COMPARING HOMOTOPY CATEGORIES
DAVID BLANC
Abstract. Given a suitable functor T : C ! D between model categories*
*, we
define a long exact sequence relating the homotopy groups of any X 2 C *
* with
those of T X, and use this to describe an obstruction theory for lifti*
*ng an object
G 2 D to C. Examples include finding spaces with given homology or homo*
*topy
groups.
0. Introduction
A number of fundamental problems in algebraic topology can be described as m*
*ea
suring the extent to which a given functor T : C ! D between model categories
induces an equivalence of homotopy categories: more specifically, which objects*
* (or
maps) from D are in the image of T , and in how many different ways. For exampl*
*e:
a) How does one distinguish between different topological spaces with the *
*same
homology groups, or with chainhomotopy equivalent chain complexes? How
can one realize a given map of chain complexes up to homotopy?
b) When do two simplyconnected topological spaces have the same rational *
*ho
motopy type?
c) When is a given topological space a suspension, up to homotopy? Dually,
how many distinct loop space structures, if any, can a given topologica*
*l space
carry?
d) Is a given algebra (that is, a graded group with an action of the pri*
*mary ho
motopy operations) realizable as the homotopy groups of a topological s*
*pace,
and if so, in how many ways?
Our goal is to describe a unified approach to such problems that works for f*
*unctors
between spherical model categories, for which several familiar concepts and con*
*struc
tions are available. These include a set A of models (to play the role of spher*
*es, in
particular determining the corresponding homotopy groups ssC*), Postnikov sys*
*tems,
and kinvariants. If a functor T : C ! D respects this additional structure*
*, we
obtain a natural long exact sequence of the form:
(0.1) . .!. nX s!ssCnX h!ssDnT X @! n1X . .,.
which generalizes the EHP sequence, J.H.C. Whitehead's "certain exact sequence*
*",
and the spiral exact sequence of Dwyer, Kan, and Stover. See (4.4) below.
Under these hypotheses, given an object G in D, we want to find an object X *
*in C
with T X ' G. The key step is to choose ssC*X which fits into (0.1). We d*
*escribe
_____________
Date: June 5, 2006.
1991 Mathematics Subject Classification. Primary: 55U35; Secondary: 55P65, 55*
*Q35, 18G55.
Key words and phrases. model category, homotopical algebra, obstruction theor*
*y, realization.
1
2 DAVID BLANC
an inductive procedure for doing this, using the Postnikov systems in both cate*
*gories,
together with an obstruction theory for lifting G to C, along the following lin*
*es:
Theorem. Given T : C ! D and G 2 D as above, for each X 2 C with
T X ' G, there is a tower of fibrations in C:
p(n+1) p(n) p(n1) p(0)
. ..! ^X ! ^X ! . . .!X^<0> ,
called the modified Postikov tower for X (Def. 5.21), with G mapping compatibly*
* to
T ^X for each n, and X ' holimn ^X.
Conversely, given such a tower up to level n, the obstruction to extending i*
*t to level
n + 1 lies in Hn+3(G; n+1X^), and the choices for ^X are classi*
*fied by:
o a class in Hn+2(G; n+1X^);
o a class in Hn+2(X^; Kn+1 ), where Kn+1 := Coker ssn+2ae(n), for *
* ae(n):
Pn+2G ! Pn+2T ^X.
See Theorem 6.8.
0.2. Related work. The comparison problems discussed above are familiar ones in
algebraic topology:
a) The question of the realizability of a graded algebra as a cohomology r*
*ing was
first raised explicitly by Steenrod in [Ste], but it goes back to Hopf *
*(in [Ho ])
in the rational case. The "Steenrod problem" of realizing a given ss1*
*action
in homology has been studied, for example, in [T , Sm ].
b) The comparison between integral and rational homotopy type was implicit*
* in
the notion of a Serre class (cf. [Se, AC ]), although an explicit formu*
*lation was
only possible after the construction of the rationalization functors of*
* Quillen
and Sullivan in [Q2 , Sul].
c) Possible loop space structures on a given Hspace were analyzed extensi*
*vely,
starting with the work of Sugawara and Stasheff (cf. [Sug , Sta]). The*
* dual
question on identifying suspensions has also been studied (see, e.g., [*
*BH ]).
d) The question of the realizability of homotopy groups goes back to J.H.C*
*. White
head, in [W2 ] (see also [W5 ]), and has reappeared in recent years i*
*n the context
of algebras (cf. [DKS1 , DKS2 ]). The relationship between homolog*
*y and
homotopy groups, which is relevant to the realization problem for both,*
* was
studied in [W3 , W4 ] (in which the "certain exact sequence" was intro*
*duced).
In [Ba4 ], H.J. Baues gave what appears to be the first general theory cove*
*ring
a wide spectrum of such realization problems. This was an outgrowth of his earl*
*ier
work on classifying homotopy types of finite dimensional CW complexes in [Ba2 ,*
* Ba3 ]
(which in turn builds on [W1 ]).
His initial setting consists of a homological cofibration category C (corres*
*ponding
to, and extending, the notion of a resolution model category) under a theory of
coactions T (corresponding to the category A of x1.2). Baues then construc*
*ts
a generalized "certain exact sequence" similar to (0.1), and provides an indu*
*ctive
obstruction theory for realizing a chain complex (or a chain map) by a Tcomplex
(corresponding to a CW complex, or more generally a cofibrant object in C)  *
*see
[Ba4 , VI, (2.22.3)]).
These results apply inter alia to the problem of realizing a chain complex b*
*y a
topological space (the motivating example for Baues's approach), as well as to *
*the
COMPARING HOMOTOPY CATEGORIES 3
realization of a algebra (cf. [Ba4 , D, (7.9)]). However, here we consider f*
*unctors
between two different model categories that are not covered by [Ba4 ]. In parti*
*cular,
our original motivating example  the realization of a simplicial algebra (*
*by a
simplicial space)  shows that in the relative context a more refined obstruc*
*tion
theory may be necessary: compare Theorem (2.3) of [Ba4 , VI] with Theorem 6.8
below.
0.3. Remark. Another set of closely related questions  which do not quite fi*
*t into
the framework described here, though they can also be stated as realization pro*
*blems
 arise in categories of structured ring spectra; see for example [R ] and [GH*
* , Cor.
5.9].
0.4. Notation and conventions. T* denotes the category of pointed connected
topological spaces; Set* that of pointed sets, and Gp that of groups. For*
* any
category C, grC denotes the category of nonnegatively graded objects over C,*
* and
sC the category of simplicial objects over C. sSet is denoted by S, sSet* *
*by S*,
and sGp by G. The constant simplicial object an an object X 2 C is writt*
*en
c(X) 2 sC.
If C has all`coproducts, then given A 2 S and X 2 C, we define X ^A 2 sC
by (X ^A)n := a2AnX, with face and degeneracy maps`induced from those of A.
For Y 2 sC, define Y A 2 sC by (Y A)n := a2An Yn (the diagonal of t*
*he
bisimplicial object Y ^A)  so that for X 2 C we have X ^A = c(X) A.
The category of chain complexes of Rmodules is denoted by Chain R (or simp*
*ly
Chain , for R = Z).
0.5. Organization: In Section 1 we define spherical model categories, having the
additional structure mentioned above. Most examples of such categories are in p*
*ar
ticular resolution model categories, which are described in Section 2; we expla*
*in how
to produce the needed structure for them in Section 3. We define spherical func*
*tors
between such categories, and construct the comparison exact sequence for them, *
*in
Section 4. This is applied in Section 5 to study the effect of a spherical fun*
*ctor on
Postnikov systems. Finally, in Section 6 we construct an obstruction theory as *
*above
for the fiber of a spherical functor. In Section 7 we indicate how the theory w*
*orks for
the above examples.
0.6. Acknowledgements. I would like to thank Paul Goerss for many hours of
discussion on various issues connected with this paper, and especially for his *
*essential
help with Sections 56, the technical core of this note. I would also like to t*
*hank Hans
Baues for explaining the relevance of his work in [Ba4 ] to me.
1.Spherical model categories
Before defining the additional structure we shall need, we briefly recapitul*
*ate the
relevant homotopical algebra:
1.1. Model categories. Recall that a model category is a bicomplete category C
equipped with three classes of maps: weak equivalences, fibrations, and cofibra*
*tions,
related by appropriate lifting properties. By inverting the weak equivalences w*
*e obtain
the associated homotopy category hoC, with morphism set [X, Y ] = [X, Y ]C. *
* We
shall concentrate on pointed model categories (with null object *). See [Q1 ] *
*or [Hi].
4 DAVID BLANC
1.2. The set of models. The additional initial data that we shall require for o*
*ur
model category consists of a set A of cofibrant homotopy cogroup objects in C, *
*called
models (playing the role of the spheres in T*). Given such a set A, let A den*
*ote the
smallest subcategory of C containing A and closed under weak equivalences, arbi*
*trary
coproducts, and suspensions. Note that every object in A is a homotopy cogro*
*up
object, too.
1.3. Example. Let C = G be the category of simplicial groups, Sk = [k]=@ [k]
the standard simplicial ksphere in S*, G : S* ! G the Kan's loop functor (cf*
*. [May ,
x26.3]), and F : S* ! G the free group functor. For each n 1, Sn := GSn 2*
* G ~=
