STABLE HOMOTOPY OF ALGEBRAIC THEORIES
Stefan Schwede 1
Abstract: The simplicial objects in an algebraic category admit an abstract ho*
*motopy theory via a
Quillen model category structure. We show that the associated stable homotopy t*
*heory is completely
determined by a ring spectrum functorially associated with the algebraic theory*
*. For several familiar
algebraic theories we can identify the parameterizing ring spectrum; for other *
*theories we obtain new
examples of ring spectra. For the theory of commutative algebras we obtain a ri*
*ng spectrum which is
related to Andre-Quillen homology via certain spectral sequences. We show that *
*the (co-)homology of
an algebraic theory is isomorphic to the topological Hochschild (co-)homology o*
*f the parameterizing ring
spectrum.
0 Introduction
The original motivation for this paper came from the attempt to generalize a ra*
*tional result about
the homotopy theory of commutative rings. For a map of commutative rings, D. Q*
*uillen [31]
defined the cotangent complex as the left derived functor of abelianization; th*
*is construction is
now referred to as Andre-Quillen homology. We wanted to obtain a topological v*
*ariant of the
cotangent complex by replacing "abelianization" by "stabilization", i.e., passa*
*ge to spectra in the
sense of stable homotopy theory. In [36] we made this precise by introducing a *
*model category of
spectra for simplicial commutative algebras. At the same time we showed that ov*
*er the rational
numbers, nothing really new is happening. More precisely, for a commutative Q-a*
*lgebra B, the
stable homotopy theory of commutative simplicial B-algebras is equivalent to th*
*e homotopy theory
of simplicial B-modules, see [36, Thm. 3.2.3]. Loosely speaking, stable homotop*
*y and homology
of commutative rings coincide rationally - just as they do for topological spac*
*es.
One aim of this paper is to understand some of the torsion phenomena in the sta*
*ble homotopy
theory of commutative rings. The main structural result is again concerned with*
* the stable homo-
topy category of commutative simplicial B-algebras, but where B is now an arbit*
*rary commutative
ring. We will see that this stable homotopy category is still a category of mod*
*ules if one allows
rings spectra rather than ordinary rings. For our purpose, the most convenient*
* notion of ring
spectrum is that of a Gamma-ring (see Definition 1.12). Gamma-rings are based o*
*n a symmetric
monoidal smash product for -spaces with good homotopical properties [38, 9, 25]*
*. The homotopy
theory of Gamma-rings and their modules is developed in [37]. The generalizatio*
*n of the rational
result [36, Thm. 3.2.3] then reads:
Theorem: Let B be a commutative ring. Then the stable homotopy theory of augmen*
*ted com-
mutative simplicial B-algebras is equivalent to the homotopy theory of modules *
*over a certain
Gamma-ring DB. The graded ring of homotopy groups of DB is isomorphic to the ri*
*ng of stable
homotopy operations of commutative augmented simplicial B-algebras. If B is a Q*
*-algebra, this
DB is stably equivalent to the Eilenberg-MacLane Gamma-ring of B.
This theorem is appropriately dealt with in a more general framework. We are le*
*d to consider
pointed simplicial algebraic theories, or just simplicial theories for short. A*
* simplicial theory T has
a category of algebras; definitions will be given later, but thinking of a T -a*
*lgebra as a simplicial set
_______________________________1
April 14, 1999; 1991 AMS Math. Subj. Class.: 55U35, 18C10
1
with certain algebraic operations and equational relations is a good guide line*
*. Examples include
simplicial sets, simplicial sets with an action of a group, (abelian) groups, m*
*odules over a simplicial
ring, augmented (commutative) algebras over a commutative ring, Lie algebras an*
*d many more.
The category of T -algebras is naturally a closed simplicial model category, th*
*us allowing one to
apply homotopy theoretic concepts. The category of spectra of T -algebras is al*
*so a closed simplicial
model category, and its homotopy theory will be referred to as the stable homot*
*opy theory of T .
The above theorem then becomes a special case of Theorem 4.4, which in particul*
*ar states
Theorem: To a simplicial theory T , there is functorially associated a Gamma-r*
*ing T s. The
graded ring of homotopy groups of T sis isomorphic to the ring of stable homoto*
*py operations of
T -algebras. The stable homotopy theory of T -algebras is equivalent to the hom*
*otopy theory of
modules over T s.
For several examples of algebraic theories the parameterizing Gamma-ring can be*
* identified with
something familiar: for the theory of sets we obtain the standard model of the *
*sphere spectrum;
the theories of monoids and groups give different, but stably equivalent models*
* for the sphere
spectrum; for sets with an action of a fixed groups one gets the spherical grou*
*p ring; the theory
of modules over a fixed ring leads to the Eilenberg-MacLane Gamma-ring. More de*
*tails on these
examples can be found in Section 7; there we also list algebraic theories - suc*
*h as the motivating
example of commutative algebras - whose associated Gamma-rings give new homotop*
*y types of
ring spectra.
With the help of Theorem 4.4 we can deduce several structural properties that t*
*he homotopy
theory of T -algebras shares with the ordinary homotopy theory of spaces. Among*
* other things we
provide Hurewicz and Whitehead theorems (Corollaries 5.3 and 5.4) as well as At*
*iyah-Hirzebruch
and universal coefficient spectral sequences (see 5.5) which relate the Quillen*
*-homology of a T -
algebra to its stable homotopy.
