AN E2 TYPE CLOSED MODEL CATEGORY FOR
BISIMPLICIAL GROUPS
ALEXANDER NOFECH
Abstract. A closed model category structure on the category of bisimpli
cial groups is defined in which the weak equivalences are isomorphisms on
bigraded homotopy groups ßk,land at the same time isomorphisms on the
E2 term of the Quillen spectral sequence. There is an analogue of the spir*
*al
exact sequence of DwyerKanStover [4]. This structure is considered as a
convenient setting for a study of the relation between bigraded homotopy
and hyperhomology.
Contents
1. Introduction and notation 2
1.1. Definition and properties of bigraded homotopy groups 2
1.2. Constant Extensions 4
1.3. Notation and preliminaries 4
2. The closed model category structure 6
2.1. The existence of a closed model category 6
3. A criterion of E2 fibrations for bisimplicial groups 9
3.1. The horizontal long exact sequence of a fibration 10
4. The simplicial structure 11
5. Attachment of Cells and Simplicial Resolutions 12
6. Properness 13
References 14
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 2
1. Introduction and notation
1.1. Definition and properties of bigraded homotopy groups
Let G be a bisimplicial group, and let cvßv0G be a vertical constant extension
of the ß0 groups of its vertical terms. Let
1 ! vG ! EvG ! IvG ! {1}, {1} ! IvG ! G ! cßv0G ! 1
be exact sequences obtained by taking the contractible path group E(_ ) for
every vertical term Gk,o. In this sequence the kernel vG is the vertical loop
group of G, and IvG = G(1) the identity component of G. Iterating this
construction of Quillen gives the exact couple of [6]:
ßp1 ( q+1vG) _______________ßp( qvG)
I@
@
@
@
@
ßp hßvqG
In what follows we denote ßp,qG def=ßp qvG, and call these groups the bigraded
homotopy groups of G. In this notation Quillen's exact couple becomes
ßp1,q+1G __________________ ßp,qG
I@
@
@
@
@
ßphßvqG
and as a long exact sequence it can be rewritten:
. .!.ßp1,q+1G ! ßp,qG ! ßphßvqG ____ ßp2,q+1G ! ßp1,qG ! . . .
Following [3] this sequence is called the spiral exact sequence.
Rewriting the exact sequence for p = 2 and keeping in mind the connection
between the vertical identity components IvG and the vertical loops vG, we
get
. . .! ß1 Iv lvG ! ß1 lvG ! ßh1ßvlG ! ß0 Iv lvG ! ß0 lvG ! ßh0ßvlG ! {1}
Hence, since ß1 Iv lvG ~=ß0 l+1vG,
. .!.ß0,l+1G ! ß1,lG ! ßh1ßvlG ! ß0 Iv lvG ! ß0,lG ! ßh0ßvlG ! {1}
where l 0.
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 3
From ßv0Iv lvG = 0 for all vertical terms it follows that ß0 Iv lvG = 0 for all
l 0.
Hence ß0 lvG ~= ßh0ßvlG. In other words, ß0,lG ~= ßh0ßvlG the 0th column
of bigraded groups coincides with the 0th column of the E2 term. The spiral
exact sequence ends as:
. .!.ß1,l+1G ! ß2,lG ! ßh2ßvlG ! ß0,l+1G ! ß1,lG ! ßh1ßvlG ! 0
The following lemma follows from the above and from the definitions:
Lemma 1.1. For any bisimplicial group G
(1). ß*,0G = ß* G (the homotopy of the diagonal of G)
(2). ßp,q(G x H) = ßp,qG x ßp,qH
(3). ßp,qG are abelian unless p = q = 0
(4). ß0,lG ~=ßh0ßvlG
(5). ß1,lG ! ßh1ßvlG is onto
Theorem 1.2. Let f : G ! H be a homomorphism of bisimplicial groups, and
let f induce isomorphisms on all fringe bigraded groups ß1,*. Then f induces
an isomorphism of the E2 terms of the Quillen spectral sequence if and only if
it induces isomorphisms of the bigraded homotopy groups
~=
ßp,qf : ßp,qG ____ ßp,qH
for all (p, q) 0.
Proof. One direction follows from the spiral exact sequence and the five lemma.
For the other we need to show that if
ßphßvqf : ßphßvqG '! ßphßvqH
then
ßp,qf : ßp,qG '! ßp,qH
