CALCULUS OF FUNCTORS AND MODEL CATEGORIES
GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
Abstract.The category of small covariant functors from simplicial sets to
simplicial sets supports the projective model structure [5]. In this pap*
*er we
construct various localizations of the projective model structure and al*
*so give
a variant for functors from simplicial sets to spectra. We apply these m*
*odel
categories in the study of calculus of functors, namely for classificati*
*on of poly
nomial and homogeneous functors. Finally we show that the nth derivative
induces a Quillen map between the nhomogeneous model structure on small
functors from pointed simplicial sets to spectra and the category of spe*
*ctra
with naction. We consider also a finitary version of the nhomogeneous
model structure and the nhomogeneous model structure on functors from
pointed finite simplicial sets to spectra. In these two cases the above *
*Quillen
map becomes a Quillen equivalence. This improves the classification of f*
*initary
homogeneous functors by T. G. Goodwillie [12].
1. Introduction
Calculus of homotopy functors applies to functors from spaces to spaces or s*
*pec
tra, which preserve weak equivalences. It interpolates between stable and unsta*
*ble
homotopy theory by analyzing carefully the rate of change of such functors. It *
*was
developed around 1990 by Thomas G. Goodwillie and has had spectacular appli
cations to geometric topology [10, 11] and homotopy theory [1]. Although at the
present time calculus of functors is a well developed and ramified theory, foun*
*da
tions of the subject remain technically involved. Part of the difficulty is the*
* lack of
a categorical framework. The problem is that the totality of functors from spac*
*es
to spaces does not form a legitimate category (the collections of morphisms need
not be small).
In the current work we introduce a categorical approach to the foundations of
the calculus of functors. We suggest to implement the adhoc machinery developed
by Goodwillie as a part of a model category structure, which is a standard tool*
* for
describing an abstract homotopy theory. In order to overcome the settheoretical
difficulties we consider only small functors from spaces to spaces or from spac*
*es to
spectra, i.e. the functors that are determined as a left Kan extension by their*
* values
on a small subcategory only. For technical reasons, we use simplicial sets inst*
*ead
of topological spaces. This is justified by Kuhn's overview article [15], where*
* first
steps to an axiomatization of the theory are taken.
The projective model structure was constructed in [5]. In this paper we pres*
*ent
several new model structures on the category of small functors, and each of the*
*se
reflects certain aspect of Goodwillie's calculus.
____________
Date: January 10, 2006.
1991 Mathematics Subject Classification. Primary 55U35; Secondary 55P91, 18G*
*55.
Key words and phrases. calculus of functors, small functors, homotopy functo*
*rs.
1
2 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
After necessary preliminaries on small functors in Section 2 we construct in
Section 3 a localization of the projective model structure such that the new fi*
*brant
objects are precisely the projectively fibrant homotopy functors. This is the s*
*tarting
point for calculus of functors, since Goodwillie's machinery is intended for ho*
*motopy
functors only. We make the interesting margin observation that any functor may
be approximated by a homotopy functor in a universal way. It is worth mentioning
that we do not have yet a general localization theory for model categories that*
* are
not cofibrantly generated. All localizations and colocalizations of model categ*
*ories
in this paper are constructed using the BousfieldFriedlander localization tech*
*nique,
which is restricted to produce proper model categories only.
In Section 4 we localize the homotopy model structure on the category of small
functors from spaces to spaces so that the new fibrant objects are precisely the
nexcisive fibrant homotopy functors. This result may be viewed as a classifica*
*tion
of npolynomial functors. Goodwillie's nth polynomial approximation Pn is equi*
*v
alent to a fibrant replacement in our nexcisive model structure. An immediate
advantage of having a model category structure is that the cofibrant replacement
(equivalent to Pn) is universal, up to homotopy, with respect to maps into arbi*
*trary
nexcisive functor. This is an improvement of Goodwillie's result, which verifi*
*es the
universal property only on the level of homotopy category [12, 1.8].
In the simpler category of functors from finite pointed spaces to all spaces,*
* Ly
dakis has constructed the homotopy model structure as well as the 1excisive (or
stable) model structure (see [17], as well as its generalization [6] to more ge*
*neral
model categories). Our work may be seen as a twofold generalization of this wo*
*rk,
since our results immediately apply to Lydakis' category. However, there are pl*
*enty
of interesting small functors which are not finitary in the sense that they are*
* de
termined by their values on finite spaces  for example, nonsmashing Bousfield
localizations.
In Section 5 we establish the stable projective, stable homotopy, and stable *
*n
excisive model structures for small functors from (pointed) spaces to spectra. *
*Then
we recall and adapt several important definitions in Section 6. In Section 7 we*
* colo
calize the stable nexcisive model structure in order to obtain the nhomogeneo*
*us
model structure. In this model structure, the bifibrant objects are precisely t*
*hose
projectively bifibrant homotopy functors which are nhomogeneous. This model
structure may also be considered as a way to classify the nhomogeneous functors
up to homotopy. T. Goodwillie has found another, simpler classification, but it
applies only for finitary nhomogeneous functors or for a restriction of an arb*
*itrary
functor to finite spaces. Any such functor is determined by its nth derivativ*
*e,
which is a spectrum with naction.
In the final Section 8, we strengthen Goodwillie's classification. We introdu*
*ce a
finitary version our nhomogeneous model structure and an nhomogeneous model
structure on the category of functors from pointed finite simplicial sets to sp*
*ectra,
and establish a Quillen equivalence between each of these model categories and *
*the
projective model structure on the category of spectra with naction.
2.Preliminaries on small functors
Let S denote the category of simplicial sets. The object of study of this pa*
*per is
homotopy theory of functors from simplicial sets to simplicial sets. The totali*
*ty of
these functors does not form a category in the usual sense  natural transforma*
*tions
CALCULUS OF FUNCTORS 3
between two functors need not form a set in general, but rather a proper class.*
* On
the other hand we are not eager to consider all such functors and would be sati*
*sfied
with a treatment of a sufficiently large subcategory, which will be a category *
*in
the usual sense (with small homsets). The purpose of this section is to descri*
*be a
satisfactory subcategory.