F Sn1 will be called the ndimensional Gsphere, with kSn ' Sn+k. These, *
*and
their coproducts, are cofibrant strict cogroup objects for G. Here A := {S1 =*
* c(Z)};
in fact, throughout this paper A will be either a singleton, or countable.
1.4. Remark. The adjoint pairs of functors:
S G
T* kk S* ~ G
W
induce equivalences of the corresponding homotopy categories  where ~W : G *
*! S*
is the EilenbergMac Lane classifying space functor, S : T* ! S* is the singu*
*lar set
functor, and k  k : S* ! T* is the geometric realization functor (cf. [May , *
*x14,23]).
Thus to study the usual homotopy category of (pointed connected) topological sp*
*aces,
we can work in G (or S*), rather than T*.
1.5. Definition. If A is a set of models for C, then given X 2 C, for each A*
* 2 A
let ssCA,k(X) := [ kA, X0]C, where X0 ! X is a (functorial) fibrant replace*
*ment.
We write ssCkX for (ssCA,kX)A2A , and ssC*X := (ssCkX)1k=0.
1.6. Theories and algebras. Recall that a theory is a small category with fin*
*ite
products (so in particular, an FPsketch  cf. [Bor , x5.6]), and a algebra (*
*or model)
is a productpreserving functor ! Set. Think of as encoding the operations
and relations for a "variety of universal algebras", the category Alg of *
*algebras
(which is sketched by ).
For example, the obvious category G, which sketches groups, is equivalent to*
* the
opposite of the homotopy category of (finite) wedges`of circles. An Gtheory *
*(cf.
[BP , x2]) is one equipped with a map of theories S G ! (coproduct taken *
*in
the category of theories, over some index set S) which is bijective on objects.*
* This
implies that each algebra has the underlying structure of an Sgraded group, *
*so
that Alg can be thought of as a "variety of (graded) groups with operators"*
* (cf.
[Ba4 , I, (2.5)]).
1.7. Remark. We will assume that all the functors ssCn(n 0) take value in a c*
*ategory
C Alg sketched by a Gtheory , and thus equipped with a faithful forgetful *
*functor
UC : C Alg ! GpA into the category of Agraded groups. The objects of CAlg
are called Calgebras.
For topological spaces, with A = {S1}, the Calgebras are simply groups. *
*If we
use rational spheres as the models, then CAlg is the category of Qvector s*
*paces.
A more interesting example appears in x2.9 below.
1.8. Constructions based on models. There are a number of familiar construc
tions for topological spaces which we require for our purposes. We can define t*
*hem
once we are given a set of models A as above, although they do not always exist*
* (see
x3.10 below).
COMPARING HOMOTOPY CATEGORIES 5
1.9. Definition. A Postnikov tower (with respect to A) is a functor that assign*
*s to
each Y 2 C a tower of fibrations:
p(n) A p(n1) A
. .!.PnAY  ! Pn1Y  ! . .!.P0 Y ,
*
* (n)
as well as a weak equivalence r : Y ! P1AY := limnPnAY and fibrations P1AY *
*r!
PnAY such that r(n1)= p(n)O r(n) for all n. Finally, (r(n)O r)# : ssCkY ! *
*ssCk(PnAY )
is an isomorphism for k n, and ssCk(PnAY ) is zero for k > n.
When A is clear from the context, we denote PnA simply by Pn.
1.10. Example. For a free chain complex C* 2 Chain R of modules over a ring R, *
*we
may take C0*:= PnC* where C0i= Ci for i n + 1, C0n+2= Zn+1C*), and C0i=*
* 0
for i n+3. The map r(n): C* ! C0*is defined by r(n)n+2:= @n+2 : Cn+2 ! Zn+1C*
**.
1.11. Definition. Given an Calgebra , a classifying object BC (or simply
B ) for is any B 2 sC such that B ' P0K and ssC0B ~= .
The name is used by analogy with the classifying space of a group, which cla*
*ssifies
Gbundles. One can interpret BC similarly, though perhaps less naturally (s*
*ee,
e.g., [BJT , x4.6]).
1.12. Definition. A module over a Calgebra is an abelian group object in
C Alg = (cf. [Q3 , x2]), and the category of such is denoted by Mod .
1.13. Remark. Since any Calgebra is in particular a (graded) group, if p : Y*
* !
is a module, then Y = K x (as sets!) for K := Ker (p), with an appropriat*
*e C
algebra structure (cf. [Bl3, x3]). We may call K itself a module (which corr*
*esponds
to the traditional description of an Rmodule, for a ring R).
1.14. Example. For any object X 2 C as above, the A x Ngraded group ssC*X
has an action of the Aprimary homotopy operations, corepresented by the maps in
ho A (see x2.9 below). In particular, one of these operations, corresponding *
*to the
action of the fundamental group on the higher homotopy groups, makes each ssCnX
(n 1) into a module over ssC0X (see Fact 3.6 below).
1.15. Definition. Given an abelian Calgebra M and an integer n 1, an n
dimensional MEilenbergMac Lane object EC(M, n) (or simply E(M, n)) is any
E 2 sC such that ssCnE ~=M and ssCkE = 0 for k 6= n.
1.16. Definition. Given a Calgebra , a module M over , and an integer n
1, an ndimensional extended MEilenbergMac Lane object EC (M, n) (or simply
E (M, n)) is any homotopy abelian group object E 2 sC= , equipped with a
section s for p(0): E ! P0E ' B , such that ssCnE ~=M as modules over ; a*
*nd
ssCkE = 0 for k 6= 0, n.
1.17. Definition. Given a Postnikov tower functor as in x1.9, an nth kinvaria*
*nt
square (with respect to A) is a functor that assigns to each Y 2 C a homotopy
pullback square:
p(n+1)
(1.18) PnA+1Y_________//PnAY
 _PB_ k
  n
fflffl fflffl
B ______//E (M, n + 2)
6 DAVID BLANC
for := ssC0Y and M := ssCn+1Y , where p(n+1): Pn+1Y ! PnY is the g*
*iven
fibration of the Postnikov tower.
The map kn : PnY ! E (M, n + 2) is the nth (functorial) kinvariant for *
*Y .
1.19. Example. If C* is a chain complex of Rmodules, and PnC* = C0*as in x1.10,
we may take E(Hn+1C* , n + 2) = E*, where Ei= 0 for i < n+2, En+2 = Zn+1C*,
and En+3 = Bn+1C*. Then kn : C0*! E* is defined by Id: C0n+1! En+1.
Of course, if R is a principle ideal domain (or a hereditary ring), such as *
*Z, then
the kinvariants for C* are trivial, since in that case any two free (or proj*
*ective)
chain complexes with the same homology are homotopy equivalent, by [D , Prop. 3*
*.5].
But this need not hold for an arbitrary ring R.
1.20. Spherical models. A set of objects A := {A}A2A in a model category C is
called a collection of spherical models if the following axioms hold:
Ax 1. Each nA (A 2 A, n 2 N) is a cofibrant homotopy cogroup object in C.
Ax 2. For any X 2 C and n 1, ssCnX has a natural structure of a module*
* over
ssC0X.
Ax 3. A map f : X ! Y is a weak equivalence if and only if ssCA,nf is a w*
*eak
equivalence for each A 2 A and n 2 N.
Ax 4. C has Postnikov towers with respect to A.
Ax 5. For every Calgebra and module M over , the classifying object B
and extended MEilenbergMac Lane object E (M, n) exist (and are unique
up to homotopy) for each n 1.
Ax 6. C has kinvariant squares with respect to A for each n 0.
If each model kA (A 2 A, k 2 N) is a cofibrant strict cogroup object *
* which
implies that every object in A is such, up to weak equivalence  we call A*
* a
collection of strict spherical models.
A pointed simplicial model category C equipped with a collection A := {A}A*
*2A
of spherical models is called a spherical model category, and we denote it by *
* .
Such a category is stratified in the sense of Spali'nski (cf. [Sp ]).
1.21. Example. The category S* of pointed simplicial sets, as well as the cat*
*egory
T* of pointed connected topological spaces, have spherical model category struc*
*tures
with A = {S1}. (Functorial kinvariants in these categories are provided by *
*the
construction of [BDG , x5]; in both cases CAlg Gp). Similarly for the c*
*ategory
Chain R of chain complexes over R, with the constructions indicated in x1.10 *
*and
x1.19.
In the examples we have in mind, our model categories enjoy additional useful
properties, which we can summarize in the following:
1.22. Definition. A spherical model category as above is called strict*
* if the
following axioms hold:
Ax 1. C is a pointed rightproper cofibrantly generated simplicial model cat*
*egory
(cf. [Hi, 11.1, 13.1]), in which every object is fibrant.
Ax 2. C is equipped with a faithful forgetful functor U^ : C ! D, with left a*
*djoint ^F
 where D is one of the "categories of groups" D = Gp, grGp , G, *
*RMod ,
or sRMod , for some ring R.
COMPARING HOMOTOPY CATEGORIES 7
Ax 3. The adjoint pair (U^, ^F) create the model category structure on C in t*
*he sense
of [Bl1, x4.13]  so in particular ^Ucreates all limits in C.
Ax 4. A is a collection of strict spherical models, each of which lies in the*
* image of
the composite ^FO F 0: S ! C, where F 0: S ! D is adjoint to the fo*
*rgetful
functor U0 : D ! S, with the group structure on Hom C(A, X) induced fr*
*om
that of ^U(X).
2. Resolution model categories
Many examples of spherical model categories fit into the framework originall*
*y con
ceived by Dwyer, Kan and Stover in [DKS2 ] under the name of "E2 model categor*
*ies,"
and later generalized by Bousfield (see [Bou , J]. A slightly different general*
*ization is
given by Baues in [Ba4 , Ch. D, x2] under the name of spiral model categories.
First, some preliminary concepts:
2.1. Definition. The nth matching object for a simplicial object X over C is d*
*efined
by
MnX = {(x0, . .,.xn) 2 (Xn1)n+1  dixj = dj1xi for all0 i < j n}
(see [BK , X,x4.5]). Note that each face map dk : Xn ! Xn1 factors through t*
*he
obvious map ffin : Xn ! MnX.
2.2. Definition.`The nth latching object of a simplicial object X over C is de*
*fined
LnX := 0 i n1Xn1= ~, where for any x 2 Xnk1 and 0 i j n  1 we
set sj1sj2. .s.jkx in the ith copy of Xn1 equivalent to si1si2. .s.ikx *
*in the jth