In [20, Sec. 4] M. Jibladze and T. Pirashvili defined the cohomology of a theor*
*y with coefficients in a
functor that takes values in abelian group objects. We provide the link between*
* the (co-)homology
of an algebraic theory and its stable homotopy. Any coefficient functor G for t*
*he (co-)homology of
the theory T gives rise to a bimodule G!over the Gamma-ring T s. If T is the th*
*eory of modules
over some ring, a theorem of T. Pirashvili and F. Waldhausen [30, Thm. 3.2] ide*
*ntifies theory
homology with topological Hochschild homology (THH). In Theorem 6.7 we generali*
*ze this from
rings to arbitrary algebraic theories and provide a cohomological analogue:
Theorem: Let T be a pointed discrete algebraic theory and G a coefficient funct*
*or. Then there
is a natural isomorphism
H *(T ; G) ~= THH *(T s; G!) :
If G is additive, then there is a natural isomorphism
H *(T ; G) ~= THH *(T s; G!) :
The paper is organized as follows. In Section 1 we review -spaces, Gamma-rings*
* and their
modules. Section 2 recalls algebraic theories. The unstable homotopy of algeb*
*raic theories is
discussed in Section 3. Section 4 deals with the stable homotopy of an algebra*
*ic theory and
proves the equivalence with modules over the associated Gamma-ring, Theorem 4.4*
*. In Section 5
we describe the relationship between stable homotopy and Quillen homology for a*
*lgebras over a
theory. Section 6 establishes the equivalence of theory (co-)homology with topo*
*logical Hochschild
(co-)homology. Examples and applications are given in Section 7. The reader wit*
*h little experience
with the abstract notion of an algebraic theory might want to browse through th*
*e examples first,
2
to get an idea of what the general theory is all about. We assume familiarity w*
*ith the language
of homotopical algebra [32, 16].
This paper is a substantially modified version of the main part of the author's*
* thesis. It could
not have been written without many discussions that I had with various colleagu*
*es. I would
like to thank Greg Arone, John Klein and Manos Lydakis for many conversations o*
*n related and
unrelated topics during my time as a graduate student at Bielefeld University. *
*I am particularly
indebted to Manos Lydakis for bringing to my attention the smash product of -sp*
*aces (cf. [25]);
this device was crucial in turning ideas into theorems. It is a pleasure to ac*
*knowledge various
helpful discussions I had with Bjorn Dundas, Tom Goodwillie, Teimuraz Pirashvil*
*i, Charles Rezk,
Brooke Shipley and Jeff Smith. Lastly, I would like to thank my adviser, Friedh*
*elm Waldhausen,
for his continuous support.
1 Review of -spaces and Gamma-rings
The category of -spaces was introduced by G. Segal [38], who showed that it has*
* a homotopy
category equivalent to the stable homotopy category of connective spectra. A. K*
*. Bousfield and
E. M. Friedlander [9] considered a bigger category of -spaces in which the ones*
* introduced by
Segal appeared as the special -spaces (see 1.4). Their category admits a closed*
* simplicial model
category structure with a notion of stable weak equivalences giving rise again *
*to the homotopy
category of connective spectra. Then M. Lydakis [25] showed that -spaces admit *
*internal function
objects and a symmetric monoidal smash product with good homotopical properties.
1.1 -spaces. The category opis a skeletal category of the category of finite po*
*inted sets. There
is one object n+ = {0; 1; : :;:n} for every non-negative integer n, and morphis*
*ms are the maps of
sets which send 0 to 0. op is equivalent to the opposite of Segal's category [*
*38]. A -space is
a covariant functor from opto the category of simplicial sets taking 0+ to a on*
*e point simplicial
set. A morphism of -spaces is a natural transformation of functors. We denote t*
*he category of
-spaces by GS. We sometimes need to talk about -sets, by which we mean pointed *
*functors from
opto the category of pointed sets. Every -space can be viewed as a simplicial o*
*bject of -sets.
A symmetric monoidal smash product functor ^: opx op-! opis given by lexicograp*
*hically
ordering the elements of the set n+^ m+. This smash product extends to a smash *
*product for
all pointed sets. We denote by S the -space which takes n+ to n+, considered a*
*s a constant
simplicial set. The spectrum associated to the -space S (see 1.2) is the sphere*
* spectrum. The
representable -spaces n = op(n+; -) play a role analogous to that of the standa*
*rd simplices in
the category of simplicial sets. 1 is isomorphic to S. If X is a -space and K a*
* pointed simplicial
set, a new -space X^ K is defined by setting (X^ K)(n+) = X(n+)^ K.
There are three kinds of hom objects for -spaces X and Y . There is the set of *
*morphisms (natural
transformations) GS(X; Y ). Then there is a simplicial hom set hom(X; Y ), defi*
*ned by
hom(X; Y )i = GS (X ^(i)+; Y ) ;
where the `+' denotes a disjoint basepoint. In this way GSbecomes a simpliciall*
*y enriched category.
Finally there is an internal hom -space Hom(X; Y ) defined by
Hom (X; Y ) (n+) = hom (X; Yn+^) ;
where Yn+^(m+) = Y (n+^ m+).
3
A -space X can be prolonged, by direct limit, to a functor from the category of*
* pointed sets to
pointed simplicial sets. By degreewise evaluation and formation of the diagonal*
* of the resulting
bisimplicial sets, it can furthermore be promoted to a functor from the categor*
*y of pointed simpli-
cial sets to itself [9, x4]. This prolongation process has another description *
*as the following coend
[27, IX.6]. If X is a -space and K a pointed simplicial set, the value of the e*
*xtended functor on
K is given by Z
n+2op
Kn^ X(n+) :
The extended functor preserves weak equivalences of simplicial sets [BF, Prop. *
*4.9] and is auto-
matically simplicial, i.e., it comes with coherent natural maps K ^X(L) -! X(K *
*^L). We will
not distinguish notationally between the prolonged functor and the original -sp*
*ace.
1.2 Spectra. A spectrum X in the sense of [9, Def. 2.1] consists of a sequence *
*of pointed simplicial
sets Xn for n 0, together with maps S1^Xn -! Xn+1. A map of spectra X -! Y con*
*sists
of maps Xn -! Yn strictly commuting with the suspension maps. The homotopy gro*
*ups of a
spectrum X are defined as
ssnX = colimissn+i|Xi| :
A map of spectra is a stable equivalence if it induces isomorphisms on homotopy*
* groups. A -
space X extends to a simplicial functor from all pointed simplicial sets, so it*
* defines a spectrum
X(S) whose n-th term is the value of the prolonged -space at Sn = S1^: :^:S1 (n*
* factors).
For example, the -space S becomes isomorphic to the identity functor of the cat*
*egory of pointed
simplicial sets after prolongation. So the associated spectrum is given by S(S*
*)n = Sn, i.e., S
represents the sphere spectrum. The homotopy groups of a -space are those of th*
*e associated
spectrum, and they are always trivial in negative dimensions. A map of -spaces *
*is called a stable
equivalence if it induces isomorphisms of homotopy groups.
