BROWN REPRESENTABILITY FOR SPACE-VALUED FUNCTORS
BORIS CHORNY
Abstract.In this paper we prove two theorems which resemble the classic*
*al cohomo-
logical and homological Brown representability theorems. The main diffe*
*rence is that our
results classify small contravariant functors from spaces to spaces up *
*to weak equivalence
of functors.
In more detail, we show that every small contravariant functor from s*
*paces to spaces
which takes coproducts to products up to homotopy and takes homotopy pu*
*shouts to
homotopy pullbacks is naturally weekly equivalent to a representable fu*
*nctor.
The second representability theorem states: every contravariant conti*
*nuous functor
from the category of finite simplicial sets to simplicial sets taking h*
*omotopy pushouts
to homotopy pullbacks is equivalent to the restriction of a representab*
*le functor. This
theorem may be considered as a contravariant analog of Goodwillie's cla*
*ssification of
linear functors [12].
1.Introduction
The classical Brown representability theorem [3] classifies contravariant h*
*omotopy func-
tors F :Sop ! Setsfrom the category of spaces to the category of sets satisfyi*
*ng Milnor's
wedge axiom (W) and Mayer-Vietoris property (MV).
` Q
(W): F ( Xi) = F (Xi);
*
* A __//B
(MV): F (D) ! F (B)xF(A)F (C) is surjective for every homotopy pushout square*
* fflffl|fflffl|.
*
* C __//D
In this paper we address a similar classification problem, but the functors*
* we want to
classify are homotopy functors from spaces to spaces, satisfying (hW) and (hMV*
*), the
higher homotopy versions of (W) and (MV).
` Q
(hW): F ( Xi) ' F (Xi);
F (D)__//F (B) *
* A __//B
(hMV): fflffl|fflffl|is a homotopy pullback for every homotopy pushout squa*
*re fflffl|fflffl|.
F (C)__//F (A) *
* C __//D
Functors F :Sop ! S satisfying (hW) and (hMV) are called cohomological in t*
*his paper
(note that any functor satisfying (hMV) is automatically a homotopy functor; c*
*f 3.1).
Perhaps this terminology is not quite adequate, but we consider it as a tempor*
*ary notation,
since we are going to prove that these functors are equivalent to representabl*
*e functors, so
no alternative name is required.
___________
Date: July 6, 2007.
1
2 BORIS CHORNY
We should mention right away, that by spaces we always mean simplicial set in*
* this
paper. The results are formulated for unpointed spaces, but they remain valid *
*in the
pointed situation too.
The classical cohomological Brown representability theorem ensures that every*
* con-
travariant homotopy functor satisfying (W) and (MV) is representable on the hom*
*otopy
category. In this work we show that every space-valued functor satisfying (hW) *
*and (hMV)
is naturally weakly equivalent to a functor representable in the enriched sense*
* on the cat-
egory of spaces.
Note however, that neither our theorem implies Brown representability, nor th*
*e converse.
We assume stronger (higher homotopy) conditions about the functor, but we also *
*obtain
an enriched representability result.
Nevertheless, our result has a natural predecessor from the Calculus of homot*
*opy func-
tors. Goodwillie's classification of linear functors [12] is related to the hom*
*ological Brown
representability in the same way as our representability theorem related to the*
* cohomolog-
ical Brown representability.
The second classification result proven in this paper is "essentially equival*
*ent" to Good-
willie's classification of finitary linear functors. The difference is that we *
*prove a higher
homotopy version of the homological Brown representability representability in *
*its con-
travariant form: every cohomological functor from the compact spectra to abelia*
*n groups
is a restriction of a representable functor. We consider a non-stable version o*
*f this state-
ment: every contravariant homotopy functor from finite spaces to spaces satisfy*
*ing (hMV)
is a equivalent to a restriction of a representable functor. Such functors are *
*called homo-
logical.