This follows from the convergence of Quillen spectral sequence for groups qvG,
qvH .
Remark 1.3. In particular, this implies that Reedy equivalencies induce isomor
phisms on all bigraded homotopy groups, since they are also weak equivalencies
on vertical identity components.
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 4
1.2. Constant Extensions
Let cvGo be a constant vertical extension of a simplicial group Go. Then
qvcvGo = {1} for q > 0 and we have in the zeroth row
ßp,0(cvGo) = ßpG
and for positive q
ßp,q(cvGo) = 0, q > 0
Let chGo be a constant horizontal extension of a simplicial group Go. Then
qv(chGo) = ch( qGo)
and
qv(chGo) = qGo
After taking iterated vertical loops and then vertical identity components, the
group remains horizontally constant, so its diagonal is isomorphic to a vertical
term, which is now zeroconnected. So
ß1,l(chGo) = 0
ßp,q(chGo) = ßp qGo = ßp+q Go
and homomorphisms of bigraded homotopy groups of bidegree (1, 1) are iso
morphisms.
1.3. Notation and preliminaries
Since much of the notation is selfexplanatory, such as ßv0, ßh0 for the 0th
homotopy groups of vertical or horizontal terms, it is introduced as a table
below with short explanations. For a bisimplicial group G and a simplicial set
K the operations G K and GK are horizontal (see the section on simplicial
structure) and because of this the external tensor product of a simplicial group
G and a simplicial set K is denoted K G, rather than G K . Let ooobe
~ ~
the bisimplicial cosimplicial object which is [n] x [n] in cosimplicial degree
~
n with cosimplicial operations induced by those of ~ . Then the diagonal of
a bisimplicial group G ~=Hom( ooo, G) with the group structure induced by
that of G, and the left adjoint to diagonal L = ooowhere is a simplicial
group and the colimits are in the category of groups, similarly for bisimplicial
and pointed bisimplicial sets (this recovers Lemma B.8 of [2]).
The object cZ oooand all of its vertical suspensions allow to recover the
closed model category structure as a localization of the Reedy structure with
respect to an object, or a set of such. However, this set of objects does not
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 5
belong to the category of simplicial groups, compare with the definitions of the
model structure in [5].
The iterated loops of a simplicial group G are obtained by repeated use of a
functor Hom___(S1, G) and because of this in the construction of generators of
__k
cofibrations there appear smash products with S = S1 ^ . .^.S1 which is
weakly equivalent but not isomorphic to the ordinary k sphere Sk .
When notation consistent with [6] is needed IvH = H(1) is used for the identity
component of a simplicial group.
x external product of simplicial sets [2], which induces:
~
^~ external smash product of pointed simplicial sets
n external halfsmash product of pointed and unpointed sim*
*plicial sets
~
(S0 ^~S0) constantly extended simplicial S0
K G external tensor product of a simplicial set with a simpl*
*icial group
~
K G tensor product of a simplicial set K with a bisimplicial*
* group G,
see the section on simplicial structure
G K ordinary tensor product of a bisimplicial group G and a *
*bisimplicial set K
ßk,lGoo bigraded homotopy groups of Goo, or
ßk,lG when it is clear G is bisimplicial;
j % ' j has the Left Lifting Property with respect to '
( ~ x ) the standard double cosimplicial object
~ ~
Hom____(K, G) = Hom(K o ( ~ x ), G), bisimplicial Homgroup
~ ~
with group structure induced by that of G, where
K is a pointed bisimplicial set
lvG = Hom____((S0 ^~S1), G) lth vertical loops of G,
