Fibrewise nullification and the cube theorem
David Chataur and J'er^ome Scherer *
February 26, 2003
Abstract
In this paper we explain when it is possible to construct fibrewise loc*
*alizations
in model categories. For pointed spaces, the general idea is to decompose *
*the total
space of a fibration as a diagram over the category of simplices of the ba*
*se and
replace it by the localized diagram. This of course is not possible in an *
*arbitrary
category. We have thus to adapt another construction which heavily depend*
*s on
Mather's cube theorem. Working with model categories in which the cube the*
*orem
holds, we characterize completely those who admit a fibrewise nullificatio*
*n.
Introduction
Mather's cube theorem states that the top face of a cube of spaces whose bottom*
* face
is a homotopy pushout and all vertical faces are homotopy pullbacks is again *
*a homo
topy pushout ([Mat76 , Theorem 25]). This theorem is one of the very few occu*
*rences
of a situation where homotopy limits and colimits commute. It is actually rela*
*ted to a
theorem of Puppe about commuting fibers and pushouts ([Pup74 ]), and also to Q*
*uillen's
Theorem B in [Qui73 ]. Doeraene's work on Jcategories has incorporated the cub*
*e theo
rem as an axiom in pointed model categories and allowed him to study the L.S.c*
*ategory
in an abstract setting ([Doe93 ]). Roughly speaking a Jcategory is a model cat*
*egory in
which the cube theorem holds. Such a model category is very suitable for studyi*
*ng the
relationship between a localization functor (constructed by means of certain ho*
*motopy
colimits) and fibrations.
________________________________
*The first author was partially supported by DGESIC grant PB970202.
1
Recall that a localization functor in a model category M is any coaugmented *
*idempo
tent functor L : M ! M. The coaugmentation is a natural transformation j : Id !*
* L.
We will only deal with nullification functors PA. In this context the image of *
*PA is charac
terized by the property that map(A, PAX) ' *. We are looking for an existence t*
*heorem
of fibrewise nullification, i.e. a construction which associates to any fibrati*
*on F ! E ! B
another fibration together with a natural transformation
F _____//_E___//_B
''  
fflffl fflfflfflffl
PAF ____//_~E__//_B
where E ! ~Eis a PAequivalence. This is achieved by imposing the join axiom fo*
*r the
object A: We require the join X * A to be killed by PA, i.e. PA(X * A) ' *, f*
*or any
object X.
For pointed spaces, the most elegant construction of fibrewise localization *
*is due to
E. Dror Farjoun (in [DF96 , Theorem F.3]). His idea is to decompose the total s*
*pace of
a fibration as a diagram over the category of simplices of the base and replace*
* it by the
corresponding localized diagram. In certain particular settings, some authors u*
*sed other
constructions (P. May [May80 ], W. Dwyer, H. Miller, and J. Neisendorfer in [DM*
*N89 ] for
completions, C. Casacuberta and A. Descheemaker in [CD02 ] in the category of g*
*roups),
but none of these can be adapted in model categories. We prove the following:
Theorem 3.3 Let M be a model category which is pointed, left proper, cellular *
*and
in which the cube and the join axiom hold. Then the nullification functor PA a*
*dmits a
fibrewise version.
This condition is actually necessary and we characterize completely the mode*
*l cate
gories for which fibrewise nullifications exist. This is closely related to th*
*e property of
preserving products: A nullification functor PA preserves (finite) products if *
*PA(X xY ) '
PAX x PAY .
Theorem 3.5 Let M be a model category which is pointed, left proper, cellular a*
*nd in
which the cube axiom holds. Then the following conditions are equivalent:
(i)The nullification functor PA admits a fibrewise version.
(ii)The nullification functor PA preserves finite products.
2
(iii)The canonical projection X x A ! X is a PAequivalence for any X 2 M.
(iv)The join axiom for A is satisfied.
We show in the last part of the paper that the category of algebras over an *
*admissible
operad satisfies the cube axiom. Therefore the plusconstruction developed in [*
*CRS ] has
a fibrewise analogue. Let us only say that the plusconstruction performed on a*
* Oalgebra
B kills the maximal Operfect ideal in ß0B and preserves Quillen homology. As a*
* direct
consequence we get the following result which is classical for spaces.
Theorem 4.4 Let O  alg be the category of algebras over an admissible operad *
*O.
For any Oalgebra B, denote by B ! B+ the plus construction. The homotopy fib*
*er
AB = F ib(B ! B+ ) is then acyclic with respect to Quillen homology.
Acknowledgements. We would like to thank Gustavo Granja and Sophie Reinberg for
helpful comments.
1 The cube axiom
We work in a model category M which is pointed, i.e. the terminal object coinci*
*des with
the initial one and is denoted by *. In such a category the homotopy fiber F ib*