copy of Xn1 whenever the simplicial identity sisj1sj2. .s.jk= sjsi1si2. .s.*
*ikholds.
The map oen : LnX ! Xn is defined oen(x)i= six, where (x)i2 (Xn1)i.
There are two canonical ways to extend a given model category structure on ^*
*Cto
C := sC^:
2.3. The Reedy model structure. This is defined by letting a simplicial map
f : X ! Y in C := sC^ be:
(i)a weak equivalence if fn : Xn ! Yn is a weak equivalence in C^ for ea*
*ch
n 0;
(ii)a (trivial) cofibration if fn q oen : Xn qLnX LnY ! Yn is a (trivial) *
*cofibration
in ^C for each n 0;
(iii)a (trivial) fibration if fn x ffin : Xn ! Yn xMnY MnX is a (trivial)*
* fibration
in ^C for each n 0.
See [Hi, 15.3].
2.4. The resolution model category. Let C^be a pointed cofibrantly generated
right proper model category (in our cases, every object will be fibrant, though*
* this
is not needed in general  cf. [J]). Given a set A^ of models for C^ (x1.2),*
* we let
A := {c( kA^)}k2N,A^2A^(the constant simplicial objects on kA^ 2 ^C) be the *
*set of
models for C. Note that nc( kA^) := c( kA^) ^Sn (x1.2), so we shall gener*
*ally
reserve the notation k for (internal) suspension in ^C, and  Sn for the (simp*
*licial)
suspension in C = sC^.
2.5. Remark. If we do not assume that each ^A2 A^ is a homotopy cogroup object
in ^C, we take A := {( kA^) S1}A2A as our collection of models for C.
8 DAVID BLANC
2.6. Definition. A map f : V ! Y in C = sC^ is called homotopically A^free *
*if for
each n 0, there is
a) a cofibrant object Wn in A^ ^C, and
b) a map 'n : Wn ! Yn in C inducing a trivial cofibration (Vn qLnV LnY *
*) q
Wn ! Yn.
We define the resolution model category structure on sC^ determined by A^, *
*by
letting a simplicial map f : X ! Y be:
(i)a weak equivalence if ssCA,nf is a weak equivalence of simplicial gro*
*ups for
each A 2 A and n 0.
(ii)a cofibration if it is a retract of a homotopically A^free map;
(iii)a fibration if it is a Reedy fibration (x2.3(iii)) and ssCA,nf is a f*
*ibration of
simplicial groups for each A 2 A and n 0
2.7. Definition. Given a fibrant X 2 sC^, define its ncycles object ZnX to*
* be
{x 2 Xn  dix = 0 fori = 0, . .,.n} (the fiber of ffin : Xn ! MnX of x2.1). S*
*imilarly,
the nchains object for X is CnX = {x 2 Xn  dix = 0 fori = 1, . .,.n}.
If X is fibrant, the map d0 = dn0:= d0CnX : CnX ! Zn1X fits into a fibra*
*tion
sequence:
jXn+1 dn+10
(2.8) . . .ZnX ! Zn+1X ! Cn+1X ! ZnX
(see [DKS2 , Prop. 5.7]).
2.9. Definition. A A algebra is a productpreserving functor from (ho A )o*
*p to
sets. The category of A algebras is denoted by A Alg.
Equivalently, we can think of an A algebra as an NxAgraded group equipp*
*ed
with an action of the Aprimary homotopy operations (corepresented by the maps *
*in
ho A ).
Thus we can think of the functor ssC* as taking value in A Alg . This expl*
*ains the
additional Calgebra structure on the Agraded groups ssCnX, mentioned in x1*
*.7:
when C = sC^, we have CAlg := A^Alg.
2.10. Example. When C = G, and A = {S1}  so A is the category of
wedges of Gspheres (x1.3)  then (up to indexing) A Alg is the usual cat*
*egory
of algebras (see [Sto, x2]): graded groups equipped with an action of the pr*
*imary
homotopy operations (Whitehead products and compositions).
2.11. Examples of resolution model categories. In this paper we shall be inter
ested mainly in the following instances of resolution model categories:
(a) Let C^= Gp with the trivial model category structure: i.e., only isomo*
*rphisms
are weak equivalences, and every map is both a fibration and a cofibrat*
*ion.
Let A^ = {Z} consist of the free cyclic group (whose coproducts are t*
*he
cogroup objects in Gp). The resulting resolution model category struc*
*ture
on G := sC^ is the usual one (cf. [Q1 , II,x3]. Here CAlg Gp  *
* there is
no extra structure on the individual homotopy groups of a simplicial gr*
*oup.
Note that if we tried to do the same for ^C= Set, there are no nont*
*rivial
cogroup objects, while in S not all objects are fibrant. Note also t*
*hat the
category T* of pointed topological spaces, which is one of the main ex*
*amples
we have in mind, has a spherical model category structure which is not *
*strict
(x1.22). This explains the significance of Remark 1.4 in our context.
COMPARING HOMOTOPY CATEGORIES 9
(b) The previous example extends to any category ^Cof (possibly graded) uni*
*versal
algebras with an underlying group structure  such as rings, Rmodule*
*s,
associative algebras, Lie algebras, and so on  so that C is corepres*
*ented by
a Gtheory , in the language of [BP , x4]. Here A consists of free mon*
*ogenic
algebras (one for each isomorphism class), and thus once more CAlg *
* C.
(c) We can iterate the process by taking G for ^C, and letting A^ := {Sn}1n*
*=1(x1.3).
We thus obtain a resolution model category structure on sG (or equival*
*ently,
on the category of simplicial spaces).
In this case the homotopy groups sssGk,nX, denoted briefly by ss"n*
*X, are
the "bigraded groups" of [DKS2 ], and Proposition 5.8 there shows that*
*, for a
fibrant simplicial space X 2 sG, we have ssCA,nX ~=ss0 map (A Sn, *
*X).
(d) If C is a resolution model category and I is some small category, the c*
*ategory
CI of Idiagrams in C also has a resolution model category structure, i*
*n which
the models consists of`all free Idiagrams F [A, i] for i 2 Obj I an*
*d A 2 A,
where F [A, i](j) := HomI(i,j)A. See [BJT , x1]).
2.12. Remark. In all these examples, if Y 2 C = sC^, is fibrant, then for eac*
*h n 0
we have an exact sequence:
C^ (dn+10)# C^ ^#n C
(2.13) ss*Cn+1X ! ss*ZnX ! ssnY ! 0.
3. Constructions in resolution model categories
Not all spherical model categories are resolution model categories (see x1.2*
*1), but
all known examples appear to be Quillen equivalent to such. Conversely, the exa*
*mples
of resolution model categories we are interested in are spherical *
*(though
this does not hold in general  see x3.10 below). We briefly indicate why thi*
*s is so.
3.1. Postnikov sections. Given Y 2 sC^, for each n 0 define Y (n)2 sC^ by
setting Yk(n):= Yk for k n + 1 and Yk(n):= Mk(Y (n)) (x2.1) for k n *
*+ 2.
Note that for any X 2 sC, MkX depends only on X through dimension (k  1),
so this definition is valid inductively. Denote the obvious maps by r(n): Y ! *
*Y (n)
and p(n): Y (n+1)! Y (n)(see [DK2 , x1.2]).
Now for any X 2 sC^, choose a functorial fibrant replacement Y , and set *
*PnX :=
Y (n), with '(n): X ! PnX defined to be the composite of r(n) with the triv*
*ial
cofibration i : X ! Y , and p(n): Pn+1X ! PnX defined as above.
3.2. Remark. The functor (n): C ! C is right adjoint to the (n + 1)skeleton
functor skn+1, so PnX depends only on skn+1X, even if X is not fibrant. I*
*f X is
fibrant, we can find Y ' PnX with skn+1Y = skn+1X.
3.3. Fact. In each of the examples of x2.11(ad), the tower:
p(n)
X ! . .!.Pn+1X ! PnX ! . .!.P0X
is a functorial Postnikov tower for C = sC^ with respect to A (x1.9).
Proof. From x2.3 and x3.1 it follows that if Y 2 sC^ is fibrant, then so is e*
*ach Y (n),
and for each n, Y (n+1)! Y (n) is a fibration, ZkY (n)= 0 and CkY (n)d0!Zk*
*1Y
is an isomorphism for k n + 2. The claim then follows from (2.13).
10 DAVID BLANC
3.4. Fact. In each of the examples of x2.11(ad), there is a classifying objec*
*t B
for any A algebra , and it is unique up to homotopy.
Proof. In the algebraic cases of x2.11(ab), we may take B to be (a cofibra*
*nt
model for) the constant simplicial object on . For simplicial spaces, B may
be constructed as for topological spaces, using generators and relations (see [*
*BDG ,
x8.9]). The extension to the diagram case of x2.11(d) is objectwise.
3.5. Fact. In each of the examples of x2.11(ad), for each n 1 there is an*
* n
dimensional MEilenbergMac Lane object E(M, n) for any abelian A algebra M,
and there is an ndimensional extended MEilenbergMac Lane object E (M, n) for
any A algebra and module M over . Each of these is unique up to homotopy.
Proof. In the algebraic cases of x2.11(ab), we may take E(M, n) to be the i*
*terated
EilenbergMac Lane construction W~ on BM, while E (M, n) is a semidirect
product E(M, n) n B (see [BDG , Prop. 2.2]). For simplicial spaces, use the e*
*xplicit
construction of [BDG , x8.9] The extension to the diagram case is again objec*
*twise.
3.6. Fact. In each of the examples of x2.11(ad), for each n 1 and X 2 C,*
* ssCnX
has a natural structure of a module over ssC0X.
Proof. Note that by [Q1 , II,1,(6)] we have map (A Sn, X) ~=map S (Sn, map (*
*A, X))
(unpointed maps), so ssCnX ! ssC0X associates to each f : A Sn ! X its
component in map (A, X). This defines an abelian algebra over ssC0X by [BP *
*, Prop.
6.26]).
3.7. Fact. In each of the examples of x2.11(ad), for each X 2 sC, := ssC0X
and n 1, the commutative square obtained by applying the functor Pn+2 to *
*the
pushout diagram:
p(n+1)
Pn+1X ________//PnX
 ____ kn
fflffl _PO_fflffl
B __________//_Y
is an nth kinvariant square (Def. 1.17)  that is, Pn+2Y ' E (ssCA,n+1X, *
*n + 2).
Proof. See [BDG , x5].
We may summarize these facts in the following:
3.8. Theorem. The following resolution model categories (cf. x2.11) are strict *
*spher
ical model categories:
i. The category C = s Set* of simplicial algebras for any Gtheory , *
*with
A^consisting of monogenic free algebras;
ii. In particular, the category C = G of simplicial groups, with A^= {Z};
iii.The category sG of bisimplicial groups ("simplicial spaces"), with *
*A^ =
{S1 Sk}1k=0.
iv. The category CI of Idiagrams in a strict spherical model category C.