1.3 Smash products. In [25, Thm. 2.2], M. Lydakis defines a smash product for -*
*spaces by
the formula
(X ^Y ) (n+) = colimk+^ l+-!n+X(k+ ) ^Y (l+) :
The smash product is characterized by the universal property that -space maps X*
* ^Y -! Z are
in bijective correspondence with maps
X(k+ ) ^Y (l+) -! Z(k+ ^l+)
which are natural in both variables. By [25, Thm. 2.18], the smash product of *
*-spaces is as-
sociative and commutative with unit S, up to coherent natural isomorphism. Ther*
*e is a natural
isomorphism of -spaces
Hom (X ^Y; Z) ~= Hom (X; Hom(Y; Z)) :
In other words, the category of -spaces becomes a symmetric monoidal closed cat*
*egory.
1.4 Special -spaces. A -space X is called special if the map X(k+ _l+) -! X(k+ *
*) x X(l+)
induced by the projections from k+ _l+ to k+ and l+ is a weak equivalence for a*
*ll k and l. In this
case, the weak map
X(1+) x X(1+) --~-- X(2+) --X(r)---!X(1+)
induces an abelian monoid structure on ss0(X(1+)). Here r : 2+ -! 1+ is the fol*
*d map defined
by r(1) = 1 = r(2). X is called very special if it is special and the monoid ss*
*0(X(1+)) is a group.
4
By Segal's theorem ([38, Prop. 1.4], see also [9, Thm. 4.2]), the spectrum asso*
*ciated to a very
n)| -! |X(Sn+1)*
*| adjoint
special -space X is an -spectrum in the sense that the maps |X(S
to the spectrum structure maps are homotopy equivalences. In particular, the ho*
*motopy groups
of a very special -space X are naturally isomorphic to the homotopy groups of t*
*he simplicial set
X(1+).
1.5 Eilenberg-MacLane -spaces. Simplicial abelian groups give rise to very spec*
*ial -spaces
via an Eilenberg-MacLane functor H. For a simplicial abelian group A, the -spac*
*e HA is defined
by (HA)(n+) = A eZ[n+] where eZ[n+] denotes the reduced free abelian group gen*
*erated by
the pointed set n+. HA is very special and the associated spectrum is an Eilen*
*berg-MacLane
spectrum for A. The homotopy groups of HA are naturally isomorphic to the homot*
*opy groups
of A. The functor H embeds simplicial abelian groups as a full subcategory of G*
*S and it has a
left adjoint, left inverse functor L. For a -space X, L(X) is the cokernel of t*
*he map of simplicial
abelian groups
(p1)*+ (p2)*- r* : eZ[X(2+)] ----! eZ[X(1+)] :
Here p1 and p2 are the two projections from 2+ to 1+ in op. The functor L is co*
*mpatible with the
smash product of -spaces (i.e., L is strong symmetric monoidal) and it preserve*
*s finite products,
see [37, Lemma 1.2]. For Q-cofibrant -spaces (see 1.6), L represents spectrum *
*homology [37,
Lemma 4.2].
1.6 Model category structures. A. K. Bousfield and E. M. Friedlander introduce *
*two model
category structures for -spaces called the strict and the stable model categori*
*es [9, 3.5, 5.2]. It
will be more convenient for our purposes to work with slightly different model *
*category structures,
which we call the Quillen- or Q-model category structures (see [37, App. A]). T*
*he strict and
stable Q-structures have the same weak equivalences (hence the same homotopy ca*
*tegories) as
the corresponding Bousfield-Friedlander model category structures, but differen*
*t fibrations and
cofibrations.
We call a map of -spaces a strict Q-fibration (resp. a strict Q-equivalence) if*
* it is a Kan fibration
(resp. weak equivalence) of simplicial sets when evaluated at every n+ 2 op. St*
*rict Q-cofibrations
are defined as the maps having the left lifting property with respect to all st*
*rict acyclic Q-fibrations.
The Q-cofibrations can be characterized in the spirit of [32, II.4 Remark 4] as*
* the injective maps
with projective cokernel, see [37, Lemma A3 (b)] for the precise statement. By*
* [32, II.4 Thm.
4], the strict Q-notions of weak equivalences, fibrations and cofibrations make*
* the category of
-spaces into a closed simplicial model category.
More important is the stable Q-model category structure. This one is obtained *
*by localizing
the strict Q-model category structure with respect to the stable equivalences. *
* We call a map
of -spaces a stable Q-equivalence if it induces isomorphisms on homotopy groups*
*. The stable
Q-cofibrations are the strict Q-cofibrations and the stable Q-fibrations are de*
*fined by the right
lifting property with respect to the stable acyclic Q-cofibrations. By [37, Thm*
*. 1.5], these stable
notions of Q-cofibrations, Q-fibrations and Q-equivalences make the category of*
* -spaces into a
closed simplicial model category. A -space X is stably Q-fibrant if and only if*
* it is very special
and X(n+) is fibrant as a simplicial set for all n+ 2 op. The Q-model category*
* structure is
compatible with Lydakis' smash product, see [37, Lemma 1.7]. For example, smas*
*hing with a
Q-cofibrant -space preserves stable equivalences.
Bousfield and Friedlander also introduce strict and stable model category struc*
*tures for spectra.
A map of spectra X -! Y is a strict fibration (resp. strict weak equivalence) i*
*f all the maps
5
Xn -! Yn are fibrations (resp. weak equivalences) of simplicial sets. It is a s*
*trict cofibration if
X0 -! Y0 and
Xn [Xn-1 Yn-1 -! Yn
(for n 1) are cofibrations of simplicial sets. The stable weak equivalences ar*
*e the maps which
induce isomorphisms on homotopy groups. The stable cofibrations coincide with t*
*he strict cofibra-
tions and the stable fibrations are the maps with the right lifting property wi*
*th respect to stable
acyclic cofibrations. There is a more explicit characterization of the stable f*
*ibrations in [9, 2.2]. In
[9, Thm. 2.3] it is shown that the stable notions of fibrations, cofibrations a*
*nd weak equivalences
make the category of spectra into a closed simplicial model category. We show i*
*n Lemma A.3 that
this model category structure is cofibrantly generated (see [15], [37, Def. A2]*
*).
