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 chain-homotopy equivalent chain complexes? How

        can one realize a given map of chain complexes up to homotopy?

    b)  When do two simply-connected 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 k-invariants.  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  @-! n-1X . .,.


 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(n-1)    p(0)
                . .-.--!  ^X<n + 1> --!  ^X<n> ---!  . . .--!X^<0> ,


called the modified Postikov tower for X (Def. 5.21), with G mapping compatibly*
 * to

T ^X<n>  for each n, and  X ' holimn ^X<n>.

   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^<n>),  and the choices for  ^X<n + 1>  are classi*
 *fied by:

     o  a class in  Hn+2(G;  n+1X^<n>);

     o  a class in  Hn+2(X^<n>; Kn+1 ),  where  Kn+1 := Coker ssn+2ae(n),  for *
 * ae(n):

        Pn+2G ! Pn+2T ^X<n>.


   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 H-space 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 T-complex

(corresponding to a CW complex, or more generally a cofibrant object in C)  -  *
 *see

[Ba4 , VI, (2.2-2.3)]).

   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 non-negatively 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 R-modules 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 5-6, 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 k-sphere 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 Sn-1  will be called the n-dimensional G-sphere,  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*  k-k S*  ~  G
                                             W

induce equivalences of the corresponding homotopy categories  -  where  ~W : G *
 *! S*

is the Eilenberg-Mac 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 FP-sketch - cf. [Bor , x5.6]), and a  -algebra (*
 *or model)

is a product-preserving 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 G-theory   *
 *(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 S-graded 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 G-theory  , and thus equipped with a faithful forgetful *
 *functor

UC :  C -Alg ! GpA  into the category of A-graded groups. The objects of   C-Alg

are called   C-algebras.

   For topological spaces, with  A = {S1},  the  C-algebras are simply groups. *
 *If we

use rational spheres as the models, then   C-Alg  is the category of Q-vector 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(n-1)       A
                    . .!.PnAY - -! Pn-1Y - --!  . .!.P0 Y ,

                                                                               *
 * (n)
as well as a weak equivalence  r : Y ! P1AY := limnPnAY  and fibrations  P1AY -*
 *r-!

PnAY  such that  r(n-1)= 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  C-algebra   ,  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

G-bundles.  One can interpret  BC   similarly, though perhaps less naturally (s*
 *ee,

e.g., [BJT , x4.6]).


1.12.  Definition. A module over a  C-algebra    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  C-algebra 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 R-module, for a ring R).


1.14. Example. For any object  X 2 C  as above, the  A x N-graded group  ssC*X

has an action of the A-primary 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  C-algebra M and an integer  n   1,  an n-

dimensional M-Eilenberg-Mac 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  C-algebra  ,  a module M over  , and an integer  n

1,  an n-dimensional extended M-Eilenberg-Mac 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 n-th k-invaria*
 *nt

square (with respect to A) is a functor that assigns to each  Y  2 C  a homotopy

pull-back 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 n-th (functorial) k-invariant for *
 *Y .

1.19. Example. If C* is a chain complex of R-modules, 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 k-invariants 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  C-algebra    and module M over  , the classifying object  B

        and extended M-Eilenberg-Mac Lane object E  (M, n) exist (and are unique

        up to homotopy) for each  n   1.

Ax 6.   C has k-invariant 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 *
 * <C; A>.

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 k-invariants in these categories are provided by *
 *the

construction of [BDG  , x5];  in both cases   C-Alg   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  <C; A>  as above is called strict*
 * if the

following axioms hold:


Ax 1.   C  is a pointed right-proper 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,  *
 *R-Mod ,

        or  sR-Mod ,  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 n-th matching object for a simplicial object X over C is d*
 *efined

by

      MnX = {(x0, . .,.xn) 2 (Xn-1)n+1 | dixj = dj-1xi for all0   i < j   n}

(see [BK , X,x4.5]). Note that each face map  dk : Xn ! Xn-1  factors through t*
 *he

obvious map  ffin : Xn ! MnX.