Definition 2.1. Let C be a (not necessarily small) simplicial category. A funct*
*or
Xe:C ! S is called small if Xeis a small weighted colimit of representable func*
*tors.
We denote the category of small functors as SC.
Remark 2.2. If C is a small category then any functor from C to S is small. It
is our point of view that small functors are the appropriate generalization, he*
*nce
the notation. M.G. Kelly [14] calls small functors accessible and weighted coli*
*mits
indexed. We emphasize that we are working in the enriched context: the colimits
and left Kan extension are understood in the enriched sense.
Theorem 2.3 (For the proof see Prop. 4.83 of [14]). A functor is small if and o*
*nly
if it is a left Kan extension from its restriction to a full small subcategory.
Remark 2.4. Kelly proves also that small functors form an Scategory [14, 4.41],
which is closed under small (weighted) colimits [14, Prop. 5.34]. That allows u*
*s to
talk about simplicial function spaces hom (Xe, Ye) for any small functors Xeand*
* Ye
(in fact it suffices to demand that only Xeis small). Existence of weighted col*
*imits
implies, in particular, that SC is tensored over S, as the functor  K may be
viewed as a colimit over the trivial category weighted by K 2 S. Another immedi*
*ate
corollary of [14, Prop. 4.83] is that small functors are Sfunctors, i.e. simpl*
*icial.
In order to initiate a discussion of homotopy theory we are bound to work in
a category which is not only cocomplete, but also complete (at least under fini*
*te
limits). It turns out that under some condition on C the category of small func*
*tors
SC is complete.
P. Freyd [9] introduced the notion of petty and lucid setvalued functors. A
setvalued functor is called petty if it is a quotient of a small sum of repres*
*entable
functors. Any small functor is clearly petty. A functor F :A ! Sets is called
lucid if it is petty and for any functor G: A ! Sets and any pair of natural
transformations ff, fi :G ' F , the equalizer of ff and fi is petty. P. Freyd p*
*roved [9,
1.12] that the category of lucid functors from Aop to Sets is complete if and o*
*nly if
A is approximately complete (that means that the category of cones over any sma*
*ll
diagram in A has a weakly initial set).
J. Rosick'y proved [19, Lemma 1] that if the category A is approximately com
plete, then a functor F :Aop ! Setsis small if and only if it is lucid.
These results provide a full answer for the question when the category of sm*
*all
setvalued functors from a large category is complete. Unfortunately this is n*
*ot
sufficient for doing homotopy theory. We are interested in simplicial functors *
*from
a large simplicial category to simplicial sets. These results were partly gener*
*alized
by S. Lack [16] to the enriched settings. Lack shows, in particular, that the c*
*ategory
of small functors from Kop to S is complete if K is a complete Scategory. Even
more generally, the same result holds if one replaces S by a symmetric closed
monoidal category V, which is locally finitely presentable asoapclosed category.
Existence of weighted limits implies, in particular, that SK is cotensored ov*
*er S,
as the functor ()K may be viewed as a limit over the trivial category weighted*
* by
4 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
K 2 S. The cotensor ()K is the right adjoint to the tensor functor  K by the
usual commutation rules of weighted limits with the mapping spaces [14, 3.8].
Lack's results allowouspto consider a model category structure on the categor*
*y of
small functors SK . The simplest model structure on a category of functors is *
*the
projective model structure: weak equivalences and fibrations are levelwise. Th*
*is
model structure was established in [5].
The weak equivalences and fibrations in the projective model structure are "d*
*e
tected" by the maps from the representable functors: RA = hom K(, A) is the
functor from Kop to S represented in A. The following lemma is crucial for the
proof of the existence of projective model structure.
Lemma 2.5 (For the proof seeo3.1pin [5]). The subcategory of representable func
tors is locally small in SK (i.e., the inclusion functor satisfies the cosolu*
*tion set
condition).
The category of small functors SKop carries the projective model structure w*
*ith
weak equivalences and fibrations being levelwise and cofibrations defined by th*
*e left
lifting property with respect to trivial fibrations [5, Theorem 3.2].
Recall that a model category is classcofibrantly generated if there are two*
* classes
I and J of generating cofibrations and generating trivial cofibrations, respect*
*ively,
which admit the generalized small object argument [4] and generate the model
structure in the usual sense: Iinj= {trivial fibrations} and Jinj= {fibration*
*s}.
Classcofibrantly generated model categories share many nice properties with co*
*fi
brantly generated model categories. In particular, the category of functors fr*
*om
a small category to a classcofibrantly generated model category may be equipped
with the projective model structure (which is classcofibrantly generated again*
*).
The projective model category is classcofibrantly generated with
(1) I= {RA @ n ,! RA nA 2 K, n 0}
(2) J = {RA nk",!RA nA 2 K, n > 0, 0 k n}.
being classes of generating cofibration and trivial cofibrations respectively.
We summarize the properties of the projective model structure in the followi*
*ng
Proposition 2.6. The projective model category structure on SKop is simplicial,
proper and classcofibrantly generated.
Proof.Since SKop is enriched over S, it suffices to verify SM7(a):
If p: Xe! Yeis a fibration (resp. trivial fibration) and i: K ,! L is a cofi*
*bration
of simplicial sets, then the induced map hom(i, p): XeL! XeKxYeKYeLis a fibrati*
*on
(resp. trivial fibration).
But cotensor products and pullbacksoarepcomputed levelwise (as all weighted
limits and colimits, since SK contains the representable functors), therefore*
* the
map hom(i, p) is an objectwise fibration (resp. trivial fibration), since the c*
*ategory
of simplicial sets is simplicial. Hence hom(i, p) is a fibration (resp. trivial*
* fibration)
in the projective model structure on the category of small diagrams.
Properness follows in a similar manner from the properness of simplicial set*
*s and
the fact that pushouts and pullback are computed levelwise. This involves the f*
*act
that projective cofibrations are levelwise cofibrations.
We are interested in the case Kop = S. Then the category of small functors
has another important property: it is closed under composition. We will need th*
*is
CALCULUS OF FUNCTORS 5
property in the next section. In the covariant case here and in the rest of the*
* paper
let RA = hom(A, . ) denote the enriched functor represented by A.
Lemma 2.7. The category of small functors SS is closed under composition.