Although there is no direct implications between our theorem and Goodwillie's*
* classi-
fication of linear functors, there is an additional feature that our results sh*
*are. In both
cases every small functor may be approximated by an initial, up to homotopy, re*
*pre-
sentable/linear functor. Small functors form a considerable subcollection of al*
*l functors,
so that the full subcategory of small functors is locally small. We discuss the*
*m in more
detail in Section 2. This property allows us to hope that the representability *
*theorem for
contravarian functors may be interpreted in terms of the Calculus of Functors. *
*Homotopy
Calculus is suited well for the study of covariant functors, while Embedding Ca*
*lculus pro-
vides a similar machinery for the study of contravariant functors with small do*
*mains (the
category of open sets of a manifold). We expect that there exists a calculus ma*
*chine for
contravariant functors, which generalizes Embedding Calculus.
The method of proof of our results deserves a comment. Contravariant functors*
* satisfying
(hW) and (hMV) are represented first as local objects with respect to certain c*
*lass of maps
in the category of small functors. Then the classification of objects with this*
* local property
in our model category is performed in terms comparison with a Quillen equivalen*
*t model,
in which it is easier to perform the required localization.
Goodwillie has certainly used similar considerations (at least implicitly) in*
* [13] in order
to classify homogeneous functors. But he has worked on the level of homotopy ca*
*tegories.
For the first time the idea that linear functors (or more generally n-excisive *
*functors) may
be viewed as local objects was spelled out by W. Dwyer [8]. In [1] we have tri*
*ed (with
BROWN REPRESENTABILITY FOR SPACE-VALUED FUNCTORS 3
G. Biedermann and O. R"ondigs) to enhance Goodwilie's proof by the model-catego*
*rical
machinery. The goal was partially fulfilled, but we have still used many of Goo*
*dwillie's
results in order to complete the classification on the level of model categorie*
*s.
In this work we develop tools which allow for the independent approach to cl*
*assification
theorems on the level of models. We prove a statement, which is very similar t*
*o the
classification of finitary linear functors and prove a new result, not related *
*immediately to
the Calculus of Functors.
1.1. Acknowledgment. We thank Amnon Neeman for numerous helpful conversation,
which led to the results in this paper. We also thank Tom Goodwillie for helpfu*
*l remarks
about the early version of this paper.
2. Preliminaries on small functors
The object of study of this paper is homotopy theory of contravariant functo*
*rs from the
category of spaces S to S. The totality of these functors does not form a categ*
*ory in the
usual sense, since the natural transformations between two functors need not fo*
*rm a set
in general, but rather a proper class. We are willing to be satisfied with a tr*
*eatment of
a reasonable subcollection of functors, a subcollection which does form a categ*
*ory. The
purpose of this section is to describe such a subcollection.
Definition 2.1. Let D be a (not necessarily small) simplicial category. A func*
*tor X :
D ! S is representable if there is an object D 2 D such that X is naturally equ*
*ivaleent to
RD , where RD (D0) = hom 0 e
D (D, D ). A functor Xe:C ! S is called small if Xeis a*
* small
weighted colimit of representables.
Remark 2.2. Since the category of small functors is tensored over simplicial se*
*ts, the small
weighted colimit above may be expressed as a coend of the form
Z I2I
(1) RF IG = RF(I) G(I),
where I is a small category and F :I ! D, G: I ! S are functors. Here RF :Iop !*
* SD
assigns to I 2 I the representable functor RF(I):D ! S. For the general treatm*
*ent of
weighted limits and colimits see [16].
Since the simplicial tensor structure on the category of small fuctors SD is*
* given by the
objectwise direct product, we will use X x K to denote tensor product of X 2 S*
* with
K 2 S. e e
The above coend is the (enriched) left Kan extension of the functor G over t*
*he functor
F . Using the transitivity of left Kan extensions, it is easy to see that the f*
*ollowing four
conditions are equivalent [16, Prop. 4.83]:
o X :D ! S is a small functor,
o ethere is a small simplicial category I and a functor G : I ! S, such th*
*at X is
isomorphic to the left Kan extension of G over some functor I ! D, *
* e
o there a small simplicial subcategory i: D0 ! D and a functor G : D0 ! S,*
* such
that X is isomorphic to the left Kan extension of G over i, and
e
4 BORIS CHORNY
o there is a small full simplicial subcategory i: DX ! D such that X is iso*
*morphic
to the left Kan extension of i*(X ) over i. e e
e
If D 2 D and Y is a functor D ! S, then by Yoneda's lemma the simplicial clas*
*s of
natural transfoermations RD ! Y is Y(D); in particular, this simplicial class i*
*s a simplicial
set. It follows easily that ifeX isea small functor D ! S, then the natural tra*
*msformations
X ! Y also form a simplicial seet (this also follows from 2.2 above and the adj*
*ointness
eproeperty of the left Kan extension). In particular, the collection of all sma*
*ll functors is a
simplicial category.