3.9. Theorem. The following are spherical model categories (which are not stric*
*t):
i. The category S* of pointed simplicial sets, with A = {S1};
ii. The category T* of pointed topological spaces, with A = {S1};
iii.The category sT* of simplicial pointed topological spaces, with A^ =*
* {S1
Sk}1k=1.
COMPARING HOMOTOPY CATEGORIES 11
3.10. Nonspherical model categories. Consider the trivial model category struc
ture on ^C= Gp , with A^ := {A = Z=p} (for p an odd prime). This defines a
resolution model category structure on G  or equivalently, on T* (see Rema*
*rk
2.5). Note that  Sn corresponds to suspension of simplicial sets, not sim*
*plicial
abelian group, so the model A Sn 2 G corresponds to the ndimensional mod p
Moore space Sn1 [p en.
Thus ssCA,kX := [A Sk, X] is by definition the kth mod p homotopy group*
* of X
 denoted by ssk(X; Z=p) in [Ne , Def. 1.2]  which fits into a short exact seq*
*uence:
(3.11) 0 ! sskX Z=p ! ssk(X; Z=p) ! TorZ1(ssk1X, Z=p) ! 0
for k 2 (see [Ne , Prop. 1.4]). In particular, for Y := A Sn (n 4) *
* we have
(
Z=p for i = n  1, n,
ssi(Y ; Z=p) =
0 for 2 i < n  1 or i = n + 1,
with the two nontrivial groups connected by a Bockstein (cf. [Ne , x1]).
However, the resolution model category structure on G determined by A is not
spherical: if it were, in particular there would be Postnikov functors Pk = P*
*kA for
all k 1 (Def. 1.9). From (3.11)we see that, disregarding torsion prime to p,*
* because
of the Bockstein we must have Pn1Y ' E(Z, n  1) and PnY ' E(Z=p, n  1) (*
*for
Y = Sn1 [p en). But then there is no nontrivial map PnY ! Pn1Y .
3.12. Cohomology in spherical model categories. Note that the kinvariants of
a simplicial object actually take value in cohomology groups, as expected:
3.13. Proposition. For each A algebra and module M over , the functors Dn*
* :
C=B ! Ab Gp (n > 0), defined Dn(X) := [X, E (M, n)]B , are cohomology
functors on C  that is, they are homotopy invariant, take arbitrary coproduc*
*ts to
products, vanish on the spherical models nA, except in degree n, and have Ma*
*yer
Vietoris sequences for homotopy pushouts.
We therefore denote [X, E (M, n)]B by Hn(X; M).
Proof. See [BP , Thm. 7.14].
Fact 3.5 then follows from Brown Representability, since E (M, n) represe*
*nts the
nth Andr'eQuillen cohomology group in C; see [BDG , x6.7] and [Bl3, x4].
4. Spherical functors
Our objective is to study functors between model categories, and investigate*
* the ex
tent to which they induce an equivalence of homotopy categories. Our methods wo*
*rk
only for functors between spherical model categories which take models to model*
*s, in
the following sense:
4.1. Definition. Let and be two spherical model categories. *
* A
functor T : C ! D is called spherical if
i. T defines a bijection A ! B;
ii. T  A preserves coproducts and suspensions;
iii.T induces an equivalence of categories CAlg D Alg (in fact, it *
*suffices
that D Alg be a full subcategory of CAlg ).
12 DAVID BLANC
4.2. Examples of spherical functors. In the cases we shall be considering (those
mentioned in the introduction), C and D will be strict spherical resolution mod*
*el
categories, with C = sC^ and D = sD^, and T will be prolonged from a functor
T^ : ^C! ^D.
The four examples:
(a) For = and = , let T^ = Ab : Gp*
* ! Ab Gp
be the abelianization functor.
Here C = sC^= G, so ho C is equivalent to the homotopy category of
pointed connected topological spaces (x1.4), while D = sD^, the categ*
*ory
of simplicial abelian groups, is equivalent to the category of chain co*
*mplexes
under the DoldKan correspondence (see [D , x1]). Thus T : C ! D repres*
*ents
the singular chain complex functor C* : T* ! Chain .
Note that CAlg = Gp, while D Alg = Ab Gp, in this case, so st*
*rictly
speaking T does not induce an equivalence of categories. But since A*
*b Gp
is a full subcategory of Gp, we can in fact think of ss" as taking *
*values in
groups.
(b) For = and = , where Hopf*
* is the
category of complete Hopf algebras over Q, H is the monogenic free obj*
*ect in
this category, let ^Q: Gp ! Hopf be the functor which associates to *
*a group
G the completion of the group ring Q[G] by powers of the augmentation
ideal.
Again, C = sC^ is a model category for connected topological space*
*s,
while D = sD^ is a model category for the rational simplyconnected s*
*paces
(see [Q2 ]); Q (when restricted to connected simplicial groups) repres*
*ents the
rationalization functor. Once more, C Alg = Gp , while D Alg *
*is the
subcategory of vector spaces over Q.
(c) For = (so that = , by Remark *
*2.5), and
= , let ^F: Set*! Gp be the free group functor.
Again, we think of both C = sC^ = G and D = sD^ = S* as model
categories for pointed topological spaces, (under the respective equiva*
*lences
of x1.4)  so F here represents the suspension functor : T* ! T* *
*(rather
than , as one might think at first glance).
(d) For = and = < Alg ; {ss*Sk}1k=0>, *
*let bss*: G !
Alg be the graded homotopy group functor X 7! ss*X. Here C = sG *
*is
a model category for simplicial spaces.
4.3. Theorem. Let and be spherical model categories, and let*
* T :
C ! D be a spherical functor. Then for each X 2 C and A 2 A there is a nat*
*ural
long exact sequence of Calgebras:
sXn C hXn C @Xn T
(4.4) . . .! Xff,nX ! ssA,nX ! ssT*(ff),nT X ! ff,n1X . . ..
We call (4.4)the comparison exact sequence for T . Compare [Ba4 , V, (5.4)].
Proof. If X" ! X is a functorial fibrant replacement, the functor T induces a *
*natural
transformation o : map C(A, "X) ! map D(T A, dT "X), which we may functori*
*ally
COMPARING HOMOTOPY CATEGORIES 13
change to a fibration of simplicial sets, with fiber F (X). Setting Tff,n:=*
* ssnF (X),
the corresponding long exact sequence in homotopy is (4.4).
Note that the map hXn= hX is also natural in the variable A, so the grad*
*ed map
hX*: ssCnX ! ssDnT X is a morphism of Calgebras (i.e., A^algebras).
4.5. Applications of Theorem 4.3. The Theorem is not very useful in this gen
erality. However, in all the examples of x4.2, we obtain interesting (though m*
*ostly
known) exact sequences:
(a) For ^T= Ab : Gp ! Ab Gp the abelianization functor, where T : G ! sAb*
* Gp
represents the singular chain complex functor C* : T* ! Chain (cf. x4*
*.2(a)),
the sequence (4.4)is the "certain exact sequence" of J.H.C. Whitehead:
(4.6) . .!. nX ! ssnX hn!Hn(X; Z) ! n1X . . .
(See [W4 ]). In particular, the third term in this sequence, An(X),*
* is simply
the nth homotopy group of the commutator subgroup of GX.
(b) For Q : G ! sHopf of x4.2(b), representing the rationalization func*
*tor,
we obtain a long exact sequence relating the integral and rational homo*
*topy
groups of a simplyconnected space X. The third term in (4.4)may be de
scribed in terms of the torsion subgroup of ss*X together with ss*X *
* Q=Z.
(c) The free group functor ^F: Set* ! Gp of x4.2(c) represents the suspen*
*sion
: T* ! T*, and indeed for K 2 S* the map hK , which is the compo*
*site:
ssnK = ss0 map S*(Sn, K) ! ss0 map G(F Sn, F K)
~= n
! ss0 map S*( S , K) = ssn+1 K ,
is the suspension homomorphism, so (4.4)is a generalized EHP sequence (*
*cf.
[Ba1 , G , No ]).
(d) For ss* : sG ! s Alg as in x4.2(d), it turns out that for any simpli*
*cial
space X 2 sG, the induced map hXn is the "Hurewicz homomorphism"
hn : ss"nX ! ssnss*X of [DKS2 , 7.1], while TnX is just ss"n1X*
*  that is,
Ti,nX = ss"i+1,n1X for each i. Thus (4.4) is the spiral long exact *
*sequence:
@?n+1 " sn " hn " h0
(4.7) . .s.sn+1ss*X ! ssn1X ! ssnX ! ssnss*X ! . .s.s0X ! ss0ss*X *
*! 0
of [DKS2 , 8.1]. Of course, ss"1X = 0, so h0 is an isomorphism.
Note that for T : C ! D as above, the homotopy groups ssDnT X for any X 2
C = sC^ may be computed using the Moore chains C*T X as in x2.7; each ssDnT X
is a D algebra, abelian for n 1.
4.8. Explicit construction of the spiral exact sequence. It may be helpful to
inspect in detail the construction of last long exact sequence, since it is per*
*haps the
least familar of the four. Specificializing to ^C= G and T = ss*, we have:
4.9. Lemma. For fibrant X 2 C, the inclusion ' : CnX ,! Xn induces an
isomorphism '? : ss*CnX ~=Cn(ss*X) for each n 0.
Proof. See [Bl2, Prop. 2.11].
14 DAVID BLANC
Together with (2.13), this yields a commuting diagram:
(d0)# ^#n
(4.10) ss*Cn+1X ______//_ss*ZnX_____////_ss"nX
____
'?~= ^'? _hn______
fflffldss*X0 fflffl#n fflffl___
Cn+1(ss*X) ____//_Zn(ss*X)___////_ss"nss*X
which defines the dotted morphism of algebras hn : ss"nX ! ssn(ss*X). Note*
* that
for n = 0 the map ^'?is an isomorphism, so h is, too.
If X 2 sG is fibrant, applying ss* to the fibration sequence (2.8)yields a l*
*ong exact
sequence, with connecting homomorphism @n : ss*ZnX = ss* ZnX ! ss*Zn+1X;
(2.13) then implies that
(4.11) ss"nX = Coker (dn+10)# ~= Im @n ~=Ker (jXn+1)# ss*Zn+1X,
and the map sn+1 : ss"nX ! ss"n+1X in (3.11)is then obtained by composing *
*the
inclusion Ker (jXn+1)# ,! ss*Zn+1X with the quotient map #^n+1: ss*Zn+1X ! ss"*
*n+1X
of (2.13).
Similarly, hn : ss"nX ! ss"nss*X is induced by the inclusion (jXn)# : ss*
**ZnX !
Znss*X Cnss*X, and @?n+2: ss"n+2ss*X ! ss"nX is induced by the composite
(dn+20)#
Zn+2ss*X Cn+2ss*X ~=ss*Cn+2X ! Zn+1ss*X,
which actually lands in Ker (jXn+1)# ~= ss"nX by the exactness of the long e*
*xact
sequence for the fibration.
Moreover, for each n 0, (4.10)may be extended (after rotating by 90O) t*
*o a
commutative diagram with exact rows and columns:
0 0 0
  