Proof.Given two small functors Xe, Ye2 SS , we need to show that their composit*
*ion
XeO Yeis a small functor again.
It suffices to verify that RA O Yeis a small functor for any representable f*
*unctor
RA , A 2 S, since Xe is a weighted colimit of representable functors and small
functors are closed under weighted colimits. But (RA O Ye)( . ) = RA (Ye( . ))*
* =
hom (A, Ye( . )) = YeA. And YeA is a small functor, since the category of small
functors is closed under cotensor products (as under all weighted limits).
3.Homotopy model structure on SS
In this sectionowepconsider the case K = Sop and localize the projective mod*
*el
structure on SK = SS in such a way that fibrant objects after localization are
exactly the projectively fibrant homotopy functors.
Note that small functors are simplicial. That implies, in particular, that *
*sim
plicial homotopy equivalences are mapped into simplicial homotopy equivalences.
Now, all simplicial sets are cofibrant and every weak equivalence between objec*
*ts
which are both fibrant and cofibrant is a simplicial homotopy equivalence [18, *
*x2,
Prop. 5]. In other words small functors preserve weak equivalences between fibr*
*ant
objects.
We will construct the required localization by the method of BousfieldFried*
*lander
[2, A.7], which relies on existence of a coaugmented functor Q: SS ! SS , with
coaugmentation j :Id! Q .
Let fib:S ! S be a small fibrant replacement functor. To construct it it suf*
*fices
to take fib= R^*= ^Idto be a fibrant replacement of the identity functor in the
projective model structure on the category of small functors. The functor fibis
equipped with a coaugmentation ffl: Id! fib.
Define QXe = XeO fibfor all Xe2 SS , then Q: SS ! SS is a functor equipped
with a coaugmentation j given by jXf= XeO ffl. In this context a map Xe! Yeis
called a Qequivalence, if it induces a weak equivalence QXe ! QYe. Such a map
will be called a Qfibration if it has the right lifting property with respect *
*to all
projective cofibrations, that are also Qequivalences.
Proposition 3.1. Q is a coaugmented functor satisfying the following properties:
(A.4):Q is a homotopy functor, i.e. it preserves levelwise weak equivalenc*
*es;
(A.5):Q is a homotopy idempotent functor, i.e. jQXf, QjXf:QXe ' QQXe
are levelwise weak equivalences;
(A.6):For a pullback square
Ae__h__//Xe
 
 j
fflfflfflfflk
Be_____//Ye
in SS , if j is a Qfibration and k is a Qequivalence, then h is a Q
equivalence. (The dual condition was removed in [3]).
6 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
Proof.Given a levelwise weak equivalence, it is, in particular, a weak equivale*
*nce
between fibrant entries, so applying Q, we obtain a weak equivalence again. Hen*
*ce,
(A.4).
(A.5) follows from the homotopy idempotence of the fibrant replacement functor
in S.
To verify that (A.6) is true note first that Qequivalences are precisely the*
* natural
transformations of small functors which induce weak equivalences between fibrant
entries. Since every Qfibration is, in particular, a levelwise fibration, and*
* the
pullbacks in the category of small functors are computed objectwise and therefo*
*re
the result follows from the right properness of S.
Theorem 3.2. The category of small functors SS may be equipped with a proper
simplicial model structure such that weak equivalences are Qequivalences, cofi*
*bra
tions are projective cofibrations, and fibrations are Qfibrations.
Proof.It follows from Theorem A.7 of [2] and Proposition 3.1.
Definition 3.3. The model structure on SS from theorem 3.2 will be called the
homotopy model structure.
Corollary 3.4. (i)The Qequivalences are exactly those natural transformations
between small functors that are weak equivalences on fibrant spaces.
(ii)A map Xe! Yeis a Qfibration if and only if it is a projective fibration su*
*ch
that the following square
jX
Xe___f_//QXe
 
 
fflffljYfflffl
Ye_____/e/QYe
is a homotopy pullback square in the projective structure.
Proof.Part (i) follows directly from the definition of Q and part (ii) follows *
*from
the characterization theorem of Qfibrations in [2].
Corollary 3.5. Every small functor may be approximated by a homotopy functor
in a universal, up to homotopy, way. In other words: for every small functor
Xe 2 SS there exists a functor hXe and a natural transformation ': Xe! hXe such
that for every projectively fibrant homotopy functor Yeand a natural transforma*
*tion
iXe ! Yethere exists a natural transformation , :hXe ! Ye, unique up to homotop*
*y,
such that i = , O '.
Proof.The functor hXe is obtained by factorization of the map Xe! * into a triv*
*ial
cofibration followed by a fibration in the homotopy model structure.
4. The nexcisive structure
In this section we localize the homotopy model structure on the category of *
*small
endofunctors of S in such a way that the fibrant replacement becomes the nexci*
*sive
part of a functor.
We begin with recalling necessary definitions from [12].
Definition 4.1. Let P(n_) be the power set of the set n_= {1, ... , n} equipped*
* with
its canonical partial ordering. For later use we let P0(n_) be the complement o*
*f ; in
P(n_). An ncubical diagram in S is a functor P(n_) ! S. A homotopy functor F is
CALCULUS OF FUNCTORS 7
o excisive if it takes homotopy pushouts to homotopy pullbacks,
o reduced if F (*) ' *,
o linear if it is both excisive and reduced.
A cubical diagram is
o strongly homotopy cocartesian if all of its twodimensional faces are
homotopy pushouts,
o homotopy cartesian if it is a `homotopy pullback'.
A functor F is said to be nexcisive if it takes if it takes strongly homotopy
cocartesian (n+1)cubical diagrams to homotopy cartesian diagrams, see [12, 3.1*
*].
For an arbitrary homotopy functor Xe Goodwillie constructs an nexcisive ap
proximation pn,Xf:Xe ! PnXe, which is natural in Xe and universal among all
nexcisive functors under Xe. This is the nexcisive part of the Taylor tower o*
*f Xe.
Since Pn is a simplicial functor, it has a natural extension to functors with v*
*alues
in spectra. We need the following properties from [12].