*(p) of a map
p : E ! B is defined as the homotopy pullback of the diagram * ! B E. We also
assume the category is left proper, meaning that the pushout of a weak equival*
*ence along
a cofibration is again a weak equivalence. Finally we require M to be cellular *
*as defined
in [Hir, Definition 14.1.1]. Basically the small object argument applies in a c*
*ellular model
category, as one has Icells which replace the usual spheres. There exists a ca*
*rdinal ~ such
that any morphism from an Icell to a telescope of length ~ ~ factorizes thro*
*ugh an
object of this telescope. Moreover every object has a cofibrant replacement by *
*an Icell
complex by [Hir, Theorem 13.3.7]. Localization functors exist in this setting,*
* see [Hir,
Theorem 4.1.1], but in general we do not know if it is possible to localize fib*
*rewise in any
(pointed, left proper, cellular) model category. We will thus work in model ca*
*tegories
satisfying an extracondition.
Definition 1.1 A model category M satisfies the cube axiom if for every commuta*
*tive
cubical diagram in M in which the bottom face is a homotopy pushout square and*
* all
vertical faces are homotopy pullback squares, then the top face is a homotopy *
*pushout
square as well.
3
M. Mather proved the cube Theorem for spaces in [Mat76 , Theorem 25] and J.*
*P. Do
eraene introduced it as an axiom for model categories. His paper [Doe93 ] conta*
*ins a very
useful appendix with several examples of model categories satisfying this rathe*
*r strong
axiom.
Example 1.2 Any stable model category satisfies the cube axiom. Indeed homot*
*opy
pushouts coincide with homotopy pullbacks, so that this axiom is a tautology.*
* On the
other hand the category of groups does not satisfy the cube axiom. Let us give *
*an easy
counterexample by considering the pushout of (Z * ! Z), which is a free gro*
*up on
two generators a and b. The pullback along the inclusion Z < ab >,! Z < a > *Z*
* < b >
is obviously not a pushout diagram. However fibrewise localizations exist in t*
*he category
of groups as shown by the recent work of Casacuberta and Descheemaker [CD02 ].
The following proposition claims that under very special circumstances the p*
*ushout
of the fibers coincides with the fiber of the pushouts. In the category of sp*
*aces this is
originally due to V. Puppe, see [Pup74 ]. The close link between the cube Theor*
*em and
Puppe's theorem was already wellknown to M. Mather and M. Walker, as can be se*
*en in
[MW80 ].
Proposition 1.3 Let M be a pointed model category in which the cube axiom hold*
*s.
Consider natural transformations between pushout diagrams:
F = hocolim F1 oo___F0 ____//_F2
j j1 j0 j2
fflffl fflffl fflffl fflffl
E = hocolim E1 oo___E0 ____//_E2
p p1 p0 p2
fflffl fflffl fflffl fflffl
B = hocolim B _____B_______B
Assume that Fi= F ib(pi) for any 0 i 2. Then F = F ib(p).
Proof. Denote by k : G ! E the homotopy fiber of p. We show that G and F are
weakly equivalent. Let us construct a cube by pullingback Ei! E along k. The b*
*ottom
face consists thus in the middle row of the above diagram and the top face cons*
*ists in the
homotopy pullbacks of Ei ! E G, which are the same as the homotopy pullbacks
of Ei ! B *, i.e. Fi. The cube axiom now states that the top face is a homo*
*topy
pushout and we are done.
4
This result will be the main tool in constructing fiberwise localization in *
*M. In his
paper [Doe93 ] on L.S.category, J.P. Doeraene used the cube axiom in a very s*
*imilar
fashion to study fiberwise joins. Indeed Ganea's characterization of the L.S.*
*category
uses iterated fibers of pushouts over a fixed base space. The same ideas have *
*also been
used in [DT95 ].
Lemma 1.4 Let M be a model category in which the cube axiom holds. Let D be*
* the
homotopy pushout in M of the diagram A B ! C. Then, for any object X 2 M,
X x D is the homotopy pushout of the diagram X x A X x B ! X x C.
Proof. It suffices to consider the cube obtained by pulling back the mentionned*
* pushout
square along the canonical projection X x D ! D.
2 The join
We check here that we can use all the classical facts about the join in any mod*
*el category
and introduce the join axiom. Most proofs here are not new, but probably folklo*
*re. Recall
p1 *
* p2
that the join A*B of two objects A, B 2 M is the homotopy pushout of A  AxB*
* !
B. First notice that the induced maps A ! A * B and B ! A * B are trivial. Inde*
*ed the
p1
map A ! A * B can be seen as the composite A i1!A x B ! A ! A * B which by
p2
definition coincides with the obviously trivial map A i1!A x B ! B ! A * B.
Lemma 2.1 For any objects A, B 2 M, we have A * B ' (A ^ B).
Proof. We use a "classical" Fubini argument (homotopy colimit commute with itse*
*lf, cf.
for example [CS02 , Theorem 24.9]). Let P be the homotopy pushout of A A _ B*
* !
A x B and consider first the commutative diagram
A oo___A _ B _______//BOO
  