1.7 Quillen equivalences. An adjoint functor pair between model categories is c*
*alled a Quillen
pair if the left adjoint L preserves cofibrations and acyclic cofibrations. An *
*equivalent condition
is to demand that the right adjoint R should preserve fibrations and acyclic fi*
*brations. Under
these conditions, the functors pass to an adjoint functor pair on homotopy cate*
*gories (see [32, I.4
Thm. 3], [16, Thm. 9.7 (i)]). A Quillen functor pair is called a Quillen equiva*
*lence if the following
condition holds: for every cofibrant object A of the source category of L and f*
*or every fibrant
object X of the source category of R, a map L(A) -! X is a weak equivalence if *
*and only if
its adjoint A -! R(X) is a weak equivalence. Sometimes the right adjoint functo*
*r R preserves
and detects all weak equivalences. Then the pair is a Quillen equivalence if fo*
*r all cofibrant A the
unit A -! R(L(A)) of the adjunction is a weak equivalence. A Quillen equivalenc*
*e induces an
equivalence of homotopy categories (see [32, I.4 Thm. 3], [16, Thm. 9.7 (ii)]),*
* but it also preserves
higher order structure like (co-)fibration sequences, Toda brackets and the hom*
*otopy types of
function complexes. If two model categories are related by a chain of Quillen e*
*quivalences, they
can be viewed as having the same homotopy theory.
The functor that sends a -space X to the spectrum X(S) has a right adjoint [9, *
*Lemma 4.6],
and these two functors form a Quillen pair. One of the main theorems of [9] say*
*s that this Quillen
pair induces an equivalence between the homotopy category of -spaces, taken wit*
*h respect to the
stable structure of [9], and the stable homotopy category of connective spectra*
* (see [9, Thm. 5.8]).
Since every Q-cofibration is also a cofibration in the sense of Bousfield and F*
*riedlander, and since
the stable equivalences coincide in the two model category structures, the adjo*
*int functor pair of
[9, Lemma 4.6 and x5] is also a Quillen pair with respect to the stable Q-model*
* category structure
for -spaces.
1.8 The assembly map. Given two -spaces X and Y , there is a natural map X ^Y -*
*! X O Y
from the smash product to the composition product. The formal and homotopical *
*properties
of this assembly map are of fundamental importance to this paper. Since -space*
*s prolong to
functors defined on the category of pointed simplicial sets, they can be compos*
*ed. Explicitly, for
-spaces X and Y , we set (X O Y )(n+) = X(Y (n+)). This composition O is a mono*
*idal (though
not symmetric monoidal) product on the category of -spaces. The unit is the sam*
*e as for the
smash product, it is the -space S (alias 1) which as a functor is the inclusion*
* of op into all
pointed simplicial sets.
The assembly map is obtained as follows. Prolonged -spaces are naturally simpli*
*cial functors [9,
x3], which means that there are natural coherent maps X(K) ^L -! X(K^ L). This *
*simplicial
structure gives maps
X(n+) ^Y (m+) ----! X(n+^ Y (m+)) ----! X(Y (n+^ m+))
6
natural in both variables. From this the assembly map X ^Y -! X O Y is obtain*
*ed by the
universal property of the smash product of -spaces. The assembly map is associa*
*tive and unital,
S being the unit for both ^and O. In technical terms: the identity functor on t*
*he category of -
spaces becomes a lax monoidal functor from (GS; ^) to (GS; O). The crucial homo*
*topical property
of the assembly map is that it is a stable equivalence whenever X or Y is cofib*
*rant (see [25, Prop.
5.23]).
1.9 Stable excision. Every functor obtained from a -space by prolongation pres*
*erves weak
equivalences of simplicial sets and connectivity of maps [9, 4.9 and 4.10]. Bu*
*t the homotopy
functors arising as prolonged -space have further connectivity and excision pro*
*perties, such as
for example the one we prove now.
Lemma 1.10 Let F be a -space and X -! Y an injective map between simply connec*
*ted pointed
simplicial sets. Then the map
F (Y )=F (X) ----! F (Y=X)
is at least as connected as the suspension of X ^(Y=X).
Proof: Every -space admits a strict equivalence F c-! F from a Q-cofibrant -spa*
*ce. Then the
induced maps F c(X) -! F (X) are weak equivalences for all simplicial sets X, s*
*o we can assume
that F is Q-cofibrant. If F = n^ K for some simplicial set K, then F (X) ~=Xn^ *
*K and the
lemma can be verified for F by inspection.
Now we consider an injection of -spaces F - ! G, we assume that the lemma holds*
* for F
and the quotient G=F , and we claim that the lemma follows for G. Since all sp*
*aces in sight
are simply-connected, it suffices to show that the homotopy cofibre (mapping co*
*ne) of the map
G(Y )=G(X) -! G(Y=X) is as connected as X ^(Y=X). We can calculate the total ho*
*motopy
cofibre of the square
F (Y )=F (X)___wG(Y )=G(X)
| |
| |
|u |u
F (Y=X) ________G(Y=X)w
in two ways. If we take horizontal homotopy cofibres first and use that the lem*
*ma holds for the
-space G=F , we conclude that the total homotopy cofibre is as connected as X ^*
*(Y=X). By
first taking vertical homotopy cofibres and using the lemma for F we see that t*
*he lemma holds
for G.
We now know that if the lemma holds for a -space F and if G is obtained from F *
*by cobase change
along one of the generating Q-cofibrations (see proof of [37, Thm. 1.5]) n^(@i)*
*+ -! n^(i)+,
then the lemma holds for G. Also if the lemma holds for all -spaces in a (possi*
*bly transfinite)
sequence of cofibrations, then it holds for the colimit of the sequence. Finall*
*y, the property we are
interested in is preserved under retract. This finishes the proof since all -sp*
*aces can be obtained
from the trivial -space by these operations (by the small object argument [37, *
*Lemma A1]). __|_|
7
1.11 Gamma-rings and their modules. Our notion of ring spectrum is that of a Ga*
*mma-ring.