the group structure induced by that of G
F K the extension of the pointed free functor
(usually from pointed bisimplicial sets to bisimplicial *
*groups)
F (S0 ^~S0) = cZ, infinite cyclic group, constant in both indices
lvG = G (S0 ^~Sl), lth vertical suspension of G, see Lemm*
*a 2.2
IvG = Gv(1) vertical identity components of G
( G)(1)= (G(1)) The identity component of the diagonal of G
Gv(1)= IvG vertical identity components subgroup of G
Acknowledgement I wish to thank Jie Wu for his comments.
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 6
2. The closed model category structure
2.1. The existence of a closed model category
Theorem 2.1. There exists a closed simplicial model category structure on
the category Groups oo of bisimplicial groups in which
(1). The fibrations are homomorphisms of bisimplicial groups that
(a).are Reedy fibrations;
(b).induce fibrations, as simplicial groups , on the diagonals of ver*
*tical
q  loops for all q > 0.
(ii)The weak equivalences are homomorphisms of bisimplicial groups that
induce isomorphisms on all bigraded homotopy groups;
(iii)The cofibrations are defined by the LLP (the left lifting property).
Equivalently:
(1). Fib oo= { '  qv' 2 Fibo, 'k,o2 Fibo}
~=
(2). Woo = { f : G ! H  ßk,lf : ßk,lG ____ ßk,lH }
(3). Cof oo= { j  j % Fiboo \ Woo }
The generators of cofibrations are:
lvF Li[n] = F Li[n] (S0 ^~Sl) = cZ (Li[n] n (S0 ^~Sl))
The generators of trivial cofibrations are:
(1). lvF Lij[n] = cZ (Lij[n] n (S0 ^~Sl))
(2). F ( [m] x ij[n])
~
Proof. CM1, CM2, CM3 and CM4(I) follow immediately from the definitions.
Let ' be a fibration. The generators of trivial cofibrations (1) are determined
from a sequence of lifting conditions related to each other by adjunctions:
ij[k] % v'
ij[k] % Hom____(S0 ^~S1, ') (by the definition of vertical loops)
Lij[k] % Hom____(S0 ^~S1, ') (L is left adjoint to diagonal)
Lij[k] % Hom((S0 ^~S1) o ( ~ x ), ') (see Notation)
~ ~
Lij[k] x ( ~ x ) % Hom((S0 ^ S1), ')
~ ~ ~
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 7
Lij[k] % Hom((S0 ^~S1), ')
Lij[k] o (S0 ^~S1) % '
Lij[k] o (S0 ^~S1) % Forgetful(')
F (Lij[k] o (S0 ^~S1)) % ' (F is the free functor, pointed version)
cZ (Lij[k] o (S0 ^~S1)) % ', or F Lij[k] (S0 ^~S1) % '
CM4(II) Consider the diagram:
A __________X
? ..` 
 ... 
j ~ ... '
 ... 
?... ??
B __________Y
in which ' is a fibration and j is a trivial cofibration. Factor j using the sm*
*all
object argument with the generators (2):
ff
A __________C
? ~ .. 
 ...` 
j ~ ... fi
 ... 
?... ??
=
B __________B
From Lemma 2.2 it follows that ff is an equivalence and so fi is a trivial
fibration. The existence of a lifting implies that j is a retract of ff and si*
*nce
ff % ' by the definition of fibrations the lifting j % ' exists.
CM5(I) Factor the given homomorphism using both sets of generators in the
small object argument .
CM5(II) Factor the given homomorphism using only the generators of trivial
cofibrations.
Lemma 2.2. The generators of trivial cofibrations are E2  equivalences.
Proof. For type (1), we need to show that
lvF Lij[n] = F Lij[n] (S0 ^~Sl) = cZ (Lij[n] n (S0 ^~Sl))
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 8
are equivalences for all l . Recall from [2] that
L j[n] = j[n] x j[n], L [n] = [n] x [n]
~ ~
In the following diagram the vertical maps are induced by choosing a vertex
in j[n] and taking retractions. For each horizontal degree they are homo
topy equivalencies of simplicial sets, whose disjoint unions comprise the ver*
*tical
terms:
a Lij[n]
[n] x j[n] j[n] x [n] ______ [n] x [n]
~ j[n]x j ~ ~
~ [n] 
 