  
  
A oo___A _ B _____//A x B
  
  
 fflffl 
A oo___A x B ______A x B
Its homotopy colimit can be computed in two different ways. By taking first ve*
*rtical
homotopy pushouts and next the resulting horizontal homotopy pushout one gets*
* A * B.
5
By taking first horizontal homotopy pushouts one gets the homotopy cofiber of *
*P ! A.
Consider finally the commutative diagram
*________*__________*OOOOOO
  
  
  
A oo___A _ B _____//A x B
  
  
 fflffl 
A oo___A x B ______A x B
The same process as above shows that Cof(P ! A) is homotopy equivalent to (A ^*
* B).
Lemma 2.2 For any objects A, B 2 M, we have A ^ B ' (A ^ B).
Proof. Apply again the Fubini commutation rule to the following diagram
*________*________*OOOOOO
  
  
  
B oo__A__ B ____//_B
  
  
 fflffl 
B oo__A_x B ____//_B
where one uses Lemma 1.4 to identify the pushout of the bottom line.
For a fibration F ! E!! B, the holonomy action is the map m : B xF ! F indu*
*ced
on the pullbacks by the natural transformation from B ! * F to P B!! B E.
Corollary 2.3 For any fibration F ! E!! B, the homotopy pushout of B B x
F m! F is weakly equivalent to B * F .
Proof. Copy the proof above to compare this homotopy pushout to ( B ^ F ).
When working with a nullification functor PA for some object A 2 M, we say t*
*hat X
is Aacyclic or killed by A if PAX ' *. By universality this is equivalent to m*
*ap(X, Z) ' *
for any Alocal object Z, or even better to the fact that any morphism X ! Z to*
* an
Alocal object is homotopically trivial.
Definition 2.4 A cellular model category M satisfies the join axiom for the nul*
*lification
functor PA if the join of A with any Icell is Aacyclic.
6
Example 2.5 Any stable model category satisfies trivially the join axiom, as *
*pushouts
coincide with pullbacks. In such a category the join is always trivial. The *
*category of
groups satisfies the join axiom for a similar reason (but we saw in Example 1.2*
* that the
cube axiom does not hold).
Proposition 2.6 Let M be a cellular model category in which the join axiom and*
* the
cube axiom hold. Then iA * Z is Aacyclic for any i 0 and any object Z.
Proof. The join is a homotopy colimit and thus commutes with other homotopy col*
*imits.
Since any object in M has a cofibrant approximation which can be constructed as*
* a
telescope by attaching Icells, the lemma will be proven if we show that iA * *
*Z is acyclic
for any Icell Z. By assumption we know that A * Z is acyclic and we conclude *
*by
Lemma 2.2 since iA * Z ' i(A * Z) is PAacyclic.
Remark 2.7 Given a family S of Icells, we say M satisfies the restricted joi*
*n axiom if
the join of A with any Icell in S is Aacyclic. One refines then the above pr*
*oposition
to cellular model categories in which the restricted join axiom holds. Here i*
*A * Z is
Aacyclic for any i 0 and any Scellular object Z, i.e. any object weakly equ*
*ivalent to
one which can be built by attaching only Icells in S.