Gamma-rings are the monoids in the symmetric monoidal category of -spaces with *
*respect to
the smash product and they are special cases of `Functors with Smash Product' (*
*FSPs, cf. [5, 1.1],
[30, 2.2]). One can describe Gamma-rings as `FSPs defined on finite sets'. It w*
*as tempting to call
these monoids `-rings' but since that term should refer to a functor from opto *
*the category of
rings, the name `Gamma-ring' was chosen instead. A more detailed discussion of *
*the homotopy
theory of Gamma-rings can be found in [37].
Definition 1.12A Gamma-ring is a monoid in the symmetric monoidal category of -*
*spaces
with respect to the smash product. Explicitly, a Gamma-ring is a -space R equip*
*ped with maps
S -! R and R ^R -! R ;
called the unit and multiplication map, which satisfy certain associativity and*
* unit conditions
(see [27, VII.3]). A Gamma-ring R is commutative if the multiplication map is u*
*nchanged when
composed with the twist, or the symmetry isomorphism, of R ^R. A map of Gamma-r*
*ings is a
map of -spaces commuting with the multiplication and unit maps. If R is a Gamma*
*-ring, a left
R-module is a -space N together with an action map R ^N -! N satisfying associa*
*tivity and
unit conditions (see again [27, VII.4]). A map of left R-modules is a map of -s*
*paces commuting
with the action of R. We denote the category of left R-modules by R-mod.
One similarly defines right modules. The unit S of the smash product is a Gamma*
*-ring in a unique
way. The category of S-modules is isomorphic to the category of -spaces. For a *
*Gamma-ring
R the opposite Gamma-ring Rop is defined by twisting the multiplication with th*
*e symmetry
isomorphism of R ^R. Then the category of right R-modules is isomorphic to the*
* category of
left Rop-modules. The smash product of two Gamma-rings is naturally a Gamma-rin*
*g. An R-T -
bimodule is defined to be a left (R ^T op)-module. Because of the universal pro*
*perty of the smash
product of -spaces (see 1.3), Gamma-rings are in bijective correspondence with *
*lax monoidal
functors from the category op to the category of pointed simplicial sets (both *
*under smash
product). Similarly, commutative Gamma-rings correspond to lax symmetric monoid*
*al functors.
Standard examples of Gamma-rings are monoid rings over the sphere Gamma-ring S *
*and Eilenberg-
MacLane models of classical rings. If M is a simplicial monoid, we define a -sp*
*ace S[M] by
S[M] (n+) = M+ ^n+
(so S[M] is isomorphic, as a -space, to S ^M+ ). There is a unit map S -! S[M] *
*induced by the
unit of M and a multiplication map S[M] ^S[M] -! S[M] induced by the multiplica*
*tion of M
which turn S[M] into a Gamma-ring. This construction of the monoid ring over S *
*is left adjoint
to the functor which takes a Gamma-ring R to the simplicial monoid R(1+).
If B is a simplicial ring, then the Eilenberg-MacLane -space HB is naturally a *
*Gamma-ring,
simply because H is a lax monoidal functor [37, Lemma 1.2]. The functor H is fu*
*ll and faithful
when considered as a functor from the category of simplicial rings to the categ*
*ory of Gamma-rings.
The functor L is still left adjoint and left inverse to H. More examples of Gam*
*ma-rings arise from
simplicial algebraic theories and as endomorphism Gamma-rings, see 4.5 and 4.6 *
*below.
Modules over a Gamma-ring form a model category. A map of R-modules is called *
*a weak
equivalence (resp. fibration) if it is a stable Q-equivalence (resp. stable Q-f*
*ibration) as a map
of -spaces. A map of R-modules is called a cofibration if it has the left lift*
*ing property with
respect to all acyclic fibrations in R-mod. By [37, Thm. 2.2], the category of*
* left R-modules
becomes a closed simplicial model category this way. For a simplicial ring B, t*
*he functors H and
8
L are a Quillen equivalence between the model category of HB-modules and the mo*
*del category
of simplicial B-modules [37, Thm. 4.4].
The category of R-modules inherits a smash product. More precisely, let M be a *
*right R-module
and N a left R-module. Then the smash product M^R N is defined as the coequali*
*zer, in the
category of -spaces, of the two maps M ^R ^N ----!----!M ^N induced by the acti*
*on of R on
M and N respectively. If N is a cofibrant left R-module then the functor -^RN *
*takes stable
equivalences of right R-modules to stable equivalences of -spaces [37, Thm. 2.2*
*]. We define the
derived smash product M ^LRN of M and N in the usual way: we choose a cofibrant*
* left R-module
Nc and a weak equivalence Nc -~!N and set M ^LRN = M ^RNc. The derived smash pr*
*oduct is
well defined up to stable equivalence. There are certain standard Tor spectral *
*sequences converging
to the homotopy groups of M ^LRN, see [37, Lemma 3.1].
2 Algebraic theories
Algebraic theories, introduced by F. W. Lawvere [23], formalize the concept of *
*an algebraic object
as a set together with n-ary operations for various n 2 N and equational relati*
*ons. A detailed
exposition of algebraic theories can be found in [6, Sec. 3]. To do homotopy th*
*eory, we use algebraic
theories which are enriched over the category of simplicial sets; these simplic*
*ial theories have been
considered by C. L. Reedy [33]. The version of algebraic theories enriched over*
* topological spaces
can be found in [3, (6)], [4, Chpt. II] or [35, 34]. For many purposes, topolog*
*ical and simplicial
theories can be used interchangeably: the geometric realization and singular co*
*mplex functors
commute with finite products, so applying these to the simplicial hom set or th*
*e hom spaces
respectively gets one from simplicial to topological theories and vice versa. W*
*e discuss examples
of algebraic theories in 2.6 and Section 7.
An algebraic theory is essentially a category with objects indexed by the natur*
*al numbers in such
a way that the n-th object is the n-fold product of the first object, in a spec*
*ified way. The
prototypical example (and in fact the initial algebraic theory) is the category*
* opposite to the
category opof finite pointed sets.
Definition 2.1A simplicial theory is a pointed simplicial category T together w*
*ith a functor
-! T . It is required that T has the same discrete set of objects as , and tha*
*t -! T is
the identity on objects and preserves products. A morphism of simplicial theori*
*es is a product
preserving simplicial functor commuting with the functor from .