  
fflfflO fflffl(jn)* fflffl
0 _____//_Kersn_____//_Bn+1X____////_Bn+1ss*Xn+2______//0
   
   
fflffl fflffl(jn)* fflffl fflffl
0 ____//_ ss"n1XO__//ss*ZnX_______//_Znss*X_____////_Cokerhn___//_0
 
 ^ # =
 #n n 
fflfflO fflfflhn fflffl fflffl
0 _____//_Kerhn______//_ss"nX______//_ssnss*X____////_Cokerhn___//_0
   
   
fflffl fflffl fflffl fflffl
0 0 0 0
in which Bn+1X := Im (dXn+20)# ss*ZnX and Bn+1ss*Xn+2 := Im dss*Xn+20are t*
*he
respective boundary objects.
The maps @?n+1, sn, and hn, as defined above, form the spiral long exact seq*
*uence.
4.12. Inverse spherical functors. We may sometimes be interested in functors
between spherical model categories which are not quite spherical. Thus, if T *
* :
! is a spherical functor as in x4.1, a functor V : D ! C equ*
*ipped
with a natural transformation # : IdC ! V T is called an inverse spherical fu*
*nctor
to T .
COMPARING HOMOTOPY CATEGORIES 15
4.13. Example. For the free group functor F : Set* ! Gp of x4.2(c), the forge*
*tful
functor ^U: Gp ! Set* (right adjoint to F ) with the adjunction counit j : Id*
*! UF
as the natural transformation #, yields the inverse spherical functor U : G ! *
*S*.
Here we do not think of G as a model for T*  rather, U represents the forget*
*ful
functor from loop spaces (topological groups) to spaces.
Similarly, the adjoint to the abelianization functor Ab : Gp ! Ab Gp is the*
* inclusion
^I : Ab Gp ! Gp , and the corresponding functor I : sAb Gp ! G represents the
factorization of the DoldThom infinite symmetric product functor SP 1 : T* ! *
*T*
through Chain.
4.14. Proposition. If V : D ! C is an inverse spherical functor to T , then *
*for
each Y 2 D and B 2 B there is a natural long exact sequence:
V# C V
(4.15) . .!. VB,nY ! ssDB,nY ! ssV*(B),nV Y ! B,n1Y . . .
Proof. If V is an inverse spherical functor, because T A is a bijection onto*
* B, there
is an A 2 A such that B = T A. As before, V induces a natural transformati*
*on
: map D(B, "Y) ! map D(V B, dV "Y) and the natural transformation # : A !
V T A yields ## : map D (V T A, dV "Y) ! map D (A, dV "Y) so we get a compos*
*ite map
map D(B, "Y) ! map D (A, dV "Y), with homotopy fiber E(Y ). If we let Vf*
*i,nY :=
ssnE(Y ), the fibration long exact sequence is (4.15).
4.16. Remark. Note that in contradistinction to Theorem 4.3, V# of (4.15)need*
* not
respect any operations, since we only have a bijection T A : A ! B, not a fu*
*nctor.
For U : G ! S* as in x4.13, we may assume X 2 G is of the form X ' GK *
*for
K 2 S*, and then V# is the identity:
ssnK = ss"nX= ss0 map G(F Sn1, GK) ! ss0 map S*(UF Sn1, UGK)
(4.17) j# n1
! ss0 map S*(S , UGK) = ssnK ,
so (4.15)is not interesting in this case.
5.Comparing Postnikov systems
The basic problem under consideration in this paper may be formulated as fol*
*lows:
Question. Given a spherical functor T : ! and an object G 2 D,
what are the different objects X 2 C (up to homotopy) such that T X ' G?
As shown in the previous section, such a pair must be connected by*
* a
comparison exact sequence. Thus, in order to reconstruct X from G, we first try*
* to
determine ssC*X, and its relation to ssD*G.
In order to proceed further, we must make an additional assumption on T , co*
*n
tained in the following:
5.1. Definition. A spherical (or inverse spherical) functor T : C ! D is cal*
*led
special if:
i. C = sC^ and D = sD^ are spherical resolution model categories, and T *
*is
prolonged from a functor ^T: ^C! ^D.
ii. For any A algebra and module M over , T induces a homomorphism of
(graded) groups OET : ! ssD0T BC .
16 DAVID BLANC
iii.This OET induces a functor T^ : Mod ! OET Mod which is an isomorp*
*hism
on modules (see Remark 1.13).
iv. For each n 1 and ndimensional extended MEilenbergMac Lane object
E = EC (M, n), there is a natural isomorphism ssDnT E ~=M which resp*
*ects
^Tin the obvious sense.
v. The natural map
T^
(5.2) [X, EC (M, n)]BC ! [T X, ED (M, n)]BDT^L ,
defined by composition with the projection
T^
ae : T EC (M, n) ! PnT EC (M, n) = ED (M, n) ,
is an isomorphism.
5.3. Example. All the functors we have considered hitherto, except for the rati*
*onal
ization functor Q : G ! sHopf of x4.2(b), are special:
(a) For the singular chain functor T : G ! sAb Gp, induced by abelianizati*
*on,
this follows from the Hurewicz Theorem (recall that ssC0X is the fund*
*amental
group, in our indexing for X 2 G).
(b) For the suspension : T* ! T*, induced by the free group functor F *
* :
Set*! Gp, this follows (in the simply connected case) from the Freuden*
*thal
Suspension Theorem.
(c) For the homotopy groups functor ss* : sG ! s Alg , (i)(iii) follow*
* by
inspecting the spiral long exact sequence (4.7), while (iv) is [BDG , *
*Prop. 8.7].
(d) For the inverse spherical functor U : G ! S* of x4.13, induced by the f*
*orgetful
functor ^U: Gp ! Set*, this is immediate from (4.17).
5.4. Lemma. Any special spherical functor T : C ! D as above respects Postnik*
*ov
systems  that is, for any X 2 C and n 0 we have:
(5.5) PnDT PnCX ~=PnDT X 
so that ssCkT X ~=ssDkT PnX and kX ~= kPnX for k n.
Proof. This follows from the constructions in x3.1 and the proof of Theorem 4.3.
5.6. Postnikov systems and spherical functors. From now on, assume T : C !
D is a special spherical functor. Ultimately, for each object G 2 D, we woul*
*d like
find any and all X 2 C such that T X ' G. First, however, we try to discov*
*er
what can be said about T X and its Postnikov systems for a given X 2 C. Usi*
*ng
the comparison exact sequence for T and Lemma 5.4, we see that:
8
>:
k1PnX for k n + 2 .
5.8. Fact. If T : C ! D is a special spherical functor, applying ssCn+2 to *
*the
nth kinvariant kn : PnX ! EC (ssCn+1X, n + 2) yields the homomorphism sXn+*
*1:
n+1X ! ssCn+1X.
COMPARING HOMOTOPY CATEGORIES 17
Proof. Since T is special, ssDn+2T EC (ss"n+1X, n + 2) ~= ssCn+1X, and ssDn*
*+2T PnX ~=
n+1X from (5.7), so this follows from the naturality of the comparison exact *
*se
quence, applied to the maps in (1.18).
5.9. Lemma. If T : C ! D is a special spherical functor, for any X 2 C,
Pn+1T PnX _________//Pn+1T Pn1X
 
 
fflfflTkn fflffl
Pn+1T BC _____//Pn+1T EC (ssCnX, n + 2)
is a homotopy pullback square in D=T BC , where := ssC0X.
Proof. Set E := T EC (J, n + 1), Mn1 := T Pn1X, and Mn := T PnX. The
naturality of the comparison exact sequence, applied to the maps in (1.18), com*
*bined
with Fact 5.8, imply that the vertical maps in the following commutative diagra*
*m are
isomorphisms:
Tkn1 D
ssDn+2E_____//_ssDn+1Mn____//ssDn+1Mn1____//ssn+1E____//ssDnMn____//ssDnMn1
~= ~= ~= ~= ~= ~=
     
fflffl fflfflO fflfflsXn fflfflhXn fflffl fflf*
*fl
0 _______//CokerhXn+1_______//_ nX________//ssCnX____//ssDnT_X__////_Coker*
*hTn
and since the bottom row is part of the comparison long exact sequence, and the*
* rest
of the top sequence to the right is exact for by (5.5), the kinvariant square *
*(1.18)
induces a long exact sequence after applying ss" (except in the bottom dimensi*
*ons).
The obvious map from Mn to the fiber of T kn1 is thus a weak equivalence in
D=T BC through dimension n + 1.
5.10. Corollary. For T : C ! D as above, for any X 2 C and n 1 the
natural map r(n): X ! PnX of x3.1 induces an isomorphism kX ~= kPnX for
k n + 1.
Proof. For each A 2 A, take fibers vertically and horizontally of the commuta*
*tive
square:
PnX
map BC (A, PnX) _______h__________//mapBD (T A, T PnX)
(kn)* (Tkn)*
fflffl E fflffl
map BC (A, EC (ssCn+1X, n + 2))h__//mapBD (T A, T EC (ssCn+1X, n + 2)) ,
and use Lemma 5.9 and x5.1(iv).
5.11. Remark. For C = sG this follows from the fact that nX ~= ss"n1X, while*
* for
the algebraic cases of x2.11(iii), this follows from the fact that Hn+1(K(ss, *
*n); Z) = 0
for n 1.
18 DAVID BLANC
5.12. The extension. The map r(n) : X ! PnX induces a map of comparison
exact sequences:
(5.13)
hXn+2 @?n+2 sn+1 hXn+1 @?n+1
ssCn+2X____//_ssCn+2T_X___// n+1X _____//_ssCn+1X__//_ssDn+1T_X___// nX

 ssD Tr(n) =  ssD Tr(n) =
  n+2    n+1 
fflffl fflffl~= fflffl fflffl fflffl fflff*
*l
0 _______//_ssDn+2Mn___// n+1PnX _______//_0_____//_ssDn+1Mn___// nPnX
so that ssCn+1X fits into a short exact sequence of A algebras:
(5.14) 0 ! Coker ssDn+2T r(n)! ssCn+1X ! Ker ssDn+1T r(n)! 0,
where
(5.15) Coker ssDn+2T r(n)~= Ker hXn+1and Ker ssDn+1T r(n)~= Im hXn+1.
Since hXn+1 is a map of modules over := ssC0X, by Theorem 3.8, (5.14) is
actually a short exact sequence of modules over , and we can classify the poss*
*ible
values of J 2 Mod (the candidates for ssCn+1X) using the following:
5.16. Proposition. Given T r(n): T X ! T PnX, a choice for the isomorphism cl*
*ass
of ssCn+1X uniquely determines an element of
Ext Mod(Ker (T r(n))n+1, Coker (T r(n))n+2).
Proof. Since Mod is an abelian category, with a set {Aab Sn q BD }A2A,*
*n2N
of projective generators, the argument of [Mc , III] carries over to our settin*
*g.
5.17. Remark. Observe that given PnX, we know the comparison exact sequence (4.*
*4)
for X only from sn : n1X ! ssCnX down. However, if ssDiT r(n): ssDiT X !*
* ssDiMn
(for i 0) and the extension (5.14)are also known, all we need in order to det*
*ermine
(4.4) for X from @?n+3: ssDn+3T X ! n+1X down is the homomorphism
ssDn+3T r(n+1): ssDn+3T X ! ssDn+3T Pn+1X ,
which is just @?n+3, as one can see from (5.13).
5.18. Proposition. For any 2 D, J0, J002 Mod , and n 2, there is a
natural isomorphism Ext Mod(J00, J0) ~=Hn+1(ED (J00, n); J0).
In particular, this implies that Hn+1(ED ( , n);  ) is stable  i.e., *
*independent
of n.
Proof. By Proposition 3.13ff. there is a natural isomorphism
Hn+1(ED (J00, n); J0) ~=[ED (J00, n), ED (J0, n + 1)]sD=BD ,
and given a map _ : ED (J00, n) ! ED (J0, n + 1), we can form the fibration s*
*equence
over BD (that is, pullback square as in (1.18)):
_ 0 0 00 _ 0
ED (J00, n) ! ED (J , n + 1) ' ED (J , n) ! F ! ED (J , n) ! ED (J , n*
* + 1).
From the corresponding long exact sequence in homotopy for this sequence in D, *
*we
obtain a short exact sequence of modules over :
(5.19) 0 ! J0 ! J ! J00! 0.
COMPARING HOMOTOPY CATEGORIES 19
On the other hand, given a short exact sequence (5.19)in Mod , we can cons*
*truct
a map _ : ED (J00, n) ! ED (J0, n + 1) over BD as follows:
Assume E := ED (J00, n) is constructed starting with skn1 ED (J00, :) = skn*
*1BD ,
and En ' W q LnBD (cf. x2.2), where W is free, equipped with a surjection
OE : W ! J00. Because J!!J00 is a surjection, and W is free, we can lift OE*
* to
OE0 : W ! J, defining a map "OE0: ZnED (J00, n) ! J. Since ssDnED (J00, n)*
* = J00,
the restriction of "OE0to BnED (J00, n) = Ker {ZnED (J00, n) ! J00} factors *
*through
_ : BnED (J00, n) ! J0 = Ker {J ! J00}. Precomposing with d0 : Cn+1ED (J00, n*
*) !
BnED (J00, n) defines _ : ED (J00, n) ! ED (J0, n + 1), which classifies (5.*
*19) as
before.
5.20. Corollary. For , J0, and J00as above, there is a natural isomorphism:
Ext Mod(J00, J0) ~=Hn+1(EC (J00, n); J0).
Proof. This follows from (5.2)(5.7)and the naturality of PnD+1.
5.21. Definition. Given X 2 C, its nth modified Postnikov section, denoted by
P^nX, is defined as follows:
Let K := {f : A Sn+1 ! X  A 2WA, [f] 2 Ker hTn+1 ssCn+1X}, and let C be the
cofiber of the obvious map : f2K A Sn+1 ! X (so that ssCn+1C ~=Coker *
* ),
with P^nX := Pn+1C. There are then natural maps p^(n+1): Pn+1X ! ^PnX (induced
by X ! C), as well as ~p(n): ^PnX ! PnX (which is just p(n)C: Pn+1C ! PnC *
*~=
PnX), with ~p(n)O ^p(n)= p(n)X: Pn+1X ! PnX. Note that ssCn+1^PnX ~= Im hXn*
*+1,
and PnP^nX ~=PnX.
The map ^r(n):= p^(n)O r(n) : X ! P^nX induces a map of comparison exact
sequences:
hXn+2 @?n+2 sn+1 hXn+1 @n+1
ssCn+2X______//ssDn+2T_X_____// n+1X _______//ssCn+1X_____//ssDn+1T_X_____// nX