Lemma 4.2. On the full subcategory of homotopy functors of SS , the functor Pn
commutes with finite homotopy limits and filtered homotopy colimits. The exten
sion of Pn to homotopy functors with values in spectra commutes with arbitrary
homotopy colimits.
The functor Pn is not defined on all objects in SS , but just on the homotopy
functors. To remedy this we precompose Pn with our fibrant replacement functor
F 2 SS , which we used in the construction of the homotopy model structure 3.1.
This ensures that Pn gets applied to a homotopy functor. We do not want to
introduce new notation, so the reader should remember that our Pn differs from
Goodwillie's Pn.
Definition 4.3. Let Pn :SS ! SS be the functor given by
Xe7! PnXe := Pn(XeO fib).
It is a coaugmented functor with coaugmentation jn,Xf= pn O jXf.
Now we will show that we can perform the BousfieldFriedlander localization
using Pn, thus giving the desired nexcisive model structure.
Proposition 4.4. The functor Pn satisfies the properties (A.4), (A.5) and (A.6)
from proposition 3.1.
Proof.Condition (A.4) is fulfilled by construction, Pn preserves weak equivalen*
*ces
in the homotopy structure. Condition (A.5) is shown in [12, proof of 1.8]. It
remains to prove condition (A.6), but this follows directly from the fact that *
*Pn
preserves homotopy pullbacks.
Definition 4.5. We call a map Xe! Yein SS
(1)an nexcisive equivalence if PnXe ! PnYeis an equivalence in the homotopy
model structure.
(2)an nexcisive fibration if it has the right lifting property with respec*
*t to all
cofibrations, that are also nexcisive equivalences.
These classes of maps will be called the nexcisive structure on SS .
8 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
Remark 4.6. A map between two homotopy functors is an equivalence in the ho
motopy model structure if and only if it is an objectwise weak equivalence. The
functor PnXe is a homotopy functor by definition, so PnXe ! PnYeis an equivalen*
*ce
in the homotopy model structure if and only if it is an objectwise weak equival*
*ence.
The next theorem follows again from the BousfieldFriedlander localization th*
*e
orem [2].
Theorem 4.7. The nexcisive structure on SS forms a proper simplicial model
structure. A map Xe! Yeis an nexcisive fibration if and only if it is a fibrat*
*ion
in the homotopy structure, such that the diagram
jn,X
Xe ___f//_PnXe
 
 
fflffljnfflffl,Y
Ye_____//PnYe e
is a homotopy pullback square in the homotopy structure. Fibrant objects are ex*
*actly
the objectwise fibrant nexcisive homotopy functors.
5.Homotopy theory of spectrumvalued functors
In this section we introduce a model category that describes homotopy theory
of small functors with values in spectra. First of all we have to give a defini*
*tion of
small spectrumvalued functors. To streamline the exposition we will use the ca*
*t
egory of pointed spaces S* as our underlying symmetric monoidal category. Note
that all arguments of this paper go through for the category SS**of small (enri*
*ched)
endofunctors of pointed spaces if one replaces a construction by its pointed an*
*a
logue. Let Sp denote the category of spectra in the sense of BousfieldFriedlan*
*der
[2]. More generally we use Sp(M) for a pointed simplicial model category M.
Definition 5.1. An object in the category Sp(M) is a sequence (X0, X1, ...) of
objects in M together with bonding maps
Xn ! Xn+1,
for n 0, where Xn := Xn 1=@ 1.
Definition 5.2. A functor from S* to Sp is small if it is the left Kan extensio*
*n of
a functor defined on a small subcategory of S*. We remind the reader that this *
*is
to be understood in the enriched context, see remark 2.2 and theorem 2.3.
Definition 5.3. For each n 0 let Evn : Sp! S* denote the functor taking a
spectrum X = (X0, X1, ...) to its nth level Xn.
Lemma 5.4. A functor F :S* ! Sp is small if and only if it is levelwise small,
i.e. if Evn O F :S* ! S* is small for each n 0.
Proof.The evaluation functors Evn are simplicial and have enriched right adjoin*
*ts,
which therefore commute with enriched left Kan extensions.
Lemma 5.5. The evident functors give an equivalence Sp(SS**) ~=Sp S*of cate
gories.
Proof.This follows directly from lemma 5.4.
CALCULUS OF FUNCTORS 9
Using lemma 5.5 we will identify the categories Sp(SS**) and SpS*. This shows
in particular that the category SpS*is complete. Now we want to lift the projec*
*tive
model structure, where a weak equivalence is given objectwise, to the spectrum
valued setting. Our strategy is the following: We take the projective model str*
*uc
ture on SS**and then consider spectrum objects over this category. Using results
from [20] we obtain a model structure on Sp(SS**), which is the desired one.
Equally well, we could use the pointed category (SS )* ~=SS*to describe a the*
*ory
of small functors originating in unpointed spaces. This would give a model for *
*the
category of small functors from unpointed spaces to spectra, but we will not pu*
*rsue
this now.
In [20] the stable model structure on spectra is obtained analogously as in [*
*2], but
the construction Q used there is adaptable to more general situations. In lemma
1.3.2. of [20] the properties are listed, which have to be satisfied in order t*
*o make
the machinery work. Although in our case the underlying model structure on SS**
is not cofibrantly generated, we are still able to prove the statements of this*
* lemma.
The reason is that our category is classcofibrantly generated, the only defici*
*ency
being the (possible) lack of functorial factorization. Here is the adapted vers*
*ion of
the relevant part (a) of the cited lemma.
Lemma 5.6. Let X ! Y be a termwise (trivial) fibration between sequences in the
category SS**. Then the induced map colimX ! colimY is a (trivial) fibration. In
particular, sequential colimits preserve weak equivalences.
Proof.The proof for the case of fibration and trivial fibration is literally th*
*e same
except that one uses the different test classes I or J from (1). Since source *
*and
target of the generating classes I and J are small, we get the following liftin*
*gs
RA K ____//_Xk____//colimX::___
______
i  _______(')__ 
fflffl___fflfflfflfflfflffl__
RA L _____//Y k____//colimY
where i is either in I or J. This proves the statement.