3 Fibrewise nullification
Let A be any object in M. Recall that it is always possible to construct mappin*
*g spaces
up to homotopy in M eventhough we do not assume M is a simplicial model category
(see [CS02 ]). Thus we can define an object Z 2 M to be Alocal if there is a *
*weak
equivalences map(A, Z) ' *. A map g : X ! Y is a PAequivalence if it induces a*
* weak
equivalences on mapping spaces g* : map(Y, Z) ! map(X, Z) for any Alocal objec*
*t Z.
Hirschhorn shows that there exists a coaugmented functor PA : M ! M such that t*
*he
coaugmentation j : X ! PAX is a PAequivalence to an Alocal object. This funct*
*or is
called nullification or periodization.
The nullification X ! PAX can be constructed up to homotopy by imitating the
topological construction 2.8 in [Bou94 ]. One must iterate (possibly transfini*
*tely, for a
cardinal given by the smallness of any cofibrant object in M, see [Hir, Theorem*
* 14.4.4])
the process of gluing Acells, i.e. take the homotopy cofiber of a map iA ! X*
*. We
7
assume throughout this section that the model category M satisfies both the joi*
*n axiom
and the cube axiom.
Let us explain now how to adapt the fibrewise construction [DF96 , F.7] in a*
* model
category. The following lemma is the step we will iterate on and on so as to co*
*nstruct the
space ~E(in Theorem 3.3).
Proposition 3.1 Consider a commutative diagram
"''
F Ø___//PAF_____//F1
j  j2
fflfflØ fflffl"fflffl'
E _____//E0_____//E1
  
p p0 p1
fflfflfflfflfflfflfflfflfflffl
B _______B_______B
where the left column is a fibration sequence, the upper left square is a homot*
*opy pushout
square, p0: E0! B is the unique map extending p such that the composite PAF ! E*
*0! B
is trivial, p1 is a fibration, and F1 is the homotopy fiber of p1. Then the co*
*mposites
E ! E0! E1 and F ! PAF ! F1 are both PAequivalences.
Proof. We can assume that the map j : F ,! PAF is a cofibration as indicated in*
* the
diagram, so that E0is obtained as a pushout, not only a homotopy pushout. Sin*
*ce j is a
PAequivalence, so is its pushout along j by left properness (see [Hir, Propos*
*ition 3.5.4]).
To prove that F ! F1 is a PAequivalence, it suffices to analyze the map PAF ! *
*F1.
We use Puppe's Proposition 1.3 to compute F1 as homotopy pushout of the homoto*
*py
fibers of PAF F ! E over the fixed base B. This yields the diagram PAF x B
F x B ! F whose homotopy pushout is F1. We investigate more closely the map
F ! F1 by decomposing the map F ! PAF into several steps obtained by gluing Ac*
*ells.
f
Consider a cofibration of the form iA ! F ! Cf. Let Ef be the homotopy pu*
*sh
out of Cf F ! E and compute as above the homotopy fiber Ff of Ef ! B. It is
weakly equivalent to the homotopy pushout of Cf x B F x B ! F . Hence Ff is
also weakly equivalent to the homotopy pushout of B iA x B ! F , using the
definition of Cf. Decompose this pushout as follows
iA x B ______//_ iA______//_F
  