Remark 2.3. M.G. Kelly [16] calls small functors accessible and weighted colimi*
*ts in-
dexed. He proves that small functors form a simplicial category which is close*
*d under
small (weighted) colimits [16, Prop. 5.34].
In order to do homotopy theory we need to work in a category which is not onl*
*y co-
complete, but also complete (at least under finite limits). Fortunately, there *
*is a simple
sufficient condition in the situation of small functors.
Theorem 2.4. If D is cocomplete, then the category SD of small functors D ! S *
*is
complete.
Remark 2.5. There is a long story behind this theorem. P. Freyd [11] introduced*
* the notion
of petty and lucid set-valued functors. A set-valued functor is called petty if*
* it is a quotient
of a small sum of representable functors. Any small functor is clearly petty. *
* A functor
F :A ! Setsis called lucid if it is petty and for any functor G: A ! Setsand an*
*y pair
of natural transformations ff, fi :G ' F , the equalizer of ff and fi is petty.*
* Freyd proved
[11, 1.12] that the category of lucid functors from Aopto Sets is complete if a*
*nd only if A
is approximately complete (that means that the category of cones over any small*
* diagram
in A has a weakly initial set). J. Rosick'y then proved [18, Lemma 1] that if t*
*he category
A is approximately complete, a functor F :Aop ! Setsis small if and only if it *
*is lucid.
Finally, these results were partially generalized by B. Day and S. Lack [17] to*
* the enriched
setting. They show, in particular, that the category of small V-enriched functo*
*rs Kop! V
is complete if K is complete and V is a symmetric monoidal closed category whic*
*h is locally
finitely presentable as a closed category. This last condition is certainly sat*
*isfied if V = S.
3.Model categories and their localization
The main technical tool used in the prove of the classification theorem is th*
*e theory of
homotopy localizations. More specifically, weoapplypcertain homotopy localizati*
*ons in the
category of small contravariant functors SS , or in a Quillen equivalent model*
* category of
maps of spaces with the equivariant model structure [6, 4].
Before proving the main classification result, we suggest the following alter*
*native char-
acterization of functors satisfying (hW) and (hMV) as local objects with respec*
*t to some
class of maps.
BROWN REPRESENTABILITY FOR SPACE-VALUED FUNCTORS 5
3.1. Homotopy functors as local objects. First of all every cohomological funct*
*or F
is a homotopy functor, i.e., F (f): F (B) ! F (A) is a weak equivalence for eve*
*ry weak
equivalence f :A ! B. Denote by F1 the class of maps between representable func*
*tors
induced by weak equivalences:
F1 = {f*: RA ! RB|f :A ! B is a w.e.},
where RA denotes the representable functor RA = S(-, A).
Yoneda's lemma implies that F1-local functors are precisely the fibrant homo*
*topy func-
tors.
3.2. Cohomology functors as local objects. Given a homotopy functor F , it suff*
*ices to
demand two additional properties for the functor F to be cohomological: F must *
*convert
coproducts to products up to homotopy and it also must convert homotopy pushout*
*s to
homotopy pullbacks. Yoneda's lemma and the standard commutation rules of vario*
*us
(ho)(co)limits with hom (-, -) implies that both properties are local with resp*
*ect to the
following classes of maps:
na fi o
F2 = RXi ! R` Xififi8{Xi}i2I2 SI
and
8 0 1 fi 9
< RA _//_RC fifiA//_C =
F3 = hocolim@ fflffl|A - ! RD fifflffl|fflffl|- homotopy pushoutSin.
: R fifi_//_ ;
B B D
Objects which are local with respect to F = F1 [ F2 [ F3 are precisely the f*
*ibrant
homotopy functors.
Remark 3.1. For any map f :A ! B the following commutative square is a homotopy
pushout and/or homotopy pullback iff f is a weak equivalence:
A ______A
|| |
|| |f
|| fflffl|
A _f__//_B.
Therefore, any functor satisfying (hMV) is automatically a homotopy functor, so*
* it suffices
to invert F0= F2[ F3, but it does not simplify anything in our proof .