2.2. Definition.`The n-th latching object of a simplicial object X over C is de*
 *fined

LnX :=   0 i n-1Xn-1= ~,  where for any  x 2 Xn-k-1  and  0   i   j   n - 1  we

set  sj1sj2. .s.jkx  in the i-th copy of  Xn-1  equivalent to  si1si2. .s.ikx  *
 *in the j-th

copy of  Xn-1  whenever the simplicial identity  sisj1sj2. .s.jk= sjsi1si2. .s.*
 *ikholds.

The map  oen : LnX ! Xn  is defined  oen(x)i= six,  where  (x)i2 (Xn-1)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 n-cycles object  ZnX  to*
 * be

{x 2 Xn | dix = 0 fori = 0, . .,.n} (the fiber of ffin : Xn ! MnX  of x2.1).  S*
 *imilarly,

the n-chains object for X is  CnX = {x 2 Xn | dix = 0 fori = 1, . .,.n}.

   If X is fibrant, the map  d0 = dn0:= d0|CnX : CnX ! Zn-1X  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 product-preserving 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 NxA-graded group equipp*
 *ed

with an action of the A-primary 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  C-algebra structure on the A-graded groups  ssCnX,  mentioned in x1*
 *.7:

when  C = sC^,  we have   C-Alg :=  A^-Alg.

2.10. Example. When  C = G,  and  A = {S1}  -  so   A   is the category of

wedges of G-spheres (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   C-Alg   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, R-module*
 *s,

        associative algebras, Lie algebras, and so on  -  so that C is corepres*
 *ented by

        a G-theory  , in the language of [BP , x4]. Here A consists of free mon*
 *ogenic

        algebras (one for each isomorphism class), and thus once more   C-Alg  *
 * 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 I-diagrams in C also has a resolution model category structure, i*
 *n which

        the models consists of`all free I-diagrams  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 <C = sC^; A> 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(a-d), 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(a-d),  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(a-b),  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(a-d),  for each  n   1  there is an*
 * n-

dimensional M-Eilenberg-Mac Lane object  E(M, n)  for any abelian  A -algebra M,

and there is an n-dimensional extended M-Eilenberg-Mac 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(a-b),  we may take  E(M, n)  to be the i*
 *terated

Eilenberg-Mac Lane construction W~  on  BM,  while  E  (M, n)  is a semi-direct

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(a-d),  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(a-d),  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 n-th k-invariant 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 G-theory  , *
 *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 I-diagrams 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. Non-spherical 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 n-dimensional mod p

Moore space  Sn-1 [p en.

   Thus  ssCA,kX := [A   Sk, X]  is by definition the k-th 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(ssk-1X, 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 non-trivial 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  Pn-1Y ' E(Z, n - 1)  and  PnY ' E(Z=p, n - 1)  (*
 *for

Y = Sn-1 [p en).  But then there is no non-trivial map  PnY ! Pn-1Y .


3.12. Cohomology in spherical model categories. Note that the k-invariants 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

n-th Andr'e-Quillen 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  <C; A>  and  <D; B >  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   C-Alg    D -Alg  (in fact, it *
 *suffices

        that   D -Alg be a full subcategory of   C-Alg ).
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 <C^; ^A> = <Gp ; {Z}> and <D^, ^B> = <Ab Gp; {Z}>, 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 Dold-Kan correspondence (see [D , x1]). Thus T : C ! D repres*
 *ents

        the singular chain complex functor  C* : T* ! Chain .

          Note that   C-Alg = 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  <C^; ^A> = <Gp ; {Z}>  and  <D^, ^B> = <Hopf  ; {H}>,  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 simply-connected 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  <C^, ^A> = <Set*; {S0}>  (so that  <C, A> = <S; {S1}>,  by Remark *
 *2.5), and

        <C^; ^A> = <Gp ; {Z}>,  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  <C^; ^A> = <G; {Sk}1k=0>  and  <D^, ^B> = < -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  <C; A>  and  <D; B >  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  C-algebras:


                           sXn   C     hXn    C            @Xn   T
(4.4)      . . .!  Xff,nX  -!  ssA,nX  -!   ssT*(ff),nT X  -!    ff,n-1X . . ..


   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  C-algebras (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) !  n-1X . . .