For the definition of the coaugmented functor Q: Sp(SS**) ! Sp(SS**) we refer
to [20, p. 93]. For each K in S* the spectrum (QXe)(K) is weakly equivalent to
usual spectrum Q(Xe(K)) in the BousfieldFriedlander sense.
Definition 5.7. A map Xe! Yein Sp(SS**) will be called
(i)a stable projective cofibration if the map Xe0! Ye0and for each n 0 the ma*
*ps
Xen_Xfn1Yen1! Yenare projective cofibrations.
(ii)a stable projective equivalence if for all n 0 the maps QXen ! QYen are
projective equivalences.
(iii)a stable projective fibration if for all n 0 the maps QXen ! QYenare pro*
*jec
tive fibrations and the squares
Xen_____//QXen
 
 
fflffl fflffl
Yen_____//QYen
are homotopy pullback squares in the projective structure.
10 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
We call these classes of maps the stable projective model structure on Sp(SS**).
Theorem 5.8. The stable projective model structure on Sp(SS**) ~=SpS*is a sim
plicial proper model structure.
We point out again, that we do not claim that this model structure has (or h*
*as
not) functorial factorization. The proof of this theorem is as in [20].
To localize this model structure in order to obtain the homotopy model struc*
*ture
we observe that as in the unstable setting small functors in SpS* are simplicia*
*l.
Hence they preserve simplicial homotopies and therefore map weak equivalences
between fibrant spaces to weak equivalences. So we can use the same method as in
Section 3 to obtain the homotopy structure on SpS*.
Definition 5.9. A map Xe! Yein SpS* will be called
(i)a stable equivalence in the homotopy structure if Xe(K) ! Ye(K) is a stable
equivalence of spectra for all fibrant spaces K.
(ii)a stable fibration in the homotopy structure if Xe ! Yeis a stable projecti*
*ve
fibration and the square
Xe ____//_XeO fib
 
 
fflffl fflffl
Ye ____//_YeO fib
is a homotopy pullback square in the stable projective structure. Here fib:S* !*
* S*
is a small fibrant replacement functor.
We call these classes of maps the stable homotopy model structure on Sp(SS**).
As in Section 3 we obtain the following theorem.
Theorem 5.10. The stable homotopy model structure on SpS*is a simplicial proper
model structure. A functor in SpS*is a homotopy functor if and only if it is we*
*akly
equivalent in the stable projective structure to a fibrant object in the stable*
* homotopy
structure.
There are different characterizations of weak equivalences, here we give one.
Lemma 5.11. A map Xe! Yeis a weak equivalence in the stable homotopy struc
ture if and only if for each n 0 the maps QXen ! QYen are weak equivalences in
the homotopy structure on SS**.
Proof.This follows from the natural equivalence Q(XeO fib) ~=(QXe) O fib.
Again in the same way we arrive at the nexcisive structure; we localize alo*
*ng
the coaugmented functor Pn.
Theorem 5.12. The stable nexcisive model structure on SpS* is a simplicial
proper model structure. A functor in SpS* is an nexcisive homotopy functor if
and only if it is weakly equivalent in the stable projective structure to a fib*
*rant
object in the stable nexcisive structure.
CALCULUS OF FUNCTORS 11
6.The Taylor tower and homogeneous functors
The natural map pn :Xe! PnXe induces a categorical localization in the ho
motopy category associated to the homotopy model structure. The local objects
with respect to this localization functor are the nexcisive functors. Since (n*
*  1)
excisive functors are also nexcisive, there is a map PnXe ! Pn1Xe under Xein *
*the
homotopy category. But Goodwillie [12, p. 664] gives a model for this map in SS
and calls it
qn,Xf:PnXe ! Pn1Xe.
These maps fit into a tower under Xe, which is called the Taylor tower of Xe. T*
*he
fibers of this tower are of special interest. Let us give a new definition firs*
*t.
Definition 6.1. A functor Xe is called nreduced if Xe is weakly contractible in
(n  1)excisive structure, i.e. Pn1Xe ' * in the homotopy structure. A functo*
*r is
called nhomogeneous if it is nreduced and nexcisive.
To introduce the homogeneous part DnXe of a small functor Xewe have to con
sider fibers and homotopy fibers. Henceforth we will work in the category of sm*
*all
functors on S* with values in pointed spaces or spectra.
Definition 6.2. For an object Z in a pointed simplicial model category we define
the simplicial path object by
1
W Z := Z x(ZxZ)(* x Z),
where the map Z 1 ! Z xZ is induced by d0_d1: 0_ 0 ! 1. The projection
pr2: Z x Z ! Z induces a map P Z ! Z. Note, that if Z is fibrant then this map
is a fibration. Note that P Z is simplicially contractible.
Definition 6.3. We define for each small functor Xe a new functor DnXe by the
following pullback square:
DnXe _____//W (Pn1Xe)
dn,Xf 
fflffl fflffl
PnXe __qn,X//_Pn1X
f e
We call DnXe the nhomogeneous part of Xe.
Remark 6.4. The map qnXe is an equivalence in the (n  1)excisive structure, a*
*nd
therefore is DnXe (n  1)excisively contractible, hence nreduced. The map dn,*
*Xf
is the base change of an nexcisive fibration, therefore DnXe is nexcisively f*
*ibrant.
So DnXe really is nhomogeneous. We also point out that the defining square has
homotopy meaning in the homotopy structure.
We will need the following properties, which are given in [12, Prop. 1.18].
Proposition 6.5. The functor Dn :SS**! SS**commutes with finite homotopy
limits and filtered homotopy colimits in the homotopy structure. On the categor*
*y of
spectrumvalued homotopy functors Dn commutes with arbitrary homotopy colimits.
12 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
7.Classification of nhomogeneous functors
In this section we construct the nhomogeneous model structure on SpS*, where
the homotopy types correspond bijectively to nhomogeneous spectrumvalued func
tors. This model structure classifies all nhomogeneous functors as homotopy ty*
*pes
(cf. [15, Remark 4.13]). We will give an interpretation of Goodwillie's classif*
*ication
of finitary homogeneous functors in terms of Quillen equivalence between model
categories in the next section.