Remark 3.2. The indexing category I used to describe F2 is a completely arbitra*
*ry small
discrete category. In particular I can be empty. This implies that the map ; *
*! R; is
in F2. In other words, if F is a cohomological functor, then F (;) = *. This pr*
*operty is
analogous to the requirement that every linear functor is reduced in homotopy c*
*alculus.
6 BORIS CHORNY
Remark 3.3. Since homological functors are defined on the category of finite si*
*mplicial sets,
we need to adjust the definition of F3.
8 0 1 fi 9
< RA _//_RC fifiA//_C =
F03= hocolim@ fflffl|A - ! RD fifflffl|fflffl|- homotopy pushout.in Sfin
: R fifi_//_ ;
B B D
op
Then the reduced homological functors in SSfin(with the projective model struct*
*ure) are
precisely the functors which are local with respect to F03[ {; ! R;}
3.3. Localization. Of course, in order to classify objects of a model category *
*with certain
property it is not enough to nominate a convenient class of maps F, so that our*
* objects will
be F-local. One must also make sure that there exists a localization of the mod*
*el structure
with respect to F and the class of objects we a willing to classify will be rep*
*resented, up
to homotopy, by the elements of the homotopy category of the localized model ca*
*tegory.
The localization procedure is not always a routine. For example, the class F *
*of maps is
so `big' that we are unable to perform the localization in the original model c*
*ategory at the
moment. However, there exists a Quillen equivalent model for our model category*
* in which
we are able to perform the localization with respect to the class of maps corre*
*sponding to
F under the Quillen equivalence. This Quillen equivalence [4] is
op 2 O
(2) | - |2:SS AE Seq:(-) .
We hope to learn, in the future, how to localize with respect to F in the ori*
*ginal category
of small functors, since this will allow for generalizations to other functor c*
*ategories, which
do not have a more convenient model, e.g., the category of covariant functors f*
*rom spaces
to spaces.
We need to localize the model category S2eqwith respect to the class of maps *
*|F|2 =
|F1|2[ |F2|2[ |F3|2, where
aeA Bfifi oe
|F1|2 = #- !#fifiA ! B is a w.e. inS,
( * * fi )
a Xi ` Xififi
|F2|2 = #-! # fi8{Xi}i2I2 SI ,
* * fi
and
8 0 1 fi 9
>>> A C fifi >>
>>< BB #__//#C fi >>>
* *C DfiA__//C =
|F3|2 = hocolimBBfflffl|CC-!#fififflffl|fflffl|is a homotopySpushout.in
>>> B C *fi__// >>
>>: @ B# A fifiBD >>>;
* fi
Remark 3.4. The realization functor | - |2 may be viewed as a coend Inc S-, whe*
*re
Inc:S = O2 ,! S2 is the fully-faithful embedding of the subcategory of orbits [*
*4]. There-
fore, computing the realization of the representable functors is just the evalu*
*ation of Inc
at the representing object, since the dual of the Yoneda lemma applies.
BROWN REPRESENTABILITY FOR SPACE-VALUED FUNCTORS 7
We construct the localization of S2eqwith respect to F using the Bousfield-F*
*riedlander
localization technique. The role of the Q-construction will be taken by the cou*
*nit of the
adjunction
(3) L: S2eqo S :R,
`A' A `A' A
where L # = A and R(A) =#. We could have just said that Q # =#, but we would
B * B *
like the reader to notice adjunction (3), which forms also a Quillen pair and w*
*ill become a
Quillen equivalence after we perform the localization with respect to Q.
In order to apply the Bousfield-Friedlander theorem [2, A.7] we have to veri*
*fy that
(1) Q preserves weak equivalences;`'
A A
(2) Q is a coaugmented j :#!# , homotopy idempotent functor;
B *
(3) The resulting localized category must be right proper.
The verifications are routine.
Remark 3.5. This localization of the model category structure on S2eqhas the fo*
*llowing
remarkable property: every orbit is @0-small with respect to the cofibrations [*
*6]. In other
words, we have found a model of the category of spaces, in which every homotopy*
* type
is represented by an @0-small, with respect to cofibrations, object. This concl*
*usion seems
contra-intuitive in view of Hovey's proof that every cofibrant and finite relat*
*ive to cofibra-
tions object in a pointed finitely generated model category C is small in Ho(C)*
* [15, 7.4.3].