          (See [W4  ]). In particular, the third term in this sequence,  An(X),*
 * is simply

        the n-th 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 simply-connected 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"n-1X*
 *  -  that is,

         Ti,nX = ss"i+1,n-1X  for each i. Thus  (4.4) is the spiral long exact *
 *sequence:

                       @?n+1  "     sn   "   hn                 "   h0
(4.7)   . .s.sn+1ss*X ---!  ssn-1X -!  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______
                            fflffl|dss*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
                   |            |              |
                   |            |              |
                   fflffl|O     fflffl|(jn)*   fflffl|
        0 _____//_Kersn_____//_Bn+1X____////_Bn+1ss*Xn+2______//0

                   |            |              |               ||
                   |            |              |               |
                   fflffl|      fflffl|(jn)*   fflffl|         fflffl|

        0 ____//_ ss"n-1XO__//ss*ZnX_______//_Znss*X_____////_Cokerhn___//_0
                                               |               |
                   |            |^             #|              =|
                   |            |#n            |n              |
                   fflffl|O     fflffl|hn      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 *
 * :

<C; A> ! <D; B >  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 Dold-Thom 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,n-1Y . . .


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 Sn-1, GK)  !  ss0 map S*(UF Sn-1, UGK)

(4.17)                  j#              n-1
                        -!  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 : <C; A> ! <D; B >  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  <X, G >  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 n-dimensional extended M-Eilenberg-Mac 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
                        ><ssDkT X                            for k   n,

(5.7)       ssDkT PnX ~=  Coker {hXn+1: ssCn+1X ! ssDn+1T X} for k = n + 1,
                        >:
                           k-1PnX                            for k   n + 2 .
5.8. Fact. If  T  : C ! D  is a special spherical functor, applying  ssCn+2 to *
 *the

n-th k-invariant  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 Pn-1X

                            |                    |
                            |                    |
                            fflffl|Tkn           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),  Mn-1  := T Pn-1X,  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:


                                          Tkn-1   D
   ssDn+2E_____//_ssDn+1Mn____//ssDn+1Mn-1____//ssn+1E____//ssDnMn____//ssDnMn-1

      |~=           |~=             |~=           ~=|        |~=           ~=|
      |             |               |             |          |             |
      fflffl|       fflffl|O        fflffl|sXn    fflffl|hXn 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 k-invariant 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 kn-1  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"n-1X, while*
 * for

the algebraic cases of x2.11(i-ii), 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 :  n-1X ! 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 skn-1 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 n-th 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|~=      fflffl|0      |fflfflO        fflffl|       f*
 *flffl|
    0 _______//_ssDn+2T ^PnX_// n+1P^nX____//_ssCn+1^PnX__//_ssDn+1T ^PnX_//_ n*
 *P^nX


so that:                          8

                                  ><ssDkT X    for k   n + 1,

(5.22)               ssDkT ^PnX ~=   n+1X      for k = n + 2,
                                  >:
                                     k-1P^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 n-th modified k-invariant 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 k-invariant 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 k-invariant ^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 self-equivalences 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 self-map  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 well-defin*
 *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 self-equivalence

g : Pn+1T W ! Pn+1T W ,  according to Fact 5.26.  But such a g induces a canoni*
 *cal

self-equivalence  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*|
                          fflffl|kn           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

 C-algebra, 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(n-1)    p(0)
(6.2)        . .-.--!  X^<n + 1> --!  ^X<n> ---!  . . .--!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<n>.

   At the n-th stage  (n   0), we have constructed  ^X<n>  as our candidate for*
 *  ^PnX

-  so in particular if we let  X<n> := PnX^<n>,  (our candidate for the ordinar*
 *y n-th

Postnikov section of  X),  then  T X<n>  satisfies (5.7),  T ^X<n>  satisfies (*
 *5.22),  and

of course  ^X<n> = Pn+1X^<n>.

   Assume also, as part of our inductive hypothesis, a given weak equivalence:


(6.3)                       ^ae(n): Pn+1G '-!Pn+1T ^X<n>.
   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<n>.  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^<n>  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)   n+3
(6.5)                On := kTn+1  O ae   2 H   (G;  n+1X^<n>) ,


 and the different liftings are classified by  Hn+2(G;  n+1X^<n>).