 ssD Tr^(n) =  = ssD Tr^(n) =
  n+2    n+1 
fflffl fflffl~= fflffl0 fflfflO fflffl f*
*flffl
0 _______//_ssDn+2T ^PnX_// n+1P^nX____//_ssCn+1^PnX__//_ssDn+1T ^PnX_//_ n*
*P^nX
so that: 8
>:
k1P^nX for k n + 3 .
Thus ^r(n)induces a weak equivalence Pn+1T X ' Pn+1T ^PnX, which, together
^p(n) ~p(n)
with the existence of the appropriate maps Pn+1X ! ^PnX ! PnX, determin*
*es
P^nX up to homotopy. In fact we have:
5.23. Proposition. P^nX is determined uniquely (up to weak equivalence) by PnX
and the map ae := Pn+1T r(n): Pn+1T X ! Pn+1T PnX.
Proof. Note that In+1 := Ker ssDn+1ae is isomorphic to Im hXn+1 and Cn+1*
* :=
Im ssDn+1ae is isomorphic to Coker hXn+1 by (5.15).
We construct Y ' P^nX as follows, starting with skn+1 Y := skn+1 PnX; *
*by
Remark 3.2, we may assume skn+1T X = skn+1T PnX, so that PnT X ~=PnT PnX.
20 DAVID BLANC
By Fact 3.7), the lower right hand square in Figure 5.24 commutes in D, thus in*
*ducing
the rest of the diagram, in which the rows and columns are fibration sequences *
*over
BD .
^ae ^kn
F pppppppppppppppppppppppppppppppppp_Pn+1TpPnXpppppppppppp_ED (I*
*n+1, n + 2)
  (n) 
' ~ p i
 (n)  TPnX  *
? pTX ? kTXn ?
Pn+1T X ___________PnT X ~=PnT PnX _________ED (ssDn+1T X, n + 2)
  TPnX 
 kn q*
  
? ? = ?
BD _____________ED (Cn+1 , n + 2)_____ED (Cn+1 , n + 2)
Figure 5.24
In particular, the induced map ^kn: Pn+1T PnX ! E (In+1, n + 2) provides a
canonical lifting of:
kTXnO p(n)TPnX: Pn+1T PnX ! ED (ssDn+1T X, n + 2)
to ED (In+1, n + 2). Composing it with the natural map r(n+1): T PnX ! Pn+1T P*
*nX
defines an element in:
[T PnX, ED (In+1, n + 2)] ~=Hn+2(PnX ; In+1) ,
which we call the nth modified kinvariant for X.
If ^kn: PnX ! EC (In+1, n + 2) is the map corresponding to ^kn under (5.2)*
*),
then its homotopy fiber Y is (weakly equivalent to) P^nX, as one can verify *
*by
calculating ssC*Y . Note that Lemma 5.9 implies that F ' Pn+1T PnX, so th*
*at
~ is the homotopy inverse of the weak equivalence Pn+1ae : T X ! T PnX, which
completes the construction.
5.25. Remark. Note that there is a certain indeterminacy in our description of *
* ^kn,
and thus of ^kn, since we must make the lower right corner of Figure 5.24 int*
*o a
strict commuting diagram of fibrations, rather than one which commutes only up *
*to
homotopy. However,
5.26. Fact. The indeterminacy for ^knas an induced map is contained in the ind*
*e
terminacy for ^knas a kinvariant for Pn+1T X = Pn+1T Y .
Proof. Let M := T PnX. Making the lower right corner of Figure 5.24 commute on
the nose (assuming q* is already a fibration) requires the choice of a homoto*
*py
H : PnT X ! ED (Cn+1 , n + 2) = ED (Cn+1 , n + 1) ,
so the indeterminacy for ^kn as defined above is _*p*[PnT X, ED (Cn+1 , n + 1)]*
*, where
_ : ED (Cn+1 , n + 1) ! ED (In+1, n + 2) classifies the extension
0 ! In+1 ! ssDn+1T X ! Cn+1 ! 0
(Proposition 5.18), and p = p(n)M: Pn+1M ! PnM = PnT X.
On the other hand, the kinvariant ^kMn: Pn+1M ! ED (In+1, n + 2) for Pn+1T *
*PnX
(which is Pn+1T X) is determined only up to the actions of the group haut (Pn+*
*1M)
COMPARING HOMOTOPY CATEGORIES 21
of homotopy selfequivalences of Pn+1M over BD , and of Aut (In+1), the grou*
*p of
automorphisms of modules over of In+1, in [Pn+1M, ED (In+1, n + 2)]. Thus *
*given
a map f : PnM ! ED (Cn+1 , n + 1), we obtain a selfmap g : Pn+1M ! Pn+1M
such that Png = IdPnM and ssDn+1g = Id, by letting g = Id +i*p*(f), *
* for
i : ED (Cn+1 , n + 1) ! Pn+1M the inclusion of the fiber. It is readily verifi*
*ed that g
induces the identity on ssD*Pn+1M, so [g] 2 haut (Pn+1M), and that ^kn+ _**
*p*(f)
is obtained from ^knunder the action of [g] on Hn+2(Pn+1M ; In+1).
5.27. Notation. Given W ' PnX and ae : Pn+1T X ! Pn+1T W , Proposition 5.23
allows us to write ^Pn(W, ae), or simply ^PnW for ^PnX 2 C, which they de*
*termine
up to homotopy. This comes equipped with a weak equivalence ae : Pn+1T X !
Pn+1T ^PnW lifting ae.
5.28. Corollary. The weak equivalence ae : Pn+1T X ! Pn+1T ^PnW is welldefin*
*ed
up to homotopy.
Proof. The map ae is inverse to ~ in Figure 5.24, which is induced by the upper
right hand square, which is determined by ^kn and thus up to a selfequivalence
g : Pn+1T W ! Pn+1T W , according to Fact 5.26. But such a g induces a canoni*
*cal
selfequivalence g0: F 0! F , where F 0:= Fib(^knO g), and the resulting ~*
*0: F 0'
Pn+1T X satisfies ~ O g0' ~0.
5.29. Definition. For W ' PnX and ae : Pn+1T X ! Pn+1T W as above, an
extension
(5.30) 0 ! Coker ssDn+2ae ,! J!!Ker ssDn+1ae ! 0
is called allowable if its classifying cohomology class
[_] 2 Hn+3(ED (Coker ssn+2ae, n +;2)Kerssn+1ae)
(cf. Proposition 5.18) satisfies [_] O ^kn= 0.
5.31. Proposition. For any X 2 C, the extension (5.14) is allowable.
Proof. Writing V ' Pn+1X and Y ' ^PnX, by naturality we have a commutative
square:
kn C
PnV _________//EC (ssn+1V, n + 2)

= q*
fflfflkn fflffl
PnY _____//EC (Ker ssn+1T r(n), n + 2).
Lemma 5.9 and (5.2)then yield the following commuting diagram in D in which
the rows and columns are all fibration sequences over BD :
22 DAVID BLANC
ED (Ker hn, n + 1)________BD ________ED (Ker hn+1, n + 2)
  
  
 (n)  
? T r ? k ?
T Pn+1X ____________T PnX ________ED (ssCn+1X, n + 2)
 (n+2)  
r = q* 
  
? ? ^k ?
T Y _____________T PnX ________ED (Im hn+1, n + 2)
_ 

?
ED (Ker hn+1, n + 3)
The map k is induced by kn, and ^k is induced by ^kn. The claim then follo*
*ws
from the universal property for fibrations.
6. The fiber of a special spherical functor
Let T : C ! D be a special spherical functor. We would like to use the re*
*sults
of Section 5 in order to determine whether a given G 2 D is (up to homotopy) *
*of
the form T X for some X 2 C  and if so, how we can distinguish between su*
*ch
realizations, or liftings.
6.1. Lifting objects of D. Let us assume for simplicity that := ssD0G is a
Calgebra, and that the map OET : ! ssD0T BC of x5.1(i) is an isomorphism.
[In the general case, we are faced with an additional, purely algebraic, proble*
*m of
determining the fiber of the functor T* : C Alg ! D Alg (compare [BP ]); *
* we
bypassed this question in x4.1(iv).
We want a map ' : T X ! G inducing isomorphisms ssDiT X ! ssDiG for i *
* 0.
Our approach is inductive: we are trying to define a tower in C:
p(n+1) p(n) p(n1) p(0)
(6.2) . ..! X^ ! ^X ! . . .!X^<0> ' BC
which are to serve as the modified Postnikov tower of the (putative) X 2 C *
* so
that in the end we will have X := holimn ^X.
At the nth stage (n 0), we have constructed ^X as our candidate for*
* ^PnX
 so in particular if we let X := PnX^, (our candidate for the ordinar*
*y nth
Postnikov section of X), then T X satisfies (5.7), T ^X satisfies (*
*5.22), and
of course ^X = Pn+1X^.
Assume also, as part of our inductive hypothesis, a given weak equivalence:
(6.3) ^ae(n): Pn+1G '!Pn+1T ^X.
We start the induction with X<0> := BC . The natural map r(1) : G !
P1T BC = BD allows us to define ^X<0>, together with ^ae(0): P1G '!P1T *
*^X<0>,
as in Definition 5.21 (see x5.25).
COMPARING HOMOTOPY CATEGORIES 23
6.4. Lifting ae(n). The first stage in the inductive step occurs in D: we must *
*lift ^ae(n)
to ae(n): Pn+2G ! Pn+2T ^X. Note that by Remark 5.17 and Fact 5.8, we alre*
*ady
know the comparison exact sequence (4.4)for the putative X from hn+1 down; the
lifting ae := ae(n) will determine @n+2 : ssDn+2G ! n+1X^ in addition, s*
*ince this
is just ssn+2ae, so that Cn+2 := Im ssDn+2ae is our candidate for Coker hX*
*n+2, while
Kn+1 := Coker ssDn+2ae is our candidate for Ker hXn+1.
From (5.22)we see that the obstruction is the class:
X^ (n) n+3
(6.5) On := kTn+1 O ae 2 H (G; n+1X^) ,
and the different liftings are classified by Hn+2(G; n+1X^).
6.6. Constructing X. The next step is to choose a cohomology class ^kn
in Hn+2(X^; Kn+1 ). This fits into a commutative diagram with rows and fib*
*ers
all fibration sequences over BC :
BC _______//EC (In+1, n + 1)=_//_EC (In+1, n + 1)
  
