)(x_ns_nd_n+1s_n(P_θX))⁻¹=x_ns_n(P_θX)(x_ns_n(P_θX))⁻¹= 1 forθ∈∆([m],[n]),m, n∈N0. Now, given morphismsθ∈∆([m],[n]),ρ∈∆([n],[p])form, n, p∈N0, we note that(P_ρX)d_n+1= d_p+1X_ρ by proposition (2.31)(c) and obtain thus

xp+1(GρX)(GθX) = (xp+1(PρX)(xp+1dp+1sp(PρX))⁻¹)(GθX)

=x_p+1(P_ρX)(G_θX)(x_p+1d_p+1s_p(P_ρX)(G_θX))⁻¹

=xp+1(PρX)(PθX)(xp+1(PρX)dn+1sn(PθX))⁻¹

·(x_p+1d_p+1s_p(P_ρX)(P_θX)(x_p+1d_p+1s_p(P_ρX)d_n+1s_n(P_θX))⁻¹)⁻¹

=xp+1(PρX)(PθX)(xp+1(PρX)dn+1sn(PθX))⁻¹

·xp+1dp+1sp(PρX)dn+1sn(PθX)(xp+1dp+1sp(PρX)(PθX))⁻¹

=xp+1(PθρX)(xp+1dp+1Xρsn(PθX))⁻¹

·xp+1dp+1spdp+1Xρsn(PθX)(xp+1dp+1sp(PθρX))⁻¹

=x_p+1(P_θρX)(x_p+1d_p+1s_p(P_θρX))⁻¹=x_p+1(G_θρX) forxp+1∈Xp+1 as well as

xn+1(Gid_[n]X) =xn+1(Pid_[n]X)(xn+1dn+1sn(Pid_[n]X))⁻¹=xn+1idP_nX(xn+1dn+1snidP_nX)⁻¹=xn+1

forxn+1∈Xn+1. HenceGX is a simplicial group.

(4.24) Definition(Kan loop group). We letX be a reduced simplicial set and we letX−→X be a simplicial bijection given byXn→Xn, xn 7→xn for alln∈N0. The simplicial groupGX given as in remark (4.23) by

GnX =hxn+1 |xn+1∈Xn+1, xnsn= 1for allxn ∈XniGrp

and

GθX: GnX →GmX, xn+17→xn+1(PθX)(xn+1dn+1sn(PθX))⁻¹ forθ∈∆([m],[n]),m, n∈N0, is called the (Kan)loop group ofX.

In the following, we always assume given a simplicial bijectionX7−→Xwhen referring to the constructionGX.

(4.25) Proposition. We letX be a reduced simplicial set. The facesdk: GnX→G_n−1Xfork∈[0, n],n∈N, and the degeneraciessk: GnX →Gn+1X fork∈[0, n], n∈N⁰, in the loop group GX are given by

xn+1dk=

(xn+1dk fork∈[0, n−1], x_n+1d_n(x_n+1d_n+1)⁻¹ fork=n

and

xn+1sk =xn+1sk fork∈[0, n], wherex_n+1∈X_n+1.

Proof. We have

xn+1dk=xn+1(G_δkX) =xn+1P_δkX(xn+1dn+1snP_δkX)⁻¹=xn+1dk(xn+1dn+1sndk)⁻¹

=

(x_n+1d_k(x_n+1d_n+1d_ks_n−1)⁻¹ fork∈[0, n−1], xn+1dn(xn+1dn+1)⁻¹ fork=n

)

=

(x_n+1d_k fork∈[0, n−1], xn+1dn(xn+1dn+1)⁻¹ fork=n

)

forx_n+1∈X_n+1,n∈N, and

xn+1sk =xn+1(G_σkX) =xn+1P_σkX(xn+1dn+1snP_σkX)⁻¹=xn+1sk(xn+1dn+1snsk)⁻¹

=x_n+1s_k(x_n+1d_n+1s_ks_n+1)⁻¹=x_n+1s_k forxn+1∈Xn+1,n∈N0.

(4.26) Definition (simplicial free group). A simplicial groupF is called asimplicial free group, ifFn is a free group with a free generating systemXn⊆Fn for every n∈N0andXnsk⊆Xn+1 for everyn∈N0,k∈[0, n].

(4.27) Proposition. The Kan loop groupGX of a reduced simplicial setX is a simplicial free group.

Proof. This follows from proposition (4.25).

(4.28) Proposition.

(a) We letX −→^f Y be a simplicial map between reduced simplicial setsX andY. Then we have an induced morphism

GX −−→^Gf GY, given by

xn+1(Gnf) =xn+1(Pnf) =xn+1fn+1 forxn+1∈Xn+1, n∈N⁰. (b) The construction in (a) yields a functor

sSet₀−→^G sGrp.