6.6. Constructing  X<n + 1>. The next step is to choose a cohomology class  ^kn

in  Hn+2(X^<n>; 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|                  |_
                    fflffl|p^(n)      fflffl|^kn          |fflffl
                X<n + 1> _________//_^X<n>________//_EC (Kn+1 , n + 2)

                    |                 |                   |
                    |                 |                   |j*
                    fflffl|           fflffl|kn           |fflffl
                X<n + 1> _________//_X<n>___________//_____________________EC (*
 *J, n + 2)


for the bottom fibration sequence  X<n + 1> ! X<n> ! 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<n>.

This will be done in two steps:

   First, note that we obtain a commuting diagram:


                                  ae
           Pn+2G    _______________________________________-Pn+2Tp^X<n>pppp
               |     ppppppp                           *      |
               |        ppppppp                               |
                           pppppp                  "
               |       ae  j                       i*         |
         p(n+1)||             Pn+2T X<n + 1>                  ||p(n+1)
          G    |                    |                         |  TX^<n>
               |                    |p(n+1)                   |
               |                    | TX<n+1>                 |
               |?      f            |?             g          |?

           Pn+1G    __________-'Pn+1T X<n + 1> ___________-'Pn+1T ^X<n>
               |                    |                         |
               |                    | TX<n+1>                 |
               |                    |kn+1                     |
               |                    |?                        |
            G  |                                              | TX^<n>
           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<n> = ssDn+1T X



factors through  Cn+2 := Im ssDn+2ae,  so that the bottom triangle commutes.

   Since the natural K-invariant   kGn+1 is given, the other two k-invariants i*
 *n the

diagram above are determined by inverting the given homotopy equivalences  f  :

Pn+1G ! Pn+1T X<n + 1>  and  g : Pn+1G ! Pn+1T ^X<n>  (assuming all objects in
                                                                               *
 *   X^<n>
D are fibrant and cofibrant),  and letting  kTX<n+1>n+1:= q* O kGn+1O f-1  and *
 * kTn+1  :=

i* O kGn+1O g-1,  using Fact 3.7.

   Therefore, the map  ae : G ! Pn+2T ^X<n>  lifts to  ae : Pn+2G ! Pn+2T X<n +*
 * 1>

(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^<n + 1*
 *>, as well

as determining a lifting of ae to a weak equivalence a^e(n+1): Pn+2G ! Pn+2T ^X*
 *<n + 1>.

   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<n>  satisfying (6.3)for  n,  a necessary and suf*
 *ficient

condition for the existence of an  ^X<n + 1>  satisfying (6.3)for  n+1  is the *
 *vanishing

of  On 2 Hn+3(G;  n+1X^<n>).  The choices are classified by:


     o  Hn+2(G;  n+1X^<n>)  (distinguishing the liftings of  a^e(n)to  Pn+2T ^X*
 *<n>);

        and

     o  ^kn2 Hn+2(X^<n>; Kn+1 ), where Kn+1 := Coker ssn+2ae(n), up to self-hom*
 *otopy

        equivalences of  ^X<n>  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> =  n+1X^<n + 1> =  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 one-to-one 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 self-weak 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^<n>, ae0), where X^*
 *<n> 2

C  has  Pn+1X^<n> ' X^<n>  and  ae0 : Pn+1G ! Pn+1T ^X<n>  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:

                                           BFn-1
(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 Eilenerg-Ma*
 *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(a-b).  The analysis in Section 6 then simplifies considerably, in as m*
 *uch as the

categories of  C-algebras 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 (simply-connected) rational space  G 2 TQ,  for each Q-v*
 *ector

space ssnG,  we need an abelian group A = ssnX  such that A   Q ~=ssnG,  and th*
 *en

lift the rational k-invariants 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 well-know*
 *n fact