However, there is no contradiction with our result, since the localized model c*
*ategory
S2eqis very far from being finitely generated (even though the original model c*
*ategory was
pretty close to be finitely generated - it could be classified as class-finitel*
*y generated model
category, but the localization process significantly changes the class of trivi*
*al cofibrations)
We have obtained so far a new model structure on S2eqthat is Quillen equival*
*ent to
the usual model structure on S. It is easy to verify that all the maps in |F|2*
* are weak
equivalences in the new model structure. It remains to show that the class of |*
*F|2-local
objects coincides with the class of Q-local objects. We will show it in the nex*
*t section.
4.Technical preliminaries
Recall that we are going to prove two theorems in this paper. Theorem 5.1 c*
*lassifies
cohomological functors and Theorem 5.8 classifies homological functors. However*
* the tech-
nicalities behind the proofs are very similar. Therefore, while we are heading *
*towards the
proof of Theorem 5.1 first, we indicate little adjustments required to adapt th*
*e argument
for the proof of Theorem 5.8. Let us denote by E the class of maps |F|2,`as req*
*uired'for
;
classification of cohomological functors, and by E0 the union |F03|2 [ ; !# *
*needed to
*
classify the reduced homological functors.
8 BORIS CHORNY
A
The Q-local objects are precisely the orbits # with A fibrant. We need to sh*
*ow that
*
every object in S2eqis E-local equivalent to an orbit.
Every 2-diagram may be approximated by an I-cellular diagram [5], up to an eq*
*uivariant
weak equivalence [7], where
ae@ n fifi A oe
I = # TfifiT=#, A 2 S.
n e e *
Therefore, it suffices to show that every I-cellular diagram is E-equivalent to*
* an orbit. We
are going to prove it by cellular induction, but we precede the proof with the *
*following
lemma, which says that the basic building blocks of cellular complexes are E-eq*
*uivalent to
orbits.
` A' ` 0'
E A
Lemma 4.1. For every A 2 S, n 0, there exists A02 S such that @ n # ' #*
* .
* *
Proof.We`will show that A0 ' @ n A. The_proof!is by induction on n. For n =*
* 0
A' ` A' E ` ;' @ 0 A ` ;'
we have @ 0 # = ; # = ; ' # = # , since the map ; ! # is
* * * * *
in |F2|2 E. Alternatively, if one is willing to exclude F2 from F, then for t*
*he base of
induction it suffices to assume that the cohomology functor F is reduced, i.e.,*
* F (;) = *;
cf. Remark 3.2. In other words the basis for induction`holds'for`E0'equivalence*
*s as well.
A E @ n A
Suppose the statement is true for n, i.e., @ n # ' # ; we need to s*
*how it
* *
for n + 1.
0 1 0 1
AO " n A @ n A A
B @ n # ____//_ #C B # _____//#C
` A' BB " *` *CC BB * * CC
E B | C
@ n+1 # ' colimBB || CC' hocolimB | C '
* BB fflffl| CC BB fflffl| CC
@ A A @ A A
n # #
* *
0 1 0 O " 1
@ n A n A B @ n"A`__//_ nCA
BB # ____//_# C colimB@ fflffl| CA
B * * CC E n A _( n` @ n n) A! _@ n+1 A!
hocolimBB || CC' # ' # ' # ,
BB fflffl| CC * * *
@ n A A
#
*
where the first E-equivalence is induced by E-equivalences of all vertices of r*
*espective homo-
topy pushouts. (If we will map both homotopy pushouts into an arbitrary E-local*
* object W ,
we will obtain a levelwise weak equivalence of homotopy pullback squares of spa*
*ces). Thfe
BROWN REPRESENTABILITY FOR SPACE-VALUED FUNCTORS 9
last E-equivalence is induced"by the map from |F3|2 E corresponding to the ho*
*motopy
pushout square: @ n"AO_____//_` n A
fflffl| `fflffl|
n A __//_( n @ n n) A.
The above argument does not change if we consider E0 instead of E.
`A'
Proposition 4.2. Every I-cellular complex X 2 S2 is E-equivalent to an orbit #*
* for
e *
some A.