The nhomogeneous model structure is obtained by a colocalization which is d*
*ual
to the BousfieldFriedlander localization. This process has similarity to the o*
*rdinary
Postnikov tower of spaces. One obtains the stages by localizing with respect to*
* Sn,
which kills all homotopy above degree n. Then one can colocalize with respect to
Sn1, where cofibrant replacement would be given by the connected covers. This
kills everything below degree n  1. Here we are going to do the same, we will
colocalize with respect to the nreduced part of a functor.
Definition 7.1. For each small functor Xewe define a new functor MnXe by the
following pullback square:
MnXe ____//_P (Pn1Xe)
mn,Xf 
fflffl fflffl
Xe_pn1,X_//Pn1X
f e
The augmented functor Mn :SS**! SS**is called the nreduced part of Xe.
Remark 7.2. It follows that MnXe is the homotopy pullback of Xe ! Pn1Xe
P (Pn1Xe) in the projective structure, as well as in the homotopy structure and
in the n  1excisive structure. The functor MnXe is weakly contractible in the
(n  1)excisive structure, and therefore nreduced. For each Xewe have a square
MnXe _____//DnXe
mn,Xf dn,X
fflffl fflfffl
Xe __pnX_//_PnX
f e
which is a pullback as well as a homotopy pullback square in the homotopy struc
ture. The construction Mn preserves homotopy pullbacks, since it is the homotopy
fiber of functors preserving homotopy pullbacks. Of course, MnXe is not a homot*
*opy
functor unless Xeis one.
We want to colocalize along the functor Mn. In fact, we will observe that we
can colocalize the nexcisive structure, as well as the homotopy structure resu*
*lting
in the nhomogeneous structure 7.7 and the nreduced structure 7.8. We have to
prove the dual set of the BousfieldFriedlander axioms. For the proof of the l*
*eft
properness condition (A.6)dualwe have to use, that Dn commutes with homotopy
pushouts. So our construction only works for spectrumvalued functors.
Lemma 7.3. The functor Mn satisfies (A.4)dual=(A.4).
Proof.Since Mn is defined as a homotopy fiber in the homotopy model structure
of the functors id and Pn1, which preserve weak equivalences in the homotopy
CALCULUS OF FUNCTORS 13
structure, Mn preserves them also. It follows that Mn also preserves nequivale*
*nces
by noticing that PnMn ' Dn ' MnPn.
Lemma 7.4. The maps mn,MnXfand Mnmn,Xf:MnMnXe ! MnXe are weak equiv
alences in the homotopy structure and, in particular, nexcisive weak equivalen*
*ces.
This is axiom (A.5)dual.
Proof.To show that mn,MnXfis an equivalence, just apply Pn1 to the defining
square of MnXe. The resulting square is again a homotopy pullback square and
Pn1MnXe ' *. For Mnmn,Xfobserve, that Mn preserves homotopy pullbacks and
MnPn1Xe ' *. This last equivalence follows from the definition of MnPn1Xe and
the fact, that Pn1Pn1Xe ' Pn1Xe.
Lemma 7.5. The functor Mn satisfies (A.6)dual.
Proof.Consider the following diagram
MnAe ______//MnXe
"" __xxx
Ae_______//_Xe 
 fflffl_//fflffl_
j MnBe  MnYe
 
fflffl fflfflfflffl
Be________//Ye
where j is a cofibration between cofibrant objects. If the map MnA ! MnX is
a weak equivalence in the homotopy structure, the left properness of the homo
topy structure implies that MnB ! MnY is a weak equivalence in the homotopy
structure.
Now suppose MnA ! MnX is an nexcisive equivalence. We have to show
that the map MnB ! MnY is also an nexcisive equivalence. To test for n
excisive equivalence we apply Pn to the second square and by using the equivale*
*nce
PnMn ' Dn we obtain the square
DnAe__'__//DnXe
 
 
fflffl fflffl
DnBe ____//_DnYe
where DnAe! DnXe is an equivalence by assumption. Since the spectrumvalued
Dn commutes with arbitrary homotopy colimits and the original square is a pusho*
*ut
as well as a homotopy pushout square, the latter square is a homotopy pushout.
The homotopy structure is left proper, so DnBe! DnYeis an equivalence, proving
that MnBe! MnYeis an nexcisive equivalence.
Definition 7.6. Let f :Xe! Yebe a map in SpS*and let eXe! eYebe a replacement
of f between homotopy functors. We call f an nhomogeneous equivalence if the
map
DnXee! DnYee
is a weak equivalence. We call f an nhomogeneous cofibration if f has the left
lifting property with respect to all maps which are nexcisive fibrations and n
homogeneous equivalences. We call this the nhomogeneous structure on SpS*.
14 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
Observe that Mn(f) is an nexcisive weak equivalence if and only if Dn(f) is *
*an
nexcisive weak equivalence. The dual of the BousfieldFriedlander machinery th*
*en
proves the following theorem.
Theorem 7.7. On the category SpS* the nhomogeneous structure exists and is
simplicial and proper. The bifibrant objects are exactly the projectively bifi*
*brant
nhomogeneous homotopy functors. In particular, the homotopy types correspond
bijectively to the homotopy types of nhomogeneous functors from S* to Sp.
Note that this theorem applies to all, not just the finitary nhomogeneous f*
*unc
tors.
For a small functor Xethe object PnXe is not exactly the localization of Xei*
*n the
sense of [13, 3.2.16], since the coaugmentation pn,Xfis usually not a trivial c*
*ofibra
tion. But PnXe is not far away from that, it is weakly equivalent to the locali*
*zation
in the underlying model structure, here the homotopy structure. The same is true
for DnXe: The maps Xe! PnXe DnXe are not a fibrant approximation followed
by a cofibrant approximation, but DnXe is weakly equivalent in the homotopy str*
*uc
ture to a fibrant and cofibrant replacement of Xein the nhomogeneous structure.
In fact, since both functors DnXe and the replacement of Xeare homotopy functor*
*s,
they are even weakly equivalent in the projective structure on SS**.
Finally it is worth remarking that we can colocalize along the functor Mn st*
*arting
directly from the homotopy structure without going first to the nexcisive stru*
*cture.
The arguments given above can easily be seen to justify the following theorem.