 i _
fflfflp^(n) fflffl^kn fflffl
X _________//_^X________//_EC (Kn+1 , n + 2)
  
  j*
fflffl fflfflkn fflffl
X _________//_X___________//_____________________EC (*
*J, n + 2)
for the bottom fibration sequence X ! X ! EC (J, n + 2) as indicated
(though we shall not need this).
Note that J, our candidate for ssCn+1X, fits into the short exact sequence o*
*f modules
over :
0 ! Kn+1 ,! J!!In+1 ! 0,
as in (5.14), and is classified by _ := ^knO i 2 Hn+2(EC (In+1, n + 1); Kn+1 )*
*, as in
Corollary 5.20. Moreover, this extension is obviously allowable in the sense of*
* x5.29.
6.7. Lifting ae. To complete the induction on (6.3), we must lift ae : G ! Pn+2*
*T ^X.
This will be done in two steps:
First, note that we obtain a commuting diagram:
ae
Pn+2G _______________________________________Pn+2Tp^Xpppp
 ppppppp * 
 ppppppp 
pppppp "
 ae j i* 
p(n+1) Pn+2T X p(n+1)
G    TX^
 p(n+1) 
  TX 
? f ? g ?
Pn+1G __________'Pn+1T X ___________'Pn+1T ^X
  
  TX 
 kn+1 
 ? 
G   TX^
kn+1 E (C , n + 3) kn+1
 q* pppppp*Dn+2 H H i* 
 pppppppp H 
? ppppppp (ss" HH ?
pp n+2ae)* Hj D
ED (ss"n+2G, n + 3)____________________________________ED (ssn+1T X, n + 3)
24 DAVID BLANC
in which the columns are fibration sequences over BC , since by definition
ssDn+2ae : ssDn+2G ! ssDn+1T ^X = ssDn+1T X
factors through Cn+2 := Im ssDn+2ae, so that the bottom triangle commutes.
Since the natural Kinvariant kGn+1 is given, the other two kinvariants i*
*n the
diagram above are determined by inverting the given homotopy equivalences f :
Pn+1G ! Pn+1T X and g : Pn+1G ! Pn+1T ^X (assuming all objects in
*
* X^
D are fibrant and cofibrant), and letting kTXn+1:= q* O kGn+1O f1 and *
* kTn+1 :=
i* O kGn+1O g1, using Fact 3.7.
Therefore, the map ae : G ! Pn+2T ^X lifts to ae : Pn+2G ! Pn+2T X
(which is induced by q*). In fact, the lifting ae is unique up to homotopy. M*
*oreover,
from the proof of Proposition 5.23 we see that this suffices to define X^, as well
as determining a lifting of ae to a weak equivalence a^e(n+1): Pn+2G ! Pn+2T ^X*
*.
We may summarize our results in:
6.8. Theorem. Given G 2 D, there is an object X 2 C such that T X ' G if
and only if there is a tower as in (6.2), serving as the modified Postnikov tow*
*er for
X. If we have constructed ^X satisfying (6.3)for n, a necessary and suf*
*ficient
condition for the existence of an ^X satisfying (6.3)for n+1 is the *
*vanishing
of On 2 Hn+3(G; n+1X^). The choices are classified by:
o Hn+2(G; n+1X^) (distinguishing the liftings of a^e(n)to Pn+2T ^X*
*);
and
o ^kn2 Hn+2(X^; Kn+1 ), where Kn+1 := Coker ssn+2ae(n), up to selfhom*
*otopy
equivalences of ^X over BC and Aut (Kn+1). In particular, th*
*is dis
tinguishes the class of ssCn+1X in Ext Mod(Ker (T r(n))n+1, Coker (*
*T r(n))n+2).
Note that n+1X^ = n+1X^ = n+1X, by Corollary 5.10.
6.9. Moduli spaces. It is possible to refine the statement of our fundamental p*
*rob
lem of lifting G 2 D to C in terms of moduli spaces:
Given a model category C, let W be a homotopically small subcategory of C, s*
*uch
that all maps in W are weak equivalences, and if f : X ! Y is a weak equivale*
*nce
in C with either X or Y in W, then f 2 W. Recall from [DK1 , x2.1] that the
nerve BW of such a category is called a classification complex. Its component*
*s are
in onetoone correspondence with the weak homotopy types (in C) of the objects*
* of
W, and the component containing X 2 C is weakly equivalent to the classifying
space B hautX of the monoid of selfweak equivalences of X.
6.10. Definition. Given a spherical functor T : C ! D and G 2 D, we denote *
*by
M(G) the category of objects in D weakly equivalent to G (with weak equivalence*
*s as
morphisms), and by TM (G) the category of objects X 2 C such that T X 2 M(*
*G)
(again, with weak equivalences in C as morphisms). The "pointed" version is den*
*oted
by R(G)  the category of pairs (X, ae), where X 2 C and ae : G ! T X *
*is a
specified weak equivalence.
COMPARING HOMOTOPY CATEGORIES 25
In all our examples the obvious functors R(G) F! TM (G) T! M(G) preserve
fibrant and cofibrant objects, and thus induce a homotopy pullback diagram:
BR(G) _BF_//_BTM (G)
 