  
fflffl fflffl fflffl
B _______//_ iA * B___//_Ff
8
The righthand square must be a homotopy pushout square as well. But both iA *
*and
iA * B are Aacyclic (by Proposition 2.6), so that the map iA ! iA * B is a
PAequivalence. Thus so is F ! Ff by left properness. Iterating this process of*
* gluing
Acells shows that F ! F1 is a telescope of PAequivalences, hence a PAequival*
*ence.
Remark 3.2 In the category of spaces it is of course true that B x F ! B x *
*PAF is
a PAequivalence, because localization commutes with finite products. In genera*
*l we will
see in Theorem 3.5 that the join axiom is actually equivalent to the commutatio*
*n of PA
with products. With the restricted join axiom we would have to impose the addi*
*tional
restriction on B that B be Scellular.
Theorem 3.3 Let M be a model category which is pointed, left proper, cellular*
* and in
which the cube axiom and the join axiom hold. Let PA : M ! M be a nullification
functor. Then there exists a fibrewise nullification, i.e. a construction whi*
*ch associates
to any fibration F ! E ! B another fibration together with a natural transforma*
*tion
F _____//_E___//_B
''  
fflffl fflfflfflffl
PAF ____//_~E__//_B
where E ! ~Eis a PAequivalence.
Proof. We construct first by the method provided in Lemma 3.1 a natural transf*
*or
mation to the fibration F1 ! E1 ! B. We iterate then this construction and get
a fibration F~ ! ~E! B where F~ = hocolim(F ! F1 ! F2 ! . .).and E~ =
hocolim(E ! E1 ! E2 ! . .).. All maps in these telescopes are PAequivalences *
*by
the lemma, hence so are E ! ~Eand F ! ~F. Moreover any map Fn ! Fn+1 factorizes
as Fn ! PAFn ' PAF ! Fn+1 so that ~F' PAF . We obtain thus the desired fibration
PAF ! ~E! B.
Define ~PAX = F ib(X ! PAX), the fiber of the nullification. As in the case *
*of spaces
we get:
Corollary 3.4 For any object X in M we have PAP~AX ' *.
Proof. Apply the fiberwise localization to the fibration ~PAX ! X ! PAX. This y*
*ields
a fibration PAP~AX ! ~X! PAX in which the base and the fiber are Alocal. There*
*fore
X~ is Alocal as well. But then X~' PAX and so PAP~AX ' *.
9
We end this section with a complete characterization of the model categories*
* which
admit fibrewise nullifications.
Theorem 3.5 Let M be a model category which is pointed, left proper, cellular*
* and in
which the cube axiom holds. Then the following conditions are equivalent:
(i)The nullification functor PA admits a fibrewise version.
(ii)The nullification functor PA preserves finite products.
(iii)The canonical projection X x A ! X is a PAequivalence for any X 2 M.
(iv)The join axiom for A is satisfied.
Proof. We prove first that (i) implies (ii). Consider the trivial fibration X !*
* XxY ! Y
and apply the fibrewise nullification to get a new fibration PAX ! E ! Y . The *
*inclusion
of the fiber admits a retraction E ! PAX, i.e. E ' PAX x Y . Applying once agai*
*n the
fibrewise nullification to Y ! Y xPAX ! PAX, we see that the map XxY ! PAXxPAY
is a PAequivalence. As a product of local objects is local, this means precis*
*ely that
PA(X x Y ) ' PAX x PAY .
Property (iii) is a particular case of (ii). We show now that (iii) implies *
*(iv). If the
canonical projection X x A ! X is a PAequivalence, the pushout of it along th*
*e other
projection yields another PAequivalence, namely A ! X * A. Therefore the join *
*X * A
is PAacyclic. Finally (iv) implies (i) as shown in Theorem 3.3.
The construction we propose for fibrewise nullification does not translate t*
*o the setting
of general localization functors. We do not know if the cube and join axioms ar*
*e sufficient
conditions for the existence of fibrewise localizations.
4 Algebras over an operad
In this section we provide the motivating example for which this theory has bee*
*n devel
opped. For a fixed field k, we work with Zgraded differential kvector spaces*
* (kdgm)
and consider the category of algebras in kdgm over an admissible operad. This *
*is indeed
a pointed, left proper and cellular category. Weak equivalences are quasiisom*
*orphisms
and fibrations are epimorphisms.
10
We do not know if the join axiom holds in full generality for any object A. *
* It does
so however when A is acyclic with respect to Quillen homology, which is the cas*
*e we
are most interested in, or when A is a free algebra. We check that the cube ax*
*iom
always holds, following the strategy of [Doe93 , Proposition A.15], which guara*
*ntees the
existence of fibrewise versions of the plusconstruction and Postnikov sections*
*. In the case
of Ngraded Oalgebras (the case O = As is treated by Doeraene) one has to rest*
*rict to a
particular set of fibrations (the socalled Jmaps), because they must be surje*
*ctive in each
degree in order to compute pullbacks. In our context all fibrations are epimor*
*phisms, so
that the cube axiom holds in full generality.
Theorem 4.1 The cube axiom holds in the category of Oalgebras.
Proof. Let us briefly recall the key steps in Doeraene's strategy. We consider *
*a pushout
`
square of Oalgebras (along a generic cofibration B ,! B O(V )):
" `
B Ø___//_C = B O(V )