One should think of the simplicial set homT(n+; 1+) as the space of n-ary opera*
*tions in the theory
T . If all the simplicial hom sets are discrete and if one omits the condition *
*that T be pointed,
one recovers the original definition of an algebraic theory. For emphasis we re*
*fer to such theories
as discrete theories. Since morphisms of theories are always the identity on ob*
*jects, a simplicial
theory is the same as a simplicial object of pointed discrete theories. There i*
*s an initial simplicial
theory, the theory of pointed sets. Formally it is the category together with *
*the identity functor.
Definition 2.2If T is a simplicial theory, a T -algebra is a product preserving*
* simplicial functor
X from T to the category of pointed simplicial sets. A morphism of T -algebras*
* is a natural
transformation of functors. X(1+) is called the underlying simplicial set of th*
*e T -algebra X.
9
Note that even if T happens to be a discrete theory, T -algebras are still allo*
*wed to have a simplicial
*
*+; 1+) gives
direction, unless otherwise stated. For a T -algebra X, each morphism ' 2homT(n*
* 0
rise to a map
X(1+)n ~=X(n+) -X(')---!X(1+) :
This justifies thinking of a T -algebra as the underlying simplicial set togeth*
*er with n-ary operations
parameterized by the simplicial set homT(n+; 1+). A morphism OE : R -! T of sim*
*plicial theories
induces a functor OE* : T -alg-! R-algby precomposition with OE. The functor OE*
** always has a left
adjoint OE* [6, Thm. 3.7.7].
2.3 Free T -algebras. The forgetful functor T -alg-! (pt. simpl. sets), X 7! X(*
*1+) has a left
adjoint, the free T -algebra functor F T. For a pointed simplicial set Y , the *
*underlying simplicial
set of F T(Y ) is given by the coend
Z n+2
F T(Y ) (1+) = Y n^homT(n+; 1+) :
(The forgetful functor is equal to the functor j* for the unique theory morphis*
*m j : -! T .
Hence the free functor F Tis isomorphic to j*.)
For any n, the representable functor homT(n+; -) is a T -algebra isomorphic to *
*the free T -algebra
generated by n+. In fact, this gives an equivalence of simplicial categories be*
*tween T opand the
full subcategory of the finitely generated free T -algebras inside T -alg (cf. *
*[6, Prop. 3.8.5]). The
composite of the free T -algebra functor with the forgetful functor has the str*
*ucture of a triple on
the category of pointed simplicial sets. A triple also has a notion of algebras*
* over it; however the
category of algebras over the triple derived from T is equivalent to the catego*
*ry of T -algebras,
so there is no ambiguity as to what a T -algebra is. Note that the triple, cons*
*idered as a functor
from the category of pointed simplicial sets to itself, is degreewise evaluable*
*, i.e., it comes from a
-space.
This leads to an alternative characterization of algebraic theories via the fre*
*e T -algebra functor:
the category of simplicial theories is equivalent to the category of those trip*
*les on the category
of pointed simplicial sets which are degreewise evaluable and commute with filt*
*ered colimits. Yet
another way of putting this is as follows: we denote by T sthe restriction of t*
*he free T -algebra
functor to the category op. So T sis a -space which comes with maps
T sO T s-! T s and S -! T s
which are associative and unital. This just says that T sbecomes a monoid in th*
*e category of -
spaces with respect to the composition product of 1.8. The functor that sends a*
* simplicial theory
T to the -space T swith the composition product structure is an equivalence of *
*categories
(simplicial theories)~=(monoids(inGS; O)) :
This result can be found in [6, Prop. 4.6.2] for discrete theories, and in [4, *
*Prop. 2.30] for topological
theories.
2.4 Models. A T -algebra is also called a model of T in the category of pointed*
* simplicial sets.
We will also consider models of a theory in categories other than simplicial se*
*ts. Let C be a pointed
category which has finite products and which is enriched over simplicial sets. *
*A model of T in C
is a product preserving simplicial functor X : T -! C. A morphism of models in *
*C is a natural
transformation of functors. The object X(1+) 2 C is called the underlying objec*
*t of X. We will
denote by C(T ) the category of models of T in C. For example, a group object o*
*f T -algebras is the
10
same thing as a model of T in the category of simplicial groups. Group objects *
*of T -algebras can
also be viewed as models of the theory G of groups (see 2.6) in the category of*
* T -algebras. In this
paper we will consider models of T in the categories of simplicial abelian grou*
*ps Ab(T ), spectra
Sp(T ) and -spaces GS(T ).
We will later need the following lemma whose proof we omit.
Lemma 2.5 Let C and D be two categories with finite products which are enriche*
*d over simplicial
sets. Let L : C -! D and R : D -! C be a simplicial adjoint functor pair such *
*that the left
adjoint L preserves finite products. Then for any simplicial theory T , composi*
*tion with L and R
is an adjoint functor pair between the categories of models of C(T ) and D(T ).
A first instance of this lemma is the geometric realization and singular comple*
*x functor pair
between the categories of simplicial sets and compactly generated topological s*
*paces. Another
example to which we will apply the lemma is the adjoint functor pair H and L be*
*tween -spaces
and simplicial abelian groups.
2.6 Example: the theory of groups. To illustrate the above definitions, we rec*
*all how the
familiar example of the category of groups fits into the abstract framework. We*
* denote the theory
of groups by G. This is a discrete theory with homG(k+ ; n+) equal to the set *
*of n-tuples of
elements of the free group on k generators fl1; : :;:flk. The identity morphism*
* of homG(k+ ; k+ )
is the tuple (fl1; : :;:flk) whose i-th component is the word consisting only o*
*f the i-th generator.
Composition is given by substitution: if (w1; : :;:wn) and (v1; : :;:vk) are tu*
*ples of words in k
resp. m generators, their composite is the tuple
(w1(v1; : :;:vk); : :;:wn(v1; : :;:vk)) :
Here wi(v1; : :;:vk) means that in the word wieach generator flj is substituted*
* by the entire word
vj. The functor -! G is given by
hom (k+ ; n+) = {0; 1; : :;:k}n----!homG(k+ ; n+)
(i1; : :;:in)7-! (fli1; : :;:flin)
with the convention fl0 = 1.