 BT
fflffl fflffl
{IdG }______//_BM(G)
`
and there are weak equivalences BTM (G) ' X2ss0TM (G)B hautX, where BM(G) '
B Aut(G) for Aut(G) the monoid of self weak equivalences of G.
6.11. Towers of moduli spaces. Although BTM (G) is the more natural object of
interest in our context, it is more convenient to study BR(G) by means of a t*
*ower
of fibrations, corresponding to the Postnikov system of X 2 R(G):
Let Rn(G) denote the category whose objects are pairs (X^, ae0), where X^*
* 2
C has Pn+1X^ ' X^ and ae0 : Pn+1G ! Pn+1T ^X is a weak equivalenc*
*e.
The maps of Rn(G) are weak equivalences compatible with the maps p(n).
As in [BDG , Thm. 9.4], one can show that BR(G) ' holimn BRn(G), so we may
try to obtain information about the moduli space TM (G) by studying the succes*
*sive
stages in the tower:
BFn1
(6.12) . .B.Rn+1(G) BFn!BRn(G) ! . .!.BR1(G).
However, from the discussion above we see that we need several intermediate *
*steps
in the study of BRn+1(G) ! BRn(G), corresponding to the additional choices
made in obtaining ^Pn+1X and p(n+1): Pn+2G '!Pn+2T ^Pn+1X from ^PnX and
p(n): Pn+1G '!Pn+1T ^PnX. As a result one obtains a refinement of the tower (*
*6.12),
where the successive fibers F are either empty, or else generalized EilenergMa*
*c Lane
spaces, whose homotopy groups may be described in terms of appropriate Quillen
cohomology groups. We leave the details to the reader; compare [BDG , Thm. 9.6*
*].
7. Applying the theory
The approach to the lifting problem for a spherical functor T : C ! D desc*
*ribed
in the previous section is somewhat unwieldy. However, in specific application*
*s it
may simplify in various ways. We illustrate this by a number of examples:
7.1. Singular chains. Consider the singular chain functor C* : T* ! Chain , w*
*hich
in the form T : G ! sAb Gp is induced by abelianization (see x4.2(a)). Thus, *
*given
a chain complex G*, we would like to find all topological spaces X (if any) w*
*ith
C*X ' G*. Over Z, this is equivalent to the question of realizing a given sequ*
*ence
of homology groups.
Our approach uses Whitehead's exact sequence (4.6) to relate the (trivial) *
*Post
nikov system for the chain complex G* to the modified Postnikov system for the
space X, in which we attach at each stage not a single new homotopy group, but *
*a pair
of groups in adjacent dimensions, corresponding to the image and kernel respect*
*ively
of the Hurewicz homomorphism.
It should be observed that the functor T involves only "algebraic" categorie*
*s C =
sC^, where ^C in our case, Gp or Ab Gp  has a trivial model category struct*
*ure, as
26 DAVID BLANC
in x2.11(ab). The analysis in Section 6 then simplifies considerably, in as m*
*uch as the
categories of Calgebras and D algebras are simply Gp and AbGp , respect*
*ively.
As noted in the Introduction, Baues's [Ba4 , VI, (2.3)] is actually a genera*
*lization
the obstruction theory described here for this case. His earlier approach in [B*
*a3 ] (as
well as that of Benkhalifa in [Be ] is parallel to this, though not framed in t*
*he same
cohomological language. See [Man ] for another viewpoint.
7.2. Rationalization. On the other hand, the rationalization functor ()Q : T *
* !
TQ, induced by the completed group ring functor ^Q: Gp ! Hopf (cf. x4.2(b))*
*, is
spherical but not special (Def. 5.1), and so the theory described here does not*
* apply
as is. In fact, one can see why if one considers the comparison exact sequence*
* for
Q^ (x4.5(b)): given a (simplyconnected) rational space G 2 TQ, for each Qv*
*ector
space ssnG, we need an abelian group A = ssnX such that A Q ~=ssnG, and th*
*en
lift the rational kinvariants for X to integral ones. Thus, much of the indet*
*erminacy
for X is algebraic.
7.3. Suspension. The suspension functor : T* ! T*, induced by the free group
functor F^ : Set*! Gp as in x4.2(c), is similar to singular chains, with the ge*
*neralized
EHP sequence replacing the "certain long exact sequence", and the modified Post*
*niov
systems involve the kernel and image of the suspension homomorphism E : ssnX !
ssn+1 X.
7.4. Homotopy groups. The motivating example for the treatment in this paper
 and the only one which requires the full force of Section 6  is the funct*
*or
ss* : T* ! Alg , prolonged to simplicial spaces (as in as in x4.2(d)). Howe*
*ver, even
this case simplifies greatly if we want to realize a single algebra  tha*
*t is, we
take G 2 s Alg to be the constant simplicial algebra B .
Indeed, given a simplicial space X with ss*X ' B (which implies that ss**
*kXk ~=
G), from the spiral exact sequence (4.7)we find that ss"nX ~= n for all n *
* 0,
so that hn : ss"nX ! ss"nss*X is trivial for n > 0. We do not need the mod*
*ified
Postnikov system in this case: the obstructions to realizing (or G) are just*
* the
classes On 2 Hn+3( ; n+1 ), and the difference obstructions distinguishing bet*
*ween
the different realizations are ffin 2 Hn+2( ; n+1 ) (n 1). See [BDG ] an*
*d [BJT ,
x5] for two descriptions of this case.
7.5. Remark. Our obstruction theory is irrelevant, of course, for the inverse s*
*pherical
functor U : G ! S* (see x4.13)  that is, in determining loop structures on *
*a given
topological space. Nevertheless, from (4.17)we can easily recover the wellknow*
*n fact
that X ' Y is a loop space if and only if its kinvariants are suspensions o*
*f those
of Y (cf. [AHK ]).
7.6. Lifting morphisms. In all of the above examples, one can ask the analogous
question regarding the lifting of maps, or more complicated diagrams, from D to*
* C.
This can be addresses via Theorem 6.8 by transfering the spherical structure fr*
*om C
and D to the diagram categories CI and DI (cf. x2.11(d)). See [BJT , x8] f*
*or a
detailed example.
Note that the kinvariants for a map of chain complexes are not trivial (cf.*
* [D ,
(3.8)]), so the theory for realizing chain maps in T* is correspondingly more *
*compli
cated.
COMPARING HOMOTOPY CATEGORIES 27
References
[AHK] K. Aoki, E. Honma, & T. Kaneko, "On natural systems of some spaces", Pr*
*oc. Jap. Acad.
32 (1956), pp. 564567.
[AC] M. Arkowitz & C.R. Curjel, "The Hurewicz homomorphism and finite homoto*
*py invari
ants", Trans. AMS 110 (1964), pp. 538551.
[Ba1] H.J. Baues, "Relationen f"ur prim"are Homotopieoperationen und eine ve*
*rallgemeinerte
EHP Sequenz", Ann. Sc. 'Ec. Norm. Sup. 8 (1975), pp. 509533.
[Ba2] H.J. Baues, Combinatorial Homotopy and 4Dimensional Complexes, Gruyte*
*r Expositions
in Mathematics 2, Walter de Gruyter, BerlinNew York, 1991.
[Ba3] H.J. Baues, Homotopy type and homology, Oxford University Press, New Y*
*ork, 1996.
[Ba4] H.J. Baues, Combinatorial Foundation of Homology and Homotopy, Applica*
*tions to
Spaces, Diagrams, Transformation Groups, Compactifications, Differentia*
*l Algebras, Alge
braic Theories, Simplicial Objects, and Resolutions. Springer Monograph*
*s in Mathematics,
BerlinNew York, 1999.
[Be] M. Benkhalifa, "Sur le type d'homotopie d'un CWcomplexe", Homology, Ho*
*motopy Appl.
5 (2003), pp. 101120.
[BH] I. Berstein & P.J. Hilton, "On suspensions and comultiplications", Topo*
*logy 2 (1963),
pp. 7382.
[Bl1] D. Blanc, "New model categories from old", J. Pure & Appl. Alg. 109 (19*
*96), pp. 3760.
[Bl2] D. Blanc, "CW simplicial resolutions of spaces, with an application to *
*loop spaces", Top.
& Appl. 100 (2000), pp. 151175.
[Bl3] D. Blanc, "Generalized Quillen cohomology", preprint, 2006.
[BDG] D. Blanc, W.G. Dywer, & P.G. Goerss, "The realization space of a alge*
*bra: a moduli
problem in algebraic topology", Topology 43 (2004), pp. 857892.
[BJT] D. Blanc, M.J. Johnson, & J.M. Turner, "On realizing diagrams of alge*
*bras", Algebraic
& Geometric Topology, to appear.
[BP] D. Blanc & G. Peschke, "The fiber of functors between categories of alg*
*ebras", J. Pure
Appl. Alg., to appear.
[Bor] F. Borceux, Handbook of Categorical Algebra, Vol. 2: Categories and Str*
*uctures, Encyc.
Math. & its Appl., 51, Cambridge U. Press, Cambridge, UK, 1994.
[Bou] A.K. Bousfield, "Cosimplicial resolutions and homotopy spectral sequenc*
*es in model cate
gories", Geom. & Topology 7 (2003), pp. 10011053.
[BK] A.K. Bousfield & D.M. Kan, Homotopy Limits, Completions, and Localizati*
*ons, Springer
Lec. Notes Math. 304, BerlinNew York, 1972.
[D] A. Dold, "Homology of symmetric products and other functors of complexe*
*s", Ann. Math.,
Ser. 2, 68 (1958), pp. 5480.
[DK1] W.G. Dwyer & D.M. Kan, "A classification theorem for diagrams of simpli*
*cial sets", Topol
ogy 23 (1984), pp. 139155.
[DK2] W.G. Dwyer & D.M. Kan, "An obstruction theory for diagrams of simplicia*
*l sets", Proc.
Kon. Ned. Akad. Wet.  Ind. Math. 46 (1984), pp. 139146.
[DKS1] W.G. Dwyer, D.M. Kan, & C.R. Stover, "An E2 model category structure fo*
*r pointed
simplicial spaces", J. Pure & Appl. Alg. 90 (1993), pp. 137152.
[DKS2] W.G. Dwyer, D.M. Kan, & C.R. Stover, "The bigraded homotopy groups ssi,*
*jX of a pointed
simplicial space", J. Pure Appl. Alg. 103 (1995), pp. 167188.
[G] T. Ganea, "A generalization of the homology and homotopy suspensions", *
*Comm. Math.
Helv. 39 (1965), pp. 295322.
[GH] P.G. Goerss & M.J. Hopkins, "Moduli spaces of commutative ring spectra"*
*, in A. Baker
and B. Richter, eds., Structured Ring Spectra, London Math. Soc. Lect. *
*Notes 315, Cam
bridge U. Press, Cambridge, 2004, pp. 151200.
[Hi] P.S. Hirschhorn, Model Categories and their Localizations, Math. Survey*
*s & Monographs
99, AMS, Providence, RI, 2002.
[Ho] H. Hopf, ""Uber die Topologie der GruppenMannigfaltkeiten und ihre Ver*
*allgemeinerun
gen", Ann. Math. (2) 42 (1941), pp. 2252.
[J] J.F. Jardine, "Bousfield's E2 Model Theory for Simplicial Objects", in *
*P.G. Goerss & S.B.
Priddy, eds., Homotopy Theory: Relations with Algebraic Geometry, Group*
* Cohomology,
and Algebraic KTheory, Contemp. Math. 346, AMS, Providence, RI 2004, p*
*p. 305319.
28 DAVID BLANC
[Mc] S. Mac Lane, Homology, SpringerVerlag Grund. math. Wissens. 114, Berli*
*nNew York
1963.
[Man] M.A. Mandell, "E1 algebras and padic homotopy theory", Topology 40 (2*
*001), pp.. 43
94.
[May] J.P. May, Simplicial Objects in Algebraic Topology, U. Chicago Press, C*
*hicagoLondon,
1967.
[Ne] J.A. Neisendorfer, Primary homotopy theory, AMS Memoirs 232, Am. Math. *
*Soc., Prov
idence, RI, 1980.
[No] Y. Nomura, "On extensions of triads", Nagoya Math. J. 27 (1966), pp. 24*
*9277
[Q1] D.G. Quillen, Homotopical Algebra, Springer Lec. Notes Math. 43, Berlin*
*New York, 1963.
[Q2] D.G. Quillen, "Rational homotopy theory", Ann. Math. 90 (1969), pp. 205*
*295.
[Q3] D.G. Quillen, "On the (co)homology of commutative rings", Applications*
* of Categorical
Algebra, Proc. Symp. Pure Math. 17, AMS, Providence, RI, 1970, pp. 658*
*7.
[R] C.A. Robinson, "Obstruction theory and the strict associativity of Mora*
*va Ktheories", in
S.M. Salamon, B. Steer, & W.A. Sutherland, eds., Advances in Homotopy T*
*heory (Cortona,
1988), London Math. Soc. Lect. Notes 139, Cambridge U. Press, Cambridge*
*, 1989, pp. 143
152.
[Se] J.P. Serre, "Groupes d'homotopie et classes de groupes ab'eliens", Ann*
*. Math. (2) 58
(1953), pp. 258294.
[Sm] J.R. Smith, "Topological realizations of chain complexes I: the general*
* theory", Top. &
Appl. 22 (1986), pp. 301313.
[Sp] J. Spali'nski, "Stratified model categories", Fund. Math. 178 (2003), p*
*p. 217236.
[Sta] J.D. Stasheff, "Homotopy associativity of Hspaces, I,II", Trans. AMS 1*
*08 (1963) pp.
275292, 293312.
[Ste] N.E. Steenrod, "The cohomology algebra of a space", Ens. Math. 7 (1961)*
*, pp. 153178.
[Sto] C.R. Stover, "A Van Kampen spectral sequence for higher homotopy groups*
*", Topology
29 (1990), pp. 926.
[Sug] M. Sugawara, "A condition that a space is grouplike", Math. J. Okayama*
* U. 7 (1957),
pp. 123149.
[Sul] D.P. Sullivan, Geometric Topology, Part I. Localization, periodicity, a*
*nd Galois symmetry,
Massachusetts Institute of Technology, Cambridge, MA, 1970.
[T] R. Thom, "Sur un probl`eme de Steenrod", C.R. Acad. Sci., Paris 236 (19*
*53), pp. 1128
1130.
[W1] J.H.C. Whitehead, "On simply connected, 4dimensional polyhedra", Comm.*
* Math. Helv.
22 (1949), pp. 4892.
[W2] J.H.C. Whitehead, "On the realizability of homotopy groups", Ann. Math.*
* (2) 50 (1949),
pp. 261263.
[W3] J.H.C. Whitehead, "The secondary boundary operator", Proc. Nat. Acad. S*
*ci. USA 36
(1950), pp. 5560.
[W4] J.H.C. Whitehead, "A certain exact sequence", Ann. Math. (2) 52 (1950),*
* pp. 51110.
[W5] J.H.C. Whitehead, "Simple homotopy types", Amer. J. Math. 72 (1952), pp*
*. 157,
Department of Mathematics, University of Haifa, 31905 Haifa, Israel
Email address: blanc@math.haifa.ac.il