 
 
fflfflØ " fflffl`
A ___//_D = A O(V )
We need to compute the pullback of this square along a fibration p : E!! D (wh*
*ich is
hence an epimorphism of chain complexes). We have thus the following isomorphis*
*m of
chain complexes:
a
E ~=A O(V ) ker(p).
This allows to compute the successive pullbacks A xD E, C xD E, and B xD E. In*
* order
to construct the homotopy pushout P of these pullbacks (which must coincide w*
*ith E)
we factorize the morphism B xD E ! C xD E as
a ~
B xD E ,! (B kerp) O(V W ) i C xD E
`
Thus P is identified with (A ker(p)) O(V W ), which allows us to build fi*
*nally a
quasiisomorphism to E.
The plusconstruction for an Oalgebra is a nullification with respect to a *
*universal
acyclic algebra U. We refer to [CRS ] for an explicit construction and nice app*
*lications.
Proposition 4.2 The join axiom holds for any acyclic Oalgebra A. It holds in*
* par
ticular for the universal acyclic algebra U constructed in [CRS ], so that the *
*fibrewise
plusconstruction exists.
11
Proof. The join A * X is weakly equivalent to A ^ X by Lemmas 2.1 and 2.2. Sin*
*ce
A is 0connected and acyclic, it is trivial by the Hurewicz Theorem [CRS , The*
*orem 1.1].
Thus A * X ' * is always PAacyclic.
We consider next the case of Postnikov sections PO(x), where x is a generato*
*r of
arbitrary degree n 2 Z. Because [O(x), X] ~=ßnX for any Oalgebra X, the nullif*
*ication
functor PO(x)is really a Postnikov section, i.e. PO(x)X ' X[n  1]. Let us also*
* recall that
ßn(X x Y ) ~=ßnX x ßnY .
Proposition 4.3 The join axiom holds for any free Oalgebra O(x) on one genera*
*tor of
degree n 2 Z. Therefore fibrewise Postnikov sections exist.
Proof. By Theorem 3.5 we might as well check that the map X x O(x) ! X is a
PO(x)equivalence for any Oalgebra X. Clearly the nth Postnikov section of th*
*e product
X x O(x) is equivalent to X[n  1] and we are done.
Our final result is a particular case of Corollary 3.4. A direct proof (with*
*out fibrewise
techniques) seems out of reach.
Theorem 4.4 Let O  alg be the category of algebras over an admissible operad*
* O.
For any Oalgebra B, denote by B ! B+ the plus construction. The homotopy fib*
*er
AB = F ib(B ! B+ ) is then acyclic with respect to Quillen homology.
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[Pup74] Volker Puppe. A remark on öh motopy fibrations". Manuscripta Math., 1*
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[Qui73] Daniel Quillen. Higher algebraic Ktheory. I. In Algebraic Ktheory, *
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85147. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973.
David Chataur
Centre de Recerca Matem`atica, E08193 Bellaterra
email: chataur@crm.es
J'er^ome Scherer
Departament de Matem`atiques, Universitat Aut'onoma de Barcelona, E08193 Bella*
*terra
email: jscherer@mat.uab.es
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