that  X '  Y  is a loop space if and only if its k-invariants 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 k-invariants 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. 564-567.
[AC]    M. Arkowitz & C.R. Curjel, "The Hurewicz homomorphism and finite homoto*
 *py invari-
        ants", Trans. AMS 110 (1964), pp. 538-551.
[Ba1]   H.-J. Baues, "Relationen f"ur prim"are Homotopieoperationen und eine ve*
 *rallgemeinerte
        EHP -Sequenz", Ann. Sc. 'Ec. Norm. Sup. 8 (1975), pp. 509-533.
[Ba2]   H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, Gruyte*
 *r Expositions
        in Mathematics 2, Walter de Gruyter, Berlin-New 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,
        Berlin-New York, 1999.
[Be]    M. Benkhalifa, "Sur le type d'homotopie d'un CW-complexe", Homology, Ho*
 *motopy Appl.
        5 (2003), pp. 101-120.
[BH]    I. Berstein & P.J. Hilton, "On suspensions and comultiplications", Topo*
 *logy 2 (1963),
        pp. 73-82.
[Bl1]   D. Blanc, "New model categories from old", J. Pure & Appl. Alg. 109 (19*
 *96), pp. 37-60.
[Bl2]   D. Blanc, "CW simplicial resolutions of spaces, with an application to *
 *loop spaces", Top.
        & Appl. 100 (2000), pp. 151-175.
[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. 857-892.
[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. 1001-1053.
[BK]    A.K. Bousfield & D.M. Kan, Homotopy Limits, Completions, and Localizati*
 *ons, Springer
        Lec. Notes Math. 304, Berlin-New York, 1972.
[D]     A. Dold, "Homology of symmetric products and other functors of complexe*
 *s", Ann. Math.,
        Ser. 2, 68 (1958), pp. 54-80.
[DK1]   W.G. Dwyer & D.M. Kan, "A classification theorem for diagrams of simpli*
 *cial sets", Topol-
        ogy 23 (1984), pp. 139-155.
[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. 139-146.
[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. 137-152.
[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. 167-188.
[G]     T. Ganea, "A generalization of the homology and homotopy suspensions", *
 *Comm. Math.
        Helv. 39 (1965), pp. 295-322.
[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. 151-200.
[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 Gruppen-Mannigfaltkeiten und ihre Ver*
 *allgemeinerun-
        gen", Ann. Math. (2) 42 (1941), pp. 22-52.
[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 K-Theory, Contemp. Math. 346, AMS, Providence, RI 2004, p*
 *p. 305-319.
28                                 DAVID BLANC


[Mc]    S. Mac Lane, Homology, Springer-Verlag Grund. math. Wissens. 114, Berli*
 *n-New York
        1963.
[Man]   M.A. Mandell, "E1 -algebras and p-adic homotopy theory", Topology 40 (2*
 *001), pp.. 43-
        94.
[May]   J.P. May, Simplicial Objects in Algebraic Topology, U. Chicago Press, C*
 *hicago-London,
        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*
 *9-277
[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. 65-8*
 *7.
[R]     C.A. Robinson, "Obstruction theory and the strict associativity of Mora*
 *va K-theories", 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. 258-294.
[Sm]    J.R. Smith, "Topological realizations of chain complexes I: the general*
 * theory", Top. &
        Appl. 22 (1986), pp. 301-313.
[Sp]    J. Spali'nski, "Stratified model categories", Fund. Math. 178 (2003), p*
 *p. 217-236.
[Sta]   J.D. Stasheff, "Homotopy associativity of H-spaces, I,II", Trans. AMS 1*
 *08 (1963) pp.
        275-292, 293-312.
[Ste]   N.E. Steenrod, "The cohomology algebra of a space", Ens. Math. 7 (1961)*
 *, pp. 153-178.
[Sto]   C.R. Stover, "A Van Kampen spectral sequence for higher homotopy groups*
 *", Topology
        29 (1990), pp. 9-26.
[Sug]   M. Sugawara, "A condition that a space is group-like", Math. J. Okayama*
 * U. 7 (1957),
        pp. 123-149.
[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, 4-dimensional polyhedra", Comm.*
 * Math. Helv.
        22 (1949), pp. 48-92.
[W2]    J.H.C. Whitehead, "On the realizability of homotopy groups", Ann. Math.*
 * (2) 50 (1949),
        pp. 261-263.
[W3]    J.H.C. Whitehead, "The secondary boundary operator", Proc. Nat. Acad. S*
 *ci. USA 36
        (1950), pp. 55-60.
[W4]    J.H.C. Whitehead, "A certain exact sequence", Ann. Math. (2) 52 (1950),*
 * pp. 51-110.
[W5]    J.H.C. Whitehead, "Simple homotopy types", Amer. J. Math. 72 (1952), pp*
 *. 1-57,


   Department of Mathematics, University of Haifa, 31905 Haifa, Israel
   E-mail address: blanc@math.haifa.ac.il