Proof.Every I-cellular complex X has a decomposition into a colimit indexed by *
*a cardinal
~: e
X = colim(X 0! . .!.Xa ! Xa+1 ! . .).,
e a<~ e e e
where X a+1is obtained from X aby attaching a cell:
e e
A
@ n # _____//_Xa".
"*` e| `
| |
| |
fflffl| |
A fflffl|
n # ____//_Xa+1
* e
Ca
Assuming, by cellular induction, that X ais E-equivalent to an orbit # , we *
*notice, by
e ` ' ` ' *
A @ n A A A
Lemma 4.1, that @ n # is E-equivalent to # , and n # ' # , so all the
* * * *
A0
vertices of the homotopy pushout above are E-equivalent to orbits # for some A0*
*. We
*
Ca+1
conclude that X a+1is E-equivalent to an orbit # , where Ca+1 is the homotopy *
*pushout
e *
(A @ n A ! Ba), similarly to the argument of Lemma 4.1.
We obtain the following commutative ladder:
" O " O " O "
X0O____//_._._.//_Xa__//_Xa+1___//... .
e e e
fflOE|| |fflOE| fflOE||
fflffl| fflffl| fflffl|
C0 Ca Ca+1
# _____//._._._//#____//# _____//. . .
* * *
E *
* Ca
Taking homotopy colimit of the upper and the lower rows we find that X ' hoc*
*olima<~#,
e *
* *
since if we will map both homotopy colimits into an arbitrary E-local diagram W*
* , we will
obtain a weak equivalence between the homotopy inverse limits. f
10 BORIS CHORNY
_ !
Ca C0 f0 Ca faCa+1fa+1
Finally, hocolima<~ #= hocolima<~ # -! . .-.! #-! # - ! . . .may be repre-
* * * *
sented as a homotopy pushout as follows:
0 ` ' ` ' ` 1
` Ca ` ` Ca 1 f//`Ca
BB #* #* _____ #*CC
Ca BB | CC
hocolima<~# ' hocolimB r| C ,
* BB fflffl| CC
@ ` Ca A
#
*
`
where f = a<~fais the shift map and r is the codiagonal. Observe that the hom*
*otopy
pushout above is weakly equivalent to the infinite telescope construction.
All vertices of the homotopy pushout above are E-equivalent to certain orbits*
* through
the respective E-equivalences from |F2|2. Testing by mapping into an arbitrary*
* E-local
diagram W , we find that the homotopy pushout above is E-equivalent to the homo*
*topy
pushout fof the respective orbits.
A
The latter pushout is E-equivalent to an orbit #through an E-equivalence from*
* |F3|2.
*
Remark 4.3. The only place in the proof where we use the property that infinite*
* coproducts
are converted into`products'by cohomological functors is the last argument. Alt*
*ernatively,
;
for E0= |F03|2[ ; !# , as required by classification of homological functors, *
*then we may
* _ !
hocolima<~Ca
conclude that X is E0-equivalent to # in the model category on S2 ge*
*nerated
e *
aeAfifi oe
by the orbits #fifiA 2 Sfin, since these orbits are @0-small. In the case of*
* homological
*
functorsothepsame conclusion is easier to make in the Quillen-equivalent model *
*category
SSfin, but we would like to stress that the proofs differ only in few places an*
*d also to make
the exposition shorter.
5. Main results
We are ready now to prove the representability theorems.
Theorem 5.1. Let F :Sop ! S be a small, homotopy functor converting coproducts *
*to
products, up to homotopy, and homotopy pushouts to homotopy pullbacks. Then the*
*re ex-
ists a fibrant simplicial set Y , such that F (-) ' S(-, Y ). Moreover, for ev*
*ery functor
G: Sop ! S there exists an approximation of G by a universal, up to homotopy, c*
*ohomo-
logical functor, which is not necessarily representable but is equivalent to on*
*e, i.e., there
exists a natural transformation fl :G ! ^G' RA, such that any other map G ! RB *
*factors
through fl and the factorization is unique up to simplicial homotopy for every *
*representable
functor RB with fibrant simplicial sets A and B.