 
a ? ~= ?
[n] x (pt) j[n] x (pt)________ [n] x (pt)
~ j[n]x ~ ~
~ (pt)
After taking vertical halfsmash products with Sl and applying the pointed fr*
*ee
functor, the vertical maps remain Reedy equivalencies, hence E2 equivalencie*
*s.
Thus in the diagram below the top horizontal map is an equivalence:
a
[n] F ( j[n] n Sl) j[n] F ( [n] n Sl) ___ [n] F ( [n] *
*n Sl)
~ j[n] j l ~ ~
~ F ( [n]^S ) 

 
 
? ~= ?
[n] F Sl _______________________________ [n] F Sl
~ ~
Rewriting the top horizontal line as a homomorphism from
a
F ( [n] x j[n]) (S0 ^ Sl) F ( j[n] x [n]) (S0 ^ S*
*l)
~ ~ F ( j[n]x j 0 l ~ ~
~ [n]) (S ^~S )
to
F ( [n] x [n]) (S0 ^ Sl)
~ ~
we see that it is F Lij[n] (S0 ^~Sl), one of the generators of trivial cofibr*
*ations.
As for type (2), these generators are Reedy equivalences and hence equivalenc*
*es.
Remark 2.3. Taking l = 0 for generators of trivial cofibrations of type (1) g*
*ives
F Lij[n] : F ( j[n] x j[n]) ! F ( [n] x [n])
~ ~
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 9
Choosing a vertex in j[n] gives a retraction of this generator onto F (ij[n] x
~
Id(pt)) ~=ij[n] Id(cZ), which is a constant vertical extension of ij[n]. Since
~
the retract inherits the LLP with respect to fibrations, by adjunction to taking
the vertical degree zero, it implies that an E2 fibration ' : G ____ H induc*
*es
a fibration of simplicial groups 'o,0: Go,0 ____ Ho,0.
3. A criterion of E2 fibrations for bisimplicial groups
The next theorem provides a characterization of fibrations:
Theorem 3.1. A Reedy fibration ' : G ! H of bisimplicial groups is an E2
fibration if and only if the induced homomorphism
ßvk' : ßvkG ! ßvkH
is a fibration of simplicial groups for all k 0.
Proof. Ö nly if": we have a diagram with exact rows
{1} _______ Gv(1) ______ G(1) _____(ßv0G)(1)______{1}
  
  
  
  
  