Theorem 7.8. The category SpS*may be equipped with the nreduced model struc
ture. The resulting model category is simplicial and proper. The cofibrant obje*
*cts
are exactly the projectively cofibrant nreduced functors.
8. Spectra with naction and nhomogeneous functors
A functor is called finitary if it commutes with filtered homotopy colimits.*
* In
[12] it is shown that a finitary nhomogeneous functor F (X) from pointed spaces
to spectra is weakly equivalent to the functor X 7! (E ^ X^n)h n, where E is a
spectrum with naction. This spectrum is called the nth derivative of F at *.
The same relation holds for an arbitrary nhomogeneous functor F , provided that
X is a finite space. This result is true, of course, in our framework (for smal*
*l functor
from pointed simplicial sets to BousfieldFriedlander spectra). In this general*
*ity it
was proven in [15].
In this section we give an alternative construction of the nth derivative (*
*and
of the nth cross effect), which allows us to interpret the above result as a Q*
*uillen
map between the nhomogeneous model category and the category of spectra with
naction. Moreover, we supply two more model categories Quillen equivalent
to the category of spectra with naction. These model categories correspond
to the two alternative conditions of Goodwillie's theorem: in the first the bif*
*ibrant
objects are finitary nhomogeneous small functors, while the second model categ*
*ory
is analogous to the nhomogeneous model structure, but the underlying category *
*is
the category of functors from finite pointed simplicial sets to spectra.
Consider the category of small functors from pointed simplicial sets to poin*
*ted
simplicial sets SS**. The finitary projective model structure on this category *
*is
given by:
CALCULUS OF FUNCTORS 15
o A natural transformation f :Xe! Yeis a weak equivalence or a fibration
if f(K): Xe(K) ! Ye(K) is a weak equivalence or a fibration of simplicial
sets for every finite simplicial set K.
o Cofibrations are given by the left lifting property with respect to triv*
*ial
fibrations.
This model category is generated by the set {RK  K 2 S*, K finite} of orbits*
* (in
the sense of Dwyer and Kan [8]) and it is Quillen equivalent to the projective *
*model
structure on the category of functors from the pointed finite simplicial sets S*
*f*to
the pointed simplicial sets S*. We start from these two Quillen equivalent model
categories and perform all the localizations, the stabilization, and the coloca*
*liza
tion we have applied on the projective model structure on the category of small
functors. Some of these procedures may be simplified, since these model categor*
*ies
are cofibrantly generated. At the end we will obtain two nhomogeneous model
categories Quillen equivalent to the category of spectra with naction.
Say that a functor F is prefinitary if F O fibfinitary.
Every cellular object in the finitary projective model structure on SS**is pr*
*e
finitary as a homotopy colimit of prefinitary functors RK , for K 2 S* being f*
*i
nite. Every cofibrant object in this model category is a retract of a cellular *
*object,
therefore a prefinitary functor. The procedures of localization and colocaliza*
*tion
preserve this property, since they either preserve or reduce the class of cofib*
*rant
objects. Stabilization of the projective model structure also preserves the pr*
*op
erty that all cofibrant functors are prefinitary, as they are levelwise prefi*
*nitary
(therefore homotopy colimits are preserved up to strict equivalence).
The name "finitary" is justified, since after passing to the homotopy model
structure bifibrant objects become finitary functors. Therefore we will use the
adjective finitary for all the derivatives from the finitary projective model s*
*tructure.
The methods used in the preceding sections of this paper apply equally(wellf)
to the finitary projective (resp., projective) model structure on SS**, so after
stabilization, finitary homotopy (resp., homotopy) and finitary nexcisive (res*
*p.,
nexcisive) model structure we finally obtain the finitary nhomogeneous (resp.,
(f)
nhomogeneous) model structure on SpS* . We would like to note only that the
f Sf
projective model structure on SS**(or Sp**) admits a set of maps F = {RL !
RK  K!"L is a w.e. of fin. simp.}sets, such that the localization with respect*
* to
F produces a homotopy model structure. The advantage of this method is that the
resulting model category is cofibrantly generated again. The same applies to the
nexcisive model structure; the corresponding set of maps was constructed in [7*
*].
Let us denote the finitary nhomogeneous model structure on SS**by M and the
f
nhomogeneous model structure on SpS**by N.
In the rest of this section we will argue simultaneously for all three model *
*cate
gories under consideration: nhomogeneous model structure on SpS**, M and N.
We have to state explicitly what type of equivariant model structure on spect*
*ra
with naction we use. View the group n as a category with one object and then
consider presheaves on this category with values in spectra. We equip spectra w*
*ith
the BousfieldFriedlander model structure and take the projective model structu*
*re
on presheaves over it. Thus, weak equivalences or fibrations are just given by *
*weak
equivalences or fibrations of the underlying spectra. More generally one can p*
*ut
such a nequivariant model structure on presheaves with values in any cofibran*
*tly
16 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
generated model category [13], and it is easy to check that it can be promoted *
*to
any classcofibrantly generated model category.
The category of small functors with projective model structure is classcofib*
*rantly
generated [5], therefore the category of small functors with naction may be g*
*iven
the projective model structure similarly to the cofibrantly generated case. The
other two model structures under consideration are cofibrantly generated and the
category of npresheaves over them also may be equipped with the projetive mod*
*el
structure.
First(wefneed)to define a homotopy invariant version of the smash product in
SS**. The smash product can be defined formally just the same as for spaces,
but the map RK _ RL ! RK_L is not a projective cofibration in general. So the
smash product might fail to be cofibrant. We have to do this nequivariantly, *
*so
for K1, ... , Kn 2S* we define
_ n !
^ ^n
RKi ! RKi
i=1 cof i=1
to be a nequivariant projective cofibrant replacement. Here the right hand si*
*de
has the naction, which permutes the factors in the smash product. In particul*
*ar,
let _ !