We claim that the category of G-algebras is equivalent to the category of simpl*
*icial groups. So
let X : G -!(pt. simpl. sets) be a G-algebra, i.e., a product preserving functo*
*r. Then the
underlying simplicial set X(1+) has a group structure as follows. The word fl1f*
*l2 is an element of
homG(2+; 1+), so it gives rise to a multiplication map
= X(fl1fl2) : X(1+)2 ~=X(2+) -! X(1+) :
The word fl-11is an element of homG(1+; 1+), so it gives rise to an inverse map
= X(fl-11) : X(1+) -! X(1+) :
The associativity and inverse conditions are codified in the category G. We ex*
*plain this for
associativity. We consider the two elements (fl1fl2; fl3) and (fl1; fl2fl3) of*
* HomG(3+; 2+). Since
multiplication in the free group on 3 generators is associative, the equality
(fl1fl2) O (fl1fl2; fl3) = fl1fl2fl3 = (fl1fl2) O (fl1; fl2fl*
*3)
holds in homG(3+; 1+). The product preserving functor X takes this relation to *
*the associativity
condition O (xid) = O (idx). Hence the underlying simplicial set of a G-algeb*
*ra is naturally
a simplicial group.
11
For the converse let H be a simplicial group. Define a functor H : G -!(pt. sim*
*pl. sets) on objects
H(n+) = Hn. The behavior on morphisms is again given by substitution: for w *
*= (w ; : :;:w )
by *
* 1 n
from homG(k+ ; n+), define
H (w)(h1; : :;:hk) = (w1(h1; : :;:hk); : :;:wn(h1; : :;:hk)) :
Here wi(h1; : :;:hk) means that the elements hj 2 H are substituted for the gen*
*erators flj into the
word wi, and then multiplication is carried out in the group H. We omit the ver*
*ification that this
H is a functor and that we in fact described an equivalence of categories
G-alg~= (simplicial groups):
This example illustrates the general pattern: an arbitrary algebraic theory can*
* be recovered from
its category of algebras as the opposite of the full subcategory of finitely ge*
*nerated free objects.
There is also a criterion for when a category with an adjoint functor pair to t*
*he category of sets
is (equivalent to) the category of algebras over some algebraic theory, see [6,*
* Thm. 3.9.1]. The
example of the theory of groups is somewhat special because here the operations*
* are generated
by unary and binary operations, and all relations involved at most three genera*
*tors. This need
not be true for general theories. In fact, what makes Lawvere's notion of algeb*
*raic theories so
elegant is that generating operations and relations are not mentioned at all. I*
*nstead, the category
underlying the theory encodes all possible operations and their relations at th*
*e same time.
3 Unstable homotopy
The model category structure for algebras over a discrete theory is due to Quil*
*len [32, II.4 Thm.
4]. For simplicial theories it was established by Reedy [33, Thm. I] and for to*
*pological theories by
Schw"anzl and Vogt [34, Thm. B]. A map of T -algebras is a weak equivalence or *
*fibration if it is a
weak equivalence or fibration on underlying simplicial sets respectively. A map*
* of T -algebras is a
cofibration if it has the left lifting property with respect to all acyclic fib*
*rations. A map A -! B
of T -algebras is called a free map if there exists subsets Cn Bn which are st*
*able under the
simplicial degeneracy operators and such that in every simplicial dimension n, *
*the induced map
An q F Tn(Cn) -! Bn
is an isomorphism of discrete Tn-algebras.
Theorem 3.1 [33, Thm. I] Let T be a simplicial theory. Then the category of T*
* -algebras is a
closed simplicial model category. The cofibrations are precisely the retracts o*
*f free maps.
Proof: The model category structure follows from the lifting lemma A.2, applied*
* to the forgetful
and free T -algebra functors between the categories of T -algebras and the cate*
*gory of pointed
simplicial sets. The category of T -algebras is locally finitely presentable, s*
*ee Lemma A.1. As the
fibrant replacement functor Q we can take either Kan's functor Ex1 [21] or the *
*composition of
the singular complex and geometric realization functor (we then have to work in*
* the category
of compactly generated topological spaces). Each of these functors is simplici*
*al and preserves
finite products, so it passes to T -algebras. Quillen's argument [32, II.4 Rem.*
* 4] shows that the
cofibrations are the retracts of the free maps. *
* __|_|
12
3.2 The bar resolution. It will be convenient to have at our disposal the stan*
*dard cotriple
resolution, also called bar resolution, for T -algebras. With the bar resoluti*
*on, homotopical
properties of free objects can be extended to all cofibrant objects. We denote*
* by T the
composite of the forgetful functor T -alg-!(pt. simpl. sets)with the free T -al*
*gebra functor
F T: (pt. simpl. sets)-! T.-algThis T is a cotriple on the category of T -algeb*
*ras, so for any
T -algebra X a simplicial object of T -algebras B(X) is defined by B(X)n = (T)n*
*+1(X) and with
the usual simplicial face and degeneracy maps (see [28, 9.6]). By construction,*
* B(X)n is a free
T -algebra and all structure maps except one of the face maps are free maps wit*
*h respect to the
defining generators. If Y* is any simplicial object of T -algebras, then the di*
*agonal of the under-
lying bisimplicial set is naturally another T -algebra. We denote this diagonal*
* T -algebra by |Y*|
and refer to it as the geometric realization. The bar resolution is augmented o*
*ver the constant
simplicial T -algebra X, so there is a map of T -algebras |B(X)| -! X.
Lemma 3.3 The augmentation map |B(X)|-! X is a weak equivalence of T -algebras*
*. If X -! Y
is a map of T -algebras which is injective on underlying simplicial sets, then *
*|B(X)| -! |B(Y )| is
a free map of T -algebras. In particular, |B(X)| is cofibrant as a T -algebra.
Proof: The fact that the augmentation map |B(X)|-! X is a weak equivalence on u*
*nderlying
simplicial sets is a well known property of the bar construction, see e.g. [28*
*, Prop. 9.8]. It
remains to show that |B(-)| takes injective maps to free maps. In simplicial di*
*mension n, the map
B(X)n -! B(Y )n is freely generated, in the category of discrete Tn-algebras, b*
*y the injective map
underlying (T)n(X)n -! (T)n(Y )n. So if we let Cn denote the complement of the *
*image of
(T)n(X)n in (T)n(Y )n, then the sets Cn satisfy the conditions in the definitio*
*n of a free map.