? ? ?
{1} _______ Hv(1) ______ H(1) _____(ßv0H)(1)______{1}
Let ' : G ! H induce fibrations on all diagonals. Then '(1) is onto and con
sequently ' induces surjections of identity components (ßv0G)(1) ___ (ßv0H)(1*
*).
Replacing ' with its l fold vertical loops lv' proves ö nly if".
In order to prove "if" consider the same diagram. The left vertical morphism is
onto, and likewise after iterated vertical looping, since ' is a Reedy fibratio*
*n.
Hence if the right terms surject, so do the extended groups at the center. It
remains valid after iterated vertical looping, proving "if".
Lemma 3.2. If ' : G ! H is a fibration, then 'o,0: Go,0! Ho,0 is a fibration
of simplicial groups.
Proof. Since the vertical constant extension is the left adjoint of the 0row
functor, we need to show that
(ij[n] x (pt)) % '
~
Since ' is a fibration, Lij[n] = ij[n] ij[n] % '. It remains to note that
~
ij[n] x (pt) is a retract of ij[n] x ij[n].
~ ~
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 10
Remark 3.3. A Reedy fibration does not need to be an E2 fibration. Let the
pointed interval be I = [1]=(0)~* and consider j : {1} ! cvF I . Since the
vertical terms of cvF I are constant, j is a Reedy fibration. But its diagonal
j : {1} ! F I is not a fibration of simplicial groups since it is not onto for
components of identity.
Lemma 3.4. For a bisimplicial group G, the natural homomorphism G !
ßv0G induced by the adjunction homomorphism ivG : G ! cvßv0G is a fibration
~=
of simplicial groups, inducing an isomorphism: ß0 G ____ ßh0ßv0G
Proof. Since the homomorphism is surjective on each vertical term, its restric
tion to diagonals is also surjective, and the isomorphism on ß0 implies it is o*
*nto
on identity components.
3.1. The horizontal long exact sequence of a fibration
The diagonals of iterated vertical loops of a fibration again form a fibration,
resulting in a homotopy long exact sequence for each row of bigraded homotopy
groups. If in the following short exact sequence ' is a fibration of bisimplici*
*al
'
groups : {1} _____ K _____ G ____ H _____ {1} , then the following sequen*
*ce
of simplicial groups is also a fibration sequence:
lv' l
{1} ____ lvK ____ lvG _____ vH
It gives rise to a homotopy sequence
. .!.ßk lvK ! ßk lvG ! ßk lvH ! ßk1 lvK ! . . .
which can be rewritten as
. . .! ßk,lK ! ßk,lG ! ßk,lH ! ßk1,lK ! . . .
giving long exact sequences for each horizontal row of bigraded homotopy
groups. It ends as:
. . .! ß0,lK ! ß0,lG ! ß0,lH
Remark 3.5. Note that if ' is a trivial fibration this does not imply that its
restriction to the 0th row 'o,0 is a trivial fibration. For example let G =
chF I  a constant horizontal extension of a free group generated by the (verti*
*cal)
interval  and let H = {1}. Then G is homotopically trivial but its 0th row
is a constant free group on two generators, (and so is its ß0).
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 11
4. The simplicial structure
Definition 4.1. Let G be a bisimplicial group and K an unpointed simplicial
set. For each k 0, l 0 let
a
(G K)n,o = Gn,o (GK )n,o = Gn,o
Kn Kn
with horizontal simplicial operations induced by those on G and K and define
Hom___(G, H) def=Hom(G ~ , H) ~=Hom(G, H ~)
Remark 4.2. This means that simplicial operations are in the horizontal direc
tion, compare with the simplicial structure of [3]. Conditions such as
Hom___(G, H)0 = Hom(G [0], H) = Hom(G (pt), H) ~=Hom(G, H)
follow from those for simplicial groups, similarly for compositions. The con
structions are functorial and agree with relations between the free functor and
the simplicial tensoring of [8] , 1.2.
Theorem 4.3. The functors (_ ) K , (_ )K are compatible with the closed
model category structure on Groups oo in the sense of [7], Part II, 2.2
Proof. We check the condition SM7(a) of [7], Part II, 2.2. Let ' : G ____ H
be a fibration of bisimplicial groups and let n 0.
(i) First, we need to check that the following is a fibration:
'i[n]: G [n] ____ G `[n] x H [n]
H `[n]
It is convenient to use the definition of fibrations rather than the c*
*ri
terion in Th.3.4. It is given that the diagonal of any iterated verti*
*cal
loops of ' is a fibration of simplicial groups. Since for any fixed f*
*irst
index n the nth horizontal row of GKooconsists of products Gn,o,
Kn
the functors (_)K and (_) commute: (GK ) = ( G)K
Hence we can first take the diagonal and use the fact that the con
struction above preserves fibrations of simplicial groups. The constru*
*c
tion also commutes with taking iterated vertical loops, etc.
(ii)Let now ' be a trivial fibration. We need to show that the diagonal of
iterated vertical loops of
'i[n]: G [n] ____ G `[n] x H [n]
H `[n]
is a trivial fibration. This reduces to the analogous property of simp*
*li
cial groups in the same way as above.
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 12
(iii)Finally we need to show that for any generator of trivial cofibrations*
* of
simplicial sets ij[n] : j[n] ! [n] and ' : G ! H the homomorphism
'ij[n]is a trivial fibration. This is seen by an argument similar to t*
*he
previous two.
A pointed version of the simplicial structure is defined as usual: for a pointed
simplicial set K define G K as the cokernel in the diagram induced by the
inclusion of the basepoint in K , and dually for GK . To simplify the notation
the use of the subscript is avoided and the type of a simplicial operation shou*
*ld
be clear from the context.
5. Attachment of Cells and Simplicial Resolutions
Let G be a bisimplicial group (an important example is chG, a constant hori
zontal extension of a simplicial group G). The stage ff + 1 in the small object
~
argument construction of a factorization {1}____ Gcof ____ G is obtained by
taking a pushout of a generator (for now, of type (1)),
F lvLi[n] = F lv(i[n] x i[n]) = F (i[n]+ ^ (i[n]+ ^ ~Sl)):
~ ~
{1}
?