Proof.

(a) We have

x_n+1(G_θX)(G_mf) = (x_n+1(P_θX)(x_n+1d_n+1s_n(P_θX))⁻¹)(G_mf)

= (xn+1(PθX)(Gmf))(xn+1dn+1sn(PθX)(Gmf))⁻¹

=xn+1(PθX)(Pmf)(xn+1dn+1sn(PθX)(Pmf))⁻¹

=xn+1(Pnf)(PθY)(xn+1(Pnf)dn+1sn(PθY))⁻¹

=xn+1(Pnf)(GθY) =xn+1(Gnf)(GθY),

for xn+1 ∈ Xn+1 and thus (GθX)(Gmf) = (Gnf)(GθY) for θ ∈ ∆([m],[n]). Hence Gf is a simplicial group homomorphismGX −→GY.

(b) We letX,Y, Z be reduced simplicial sets andX−→^f Y, Y −→^g Z be simplicial maps. Then x_n+1(G_nf)(G_ng) =x_n+1(P_nf)(G_ng) =x_n+1(P_nf)(P_ng) =x_n+1(P_n(f g)) =x_n+1(G_n(f g)) and

xn+1(GnidX) =xn+1(PnidX) =xn+1idP_nX=xn+1

for allx_n+1∈X_n+1,n∈N0. Thus(Gf)(Gg) = G(f g)andGid_X= id_GX, and henceGis a functor.

(4.29) Theorem (cf. [20, Proposition 10.5]). The functorsSet₀−→^G sGrpis left adjoint tosGrp−→^W sSet₀. Proof. We let X ∈ObsSet0 be a reduced simplicial set, H ∈ObsGrpa simplicial group and GX −→^ϕ H a

commutes. Thus the maps(ϕΦX,H)n forn∈N⁰yield a simplicial map X −−−−→^ϕΦX,H WH.

Sinceϕ∈sGrp(GX, H)was arbitrary, we have a well-defined map ΦX,H:sGrp(GX, H)→sSet₀(X,WH).

We claim that the mapsΦX,H forX ∈ObsSet0,H ∈ObsGrp, yield a natural transformation

sGrp(G−,=)−→sSet₀(−,W=).

Indeed, given reduced simplicial sets X, Y ∈ObsSet0, simplicial groups H, K ∈ ObsGrp, a simplicial map Y −→^e X and a simplicial group homomorphismH −→^ψ K, we have

commutes, and we have a natural transformation

sGrp(G−,=)−→^Φ _sSet0(−,W=).

It remains to show thatΦis an isomorphism. To this end, for a given reduced simplicial setX, a given simplicial groupH and a given simplicial mapX−→^f WH, we define a group homomorphisms(fΨX,H)n: GnX→H by

commutes. Hence the group homomorphisms(fΨX,H)n forn∈N⁰ yield a simplicial group homomorphism fΨ_X,H: GX →H,

and sincef ∈_sSet0(X,WH)was arbitrary, we have a well-defined map Ψ_X,H:_sSet0(X,WH)→_sGrp(GX, H).

We have to show thatΦ_X,H andΨ_X,H are mutually inverse maps for each reduced simplicial setX ∈ObsSet₀ and each simplicial groupH∈ObsGrp. Indeed, it holds that

x_n+1(ϕΦ_X,HΨ_X,H)_n= (x_n+1(ϕΦ_X,H)_n+1)_n=x_n+1d_bn+1,n+2cϕ_n=x_n+1ϕ_n

for all x_n+1 ∈ X_n+1, n ∈ N0, that is, ϕΦ_X,HΨ_X,H = ϕ for every simplicial group homomorphism ϕ ∈

sGrp(GX, H). Moreover,

x_n(fΨ_X,HΦ_X,H)_n= (x_nd_bn,j+2c(fΨ_X,H)_j)_{j∈bn−1,0c} = ((x_nd_bn,j+2cf_j+1)_j)_{j∈bn−1,0c}

= ((xnfnd_bn,j+2c)j)_{j∈bn−1,0c} =xnfn

for allxn∈Xn,n∈N0, that is,fΨX,HΦX,H =f for every simplicial mapf ∈sSet₀(X,WH). This implies ΦX,HΨX,H = id_sGrp(GX,H)andΨX,HΦX,H = id

s Set0(X,WH)for allX∈ObsSet0, H ∈ObsGrp, and hence the mapsΨX,H forX∈ObsSet0,H ∈ObsGrpyield a natural transformation

sSet₀(−,W=)−→^Ψ sGrp(G−,=)

inverse toΦ. ThussGrp(G−,=)∼=sSet₀(−,W=), that is,GaW.

Im Dokument (Co)homology of crossed modules (Seite 76-84)