BROWN REPRESENTABILITY FOR SPACE-VALUED FUNCTORS 11
Proof.We have proven so far that that the Q-localization constructed in 3.3 is *
*essentially
the localization with respect to E: E-equivalences are obviously Q-equivalences*
*, and the
inverse inclusion follows from Proposition 4.2, which says, in particular, that*
* every E-local
object is also Q-local, hence any Q-equivalence is also an E-equivalence.
The Quillen equivalence (2) gives rise to the equivalence of homotopy functi*
*on complexes
of the model categories [9], therefore the realization of a cohomological funct*
*or F (i.e., F-
local functor) after a fibrant replacement in S2eqbecomes E-local. Proposition *
*4.2 implies
A
that |F |2!"d|F |2'# for some A 2 S. Considering the adjoint map we obtain the*
* week
*
equivalence:
i jO `A' O
F -"! d|F |2' # = RA = S(-, A).
*
Let us point out that this conclusion could have been obtained without the u*
*se of
the Quillen equivalence (2): Proposition 4.2 could be proven in the category o*
*f small
contravariant functors by exactly the same argument, than the conclusion of the*
* theorem
would follow from the F-local Whitehead theorem [14].
Now we will take advantage of the localized model structure constructed in 3*
*.3. Given
G: Sop ! S, apply first the realization`functor'and than the fibrant replacemen*
*t in the
A
localized model category |G|2,"!d|G|2' # . Passing to the adjoint map we obt*
*ain the
*
required approximation of G by a cohomological functor:
`A' O
G -! (|dG|2)O ' # = RA.
*
The verification of the universal property of the constructed map is performed *
*by passing
through adjunction to the category of maps of spaces and constructing the lift *
*of the trivial
cofibration to the Q-local object:
`B' O B
G _______//RB = # |G|2____//_"#
| * _B*B__`__
| lO _::____ | ___O__
| _______ oo//o//////|fflOl__O____ooooo/o
| _______ | _____O
fflffl|____ fflffl|_fflfflfflfflO__
(|dG|2)O d|G|2```//*
Remark 5.2. It is possible to formulate a stronger universal property of the co*
*homological
approximation map: it should be initial with respect to maps into all F-local, *
*i.e., cohomo-
logical functors. In order to prove it, however, we would need to consider the *
*F-localized
model structure on the category of small contravariant functors. We indicate in*
* Remark 5.4
a way to construct this localizations.
Remark 5.3. There is a different, simpler, approach to the classification of co*
*homological
functors: given a fibrant simplicial cohomological functor G: Sop ! S (no need *
*to assume
12 BORIS CHORNY
that G is small), consider the natural map q :G(X) ! S(X, G(*)) obtained by adj*
*unction
from the natural map X = S(*, X) ! S(G(X), G(*)), which exists, in turn, since *
*G is
simplicial. The map q is an equivalence if X = *, which gives a basis for indu*
*ction on
the cellular structure of X similarly to Proposition 4.2. This approach is sim*
*pler, and
seemingly more general (works for all functors, not necessarily small), but it *
*does not allow
for generalizations, where the representing object is not so obvious. Another a*
*dvantage of
using model categories is that they help to prove the most interesting part of *
*our result
that any functor may be approximated by a cohomological functor. We owe this re*
*mark
to T. Goodwillie.
Remark 5.4. There is also a different way to prove our main result. Consider th*
*e projective
model structure on the category of small contravariant functors and let Q(G) = *
*q(Gb)
be a functorial fibrant replacement of Gocomposedpwith q from Remark 5.3. Sinc*
*e the
factorizations are not functorial in SS , the existence of the functorial fibr*
*ant replacement
requires an explanation. One way to construct it is to apply the realization fu*
*nctor and
than to take the fibrant replacement in the category of maps of spaces with the*
* equivariant
model structure followed by the orbit-poins functor, so Q(G) = q(|G|O2). This Q*
* satisfies
the conditions of Bousfield-Friedlander localization theorem, therefore we coul*
*d prove the
same result using the localized category of small contravariant functors. We ob*
*tain another
interesting model for the category of spaces, where every object has a represen*
*tative, which
is @0-small. However the proof of Proposition ??is easier in S2eqand we can not*
* avoid using
the category of maps completely, so we have chosen to present this approach.