?
F ( `[n]+ ^~( `[n]+ ^ ~Sl))_________________Gff
 ?
? 
 
 
 
 
 Gff+1 
 .. 
 ` .. 
 ... 
 ... 
 .. 
? R.. ?
F ( [n] `~l __________________
+ ^~( [n]+ ^ S )) G
Let ff = 0, (chG)0 = {1}. Then the top horizontal map is trivial and the
resulting first stage is free on wedges of "thickened" l spheres, on in horizo*
*ntal
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 13
degree n and one for each of [n]s horizontal degeneracies in higher degrees.
In case l = 0 the summands are homotopy equivalent to infinite cyclic groups.
The horizontal degeneracies of the pushout take summands to summands.
Now consider the cases of other generators. If it is Jl, then each vertical term
of the source is contractible, and vertical terms of the target free on wedges
of spheres. Adjoining the first type of generators is similiar to the diagram
above, and the last one does not involve suspensions and does not change the
homotopy type of vertical terms.
The sequential colimits of the small object argument preserve this structure.
Recall the definition of algebras in the category of simplicial groups , see *
*[1],
[9]. Let the small category have`as objects, for any nonnegatively graded
pointed set K , the free products i2K{*} F ~Si, where F is the pointed v*
*er
sion of the free group functor. algebras are functors opp 7! Gr Sets* whose
values on coproducts in are naturally isomorphic to products of their values
on free factors F ~Si. As usual the free functor and the forgetful functor b*
*e
tween the categories of algebras and the category Gr Sets* form an adjoint
pair.
The considerations above sum up in the following Lemma:
Lemma 5.1. The homotopy groups of the vertical terms of a cofibrant approx
imation Gcof of a bisimplicial group G, with simplicial structure induced by
the horizontal simplicial operations, form a free simplicial algebra.
6. Properness
Theorem 6.1. The E2 closed model category on Groups oo is right proper.
Proof. Let ' be a fibration and f an equivalence. We need to show that in the
diagram below F is an equivalence.
?~
G ___________X
 F 
 
pull back '
 
?? ?
~ ?
H ___________Y
f
AN E2 TYPE CLOSED MODEL CATEGORY FOR BISIMPLICIAL GROUPS 14
Since both the vertical loops and the diagonal functors commute with pullbacks
we obtain
~ l
lvG __________________ vX
lvF
 
 
lv pullback square  lv'
?? ??
~ l
lvH __________________ vY
lvf
The functor lv(_ ) preserves equivalences and fibrations, hence the top hori
zontal homomorphism is an equivalence of simplicial groups . This implies that
F , the pullback of f along ', induces isomorphisms of bigraded groups for
p 0.
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