^n 0
(idn)cof:= RS
i=1 cof
We define a pair of adjoint functors. The left adjoint ~n :Sp n! SpS* is given
by:
~nE := (E^ (idn)cof) n
Note that for cofibrant E these are actually the homotopy orbits, since the act*
*ion
is free. To describe the right adjoint we first define a functor
(f)x n S(f) S(f)*
hom :(SS** )opx Sp * ~= (Sop*) n x Sp ! Sp n
(f) S(f)
for Ke 2 SS** and Xe 2 Sp * by levelwise prolongation of the mapping space
(f)
functor of the simplicial enrichment of SS**:
Ev khom(Ke, Xe) := map(Ke, EvkXe)
The right hand side inherits its naction from Ke. Observe that we have a natu*
*ral
adjunction
map(Le^Ke, Xe) ~=map(Le, hom(Ke, Xe))
(f)
for Le2 SS** and that there is an enriched Yoneda isomorphism
hom (RK , Xe) ~=Xe(K)
(f)
for K 2 S*. Then the right adjoint of ~n is the functor aen :SpS* ! Sp n given
by:
aenF := hom((idn)cof, F )
The spectrum aenF obtains a naction through the action on (idn)cof.
We can relate the smash product to the nth cross effect as defined in [12, p*
*. 676]
or [15, 5.8.]. Recall from 4.1 that P0(n_) = P(n_)  ;.
CALCULUS OF FUNCTORS 17
Lemma 8.1. (i) For any F there is the following natural equivalence:
_ n ! 2 _ ! 0 1 3
^ n` `
hom RKi, F ~=fib4F Ki ! lim F @ KiA 5
i=1 i=1 T2P0(n_) n_T
(ii)For projectively fibrant F we have:
_ _ n ! ! 2 _ ! 0 1 3
^ n` `
hom RKi , F ' hofib4F Ki ! holimF @ KiA 5
i=1 cof i=1 T2P0(n_) n_T
~=crnF (K1, ... , Kn)
Proof.Part (ii) follows from part (i), because here source and target have homo*
*topy
meaning. Part (i) follows by adjunction from the following representation of an
iterated smash product:
`
colim RKi _____//Qn Ki
T2P0(n_)i2n_T i=1R

 
 
fflffl V n fflffl
* ____________//i=1RKi
The spectrum @(n)F (*) for any homotopy functor F was introduced in [12,
p. 686]; see also [15, pp. 1415]. There @(n)F (*) is called the nth derivativ*
*e of F at
* and identified as crnF (S0, ... , S0). From 8.1 we deduce a natural nequiva*
*riant
weak equivalence
aenF ' crnF (S0, ... , S0) ~=@(n)F (*).
(f)
Theorem 8.2. The functors ~n :Sp n o SpS* : aen form a Quillen pair, where
Sp n has the projective equivariant model structure and SpS* has either the n
homogeneousfmodel structure, or the finitary nhomogeneous model structure, or
SpS* has the nhomogeneous model structure.
If either the finitary nhomogeneousfmodel structure on SpS*, or the nhomo
geneous model structure on SpS* is considered, then this Quillen pair becomes a
Quillen equivalence.
Proof.The functor aen maps stable projective (trivial) fibrations to (trivial) *
*fibra
tions, since (idn)cofis projectively cofibrant. Therefore ~n and aen form a Qui*
*llen
pair for the stable projective structure to the projective nequivariant Bousf*
*ield
Friedlander structure. They also from a Quillen pair for the homotopy and the
nexcisive structure, since aen still preserves (trivial) fibrations. The same *
*conclu
sion holds for the finitary version of these model structures and for the analo*
*gous
model structures on functors from pointed finite simplicial sets to spectra.
We claim that ~n maps (trivial) cofibrations to nhomogeneous (trivial) cofi
brations, which shows that ~n and aen form a Quillen pair for the nhomogeneous
model structure on SpS* (resp., M or N) and the nequivariant projective model
structure on Sp n.
We know already from the previous step that ~n maps trivial nequivariant p*
*ro
jective cofibrations to trivial nhomogeneous cofibrations (resp., trivial cofi*
*brations
18 GEORG BIEDERMANN, BORIS CHORNY, AND OLIVER R"ONDIGS
of M or N), which are the same as trivial nexcisive cofibrations. Furthermore,
~n maps nequivariant cofibrations to stable projective cofibrations between n
reduced functors, i.e. there is a projective weak equivalence Mn~nE ! ~nE. This
shows that for a cofibration A ! B of nspectra the square
Mn~nA _____//Mn~nB
' '
fflffl fflffl
~nA _______//_~nB
is a homotopy pushout and therefore ~nA ! ~nB is an nhomogeneous cofibration
(resp., a cofibration in M or N).
Now we show that this is actually a Quillen equivalence if we consider M or N.
Let E be a cofibrant spectrum, and let F be a fibrant object of M or N. Without*
* loss
of generality we may assume that F is also cofibrant (this assumption is redund*
*ant
for N), hence F 2 M is a finitary functor. It suffices to show that a map E ! a*
*enF
is a weak equivalence if and only if the corresponding map ~nE ! F is a weak
equivalence.
A map E ! aenF is a weak equivalence if and only if the map ~nE ! ~naenF
is a projective weak equivalence, since the nhomogeneous functors in the image*
* of
~n are determined by their coefficient spectrum. Any nhomogeneous homotopy
functor is projectively equivalent to its nhomogeneous part, so ~nE ! ~naenF is
a projective weak equivalence if and only if Dn~nE ! Dn~naenF is a projective
weak equivalence. By [12, p. 686] and Lemma 8.1 for every finitary F or for eve*
*ry
finite K 2 S* we have
Dn~naenF (K) ~=DnF (K).
Since one of the two conditions is necessarily satisfied in either M, or N, we *
*conclude
that E ! aenF is a weak equivalence iff Dn~nE ! DnF is a projective weak
equivalence. Finally, ~nE ! F is a weak equivalence in M or N if and only if
Dn~nE ! DnF is a projective weak equivalence.
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*otangent complex.
J. Pure Appl. Algebra, 120(1):77104, 1997.
Department of Mathematics, Middlesex College, The University of Western On
tario, London, Ontario N6A 5B7, Canada
DMATH, ETH Zentrum, 8092 Z"urich, Switzerland
Fakult"at f"ur Mathematik, Universit"at Bielefeld, Postfach 100 131, D33501*
* Biele
feld, Germany
Email address: gbiederm@uwo.ca
Email address: chorny@math.ethz.ch
Email address: oroendig@math.unibielefeld.de