Homological Brown representability for spaces is essentially Goodwillie's cla*
*ssification of
linear functors. We choose, however, to discuss the contravariant version of th*
*is theorem
in our work. Even though philosophically the two versions are the same, we are *
*doubtful
if such an implication exists, since there are several significant differences.*
* We repeat the
basic definitions first:
op
Definition 5.5. A small functor F 2 SSfinis called homological if F converts ho*
*motopy
pushouts of (finite) simplicial sets to homotopy pullbacks.
Let i: Sfin,! S be the fully faithful embedding.
Example 5.6. Any functor of the form X x i*S(-, Y )is homological, hence the ne*
*xt
definition.
Definition 5.7. A homological functor F is reduced if F (;) = *.
Theorem 5.8. Any reduced homological functor F :Sop ! S is weakly equivalent to*
* the
restriction a representable functor i*S(-, Y ) for some fibrantosimplicialpset *
*Y , unique up to
homotopy. Moreover, there exists for every functor G 2 SSfinan approximation G *
*! ^Gby
a reduced homological functor, which is initial beneath all maps into homologic*
*al functors.
Proof.The proof of this theorem is essentially the same as of Theorem 5.1. We *
*give a
sketch of the proof, pointing out the differences.
BROWN REPRESENTABILITY FOR SPACE-VALUED FUNCTORS 13
op
Consider the projective model structure on the category of functors SSfin. *
*If we will
localize this model category with respect to F0 = F03[ {; ! Sfin(-, ;)}, then w*
*e will
obtain a classification of reduced homological functors as local objects. We ha*
*ve to show
that every local object is equivalent to the restriction of a representableafun*
*ctor.efioe
A fi
Consider the model structure on S2 generated by the set of orbits O0= # fif*
*iA 2 Sfin
*
in the sense of Dwyer and Kan [10].opThis model category is Quillen equivalent *
*to the
projective model structure on SSfin[10].
Next, we perform the localization of the model structure on S2 generated by *
*O0 with
respect to the class of maps E0= F0. We could do it by the same method as in th*
*e previous
theorem, but the class of maps E0 is a small set in this situation, so nothing *
*prevents us
from applying one of the standard localization machines in the cofibrantly gene*
*rated model
category S2. Proposition 4.2 together with its adaptation to the model category*
* generated
byaO0einfRemarki4.3,oshowsethat the fibrant objects of this localization have t*
*he form
Afi
#fifiS 3 A - fibrant.
*
The conclusion of Theorem 5.1 is derived in exactly the same way.
Remark 5.9. Remarks 5.2, 5.3 and 5.4 apply without change.
If required, the same result may be reformulated for small contravariant fun*
*ctors from
spaces to spaces. We only need to introduce the concept of finitary functor:
Definition 5.10. A small functor F 2 SSop is called finitary if F is a left Kan*
* extension
from its restrictionoontopthe full subcategory of finite simplicial sets. The c*
*ategory of small
functors SS supports the finitary model structure: a natural transformation i*
*s a weak
equivalence or a fibration if it induces a weak equivalence or a fibration betw*
*een values of
the functors in finite simplicial sets. Cofibrant objects in the finitary model*
* structure are
finitary functors. The finitary model structure is Quillen equivalent to the pr*
*ojective model
structure on the category of contravariant functors from finite simplicial sets*
* to simplicial
sets.
Remark 5.11. Origin of this terminology and the discussion of the elementary pr*
*operties
of the finitary model structure on the category of small covariant functors may*
* be found
in [1, Section 9].
The definition of a small homological contravariant functor is the same: it *
*takes ho-
motopy pushouts of finite simplicial sets to homotopy pullbacks. In the finita*
*ry model
structure any reduced homological functor is equivalent to a representable func*
*tor. Rele-
vant approximation result also applies. The details are left to the interested *
*reader.
Finally we would like to point out one crucial difference between the classi*
*fication of
contravariant homological functors in this paper and Goodwillie's classificatio*
*n of linear
functors: it is usually not a simple task to compute the linear approximation o*
*f a functor,
whether in our framework the answer is ready - the functor represented in the v*
*alue of the
original functor in *.
14 BORIS CHORNY
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Centre for Mathematics and its Applications, Mathematical Sciences Institute,*
* Australian
National University, Canberra, ACT 0200, Australia
E-mail address: Boris.Chorny@maths.anu.edu.au