__|_|
As an application of the bar resolution we obtain a homotopy invariance propert*
*y, which can be
found in [33, Cor. p. 37]. In the context of topological theories, results of a*
* similar kind can be
found in [3, Thm. (8)] and [4, Thm. 4.58].
Lemma 3.4 Let OE : T -! R be a morphism of simplicial theories which is a weak*
* equivalence
on all simplicial hom sets. Then the adjoint functors OE* and OE* are a Quillen*
* equivalence between
the model categories of T -algebras and R-algebras.
Proof: OE* preserves underlying simplicial sets, hence weak equivalences and f*
*ibrations, so OE*
and OE* form a Quillen pair. Since OE* detects and preserves all weak equivalen*
*ces, it remains to
check that for every cofibrant T -algebra X, the unit of the adjunction X -! OE*
**OE*X is a weak
equivalence. If X is freely generated by a finite set, this unit map is a weak*
* equivalence by
assumption. If X is freely generated by an arbitrary set, it is the filtered co*
*limit, over cofibrations,
of finitely generated free T -algebras. If X is freely generated by a simplicia*
*l set, the realization
lemma (degreewise weak equivalences of bisimplicial sets induce weak equivalenc*
*es on the diagonal
simplicial sets, see [19, Prop. 2.4]) reduces to the discrete case. The bar re*
*solution and the
realization lemma reduce the general case to the free case. *
* __|_|
3.5 Homotopy fibre sequences. Since T -algebras have underlying simplicial sets*
*, we can in-
troduce the usual notions of connectivity of maps and objects. By the homotopy*
* groups of a
T -algebra we always mean the homotopy groups of the geometric realization of t*
*he underlying
simplicial set. A T algebra will be called n-connected if all homotopy groups b*
*elow and including
dimension n are trivial. A map of T -algebras is called n-connected if it induc*
*es isomorphisms on
homotopy groups below dimension n and an epimorphism in dimension n (for all ch*
*oices of base-
point in the underlying simplicial set of the source). The homotopy fibre of a *
*map of T -algebras
X -! Y is defined by choosing a factorization in the category of T -algebras
X _____wW~ _____Yww
13
of the map into a weak equivalence and a fibration and then taking the categori*
*cal fibre of the
fibration W -! Y . The homotopy fibre is independent up to weak equivalence of*
* the choice
of factorization and it is also a homotopy fibre in the underlying category of *
*simplicial sets. If
X -! Y -! Z are two maps of T -algebras whose composite is trivial, there is an*
* induced map
from X to the homotopy fibre of Y -! Z. We call the sequence X -! Y -! Z an n-h*
*omotopy
fibre sequence if the map X -! hofibre(Y -! Z) is n-connected.
Theorem 3.6 Let X -! Y be an (n + k)-connected cofibration between n-connected*
* cofibrant
T -algebras (with n 1, k 0). Then the cofibration sequence of T -algebras
X ----! Y ----! Y==X
is a (2n+k)-homotopy fibre sequence. Here Y==X denotes the quotient in the cate*
*gory of T -algebras
which has to be distinguished from the quotient of the underlying simplicial se*
*ts.
Proof: We first prove the theorem in the special case where the map X -! Y is o*
*btained from
an (n + k)-connected cofibration A -! B between n-connected simplicial sets by *
*application of
the free T -algebra functor. In this case the quotient of T -algebras Y==X is *
*isomorphic to the
free T -algebra generated by the quotient of simplicial sets B=A. The free T -a*
*lgebras functor is a
prolonged -space, so by Lemma 1.10 the map
Y=X = T (B)=T (A) ----! T (B=A) = Y==X
is (2n + k + 2)-connected. By the Blakers-Massey homotopy excision theorem, th*
*e sequence
X -! Y -! Y=X is a (2n + k)-homotopy fibre sequence of simplicial sets. The the*
*orem follows
in the special case.
In the general case we use the bar resolution. The cofibration sequence of T -*
*algebras X -!
Y -! Y==X admits a map from the cofibration sequence |B(X)| -! |B(Y )| -! |B(X)*
*|==|B(Y )|.
Since all objects in sight are cofibrant and the maps |B(X)| -! X and |B(Y )| -*
*! Y are weak
equivalences, the map induced on cofibres |B(X)|==|B(Y )| -! Y==X is also a wea*
*k equivalence.
So it suffices to show that the sequence
|B(X)| ----! |B(Y )| ----! |B(X)|==|B(Y )|
is a (2n + k)-homotopy fibre sequence. Geometric realization of simplicial T -a*
*lgebras commutes
with taking quotients, so all three T -algebras are realizations of simplicial *
*objects. In a fixed
simplicial dimension, all objects are freely generated by certain simplicial se*
*ts, and all maps are
free maps. So by the previous paragraph, the sequence of bar resolutions is a (*
*2n + k)-homotopy
fibre sequence in every simplicial degree. But geometric realization preserves*
* connectivity and
homotopy fibre sequences of connected objects [9, Thm. B.4], which finishes the*
* proof. __|_|
4 Stable homotopy
The goal of this section is to show that the stable homotopy theory of T -algeb*
*ras is equivalent to
the homotopy theory of modules over a certain Gamma-ring T s. The theorem we ar*
*e heading for
has a well known algebraic analog which we first recall. The notion of an abeli*
*an group object in
the category of T -algebras coincides with that of a model of T in the category*
* Ab of simplicial
abelian groups. The forgetful functor from abelian group objects to T -alg has *
*a left adjoint and
the category Ab(T ) of abelian group objects is in fact an abelian category. Mo*
*reover, the category
14
Ab(T ) is equivalent to the category of modules over a certain simplicial ring *
*T ab, namely the
endomorphism ring of the free abelian group object on one generator:
Ab(T ) ~= T ab-mod:
In Theorem 4.4 we are proving the homotopy theoretic analog of this fact. The i*
*dea is to replace
abelian group objects by its homotopy analogue, namely infinite loop objects or*
* spectra. Spectra
of T -algebras form a model category Sp(T ). Then there is another `ring' T swh*
*ose modules are
the spectra of T -algebras. We only have to allow Gamma-rings instead of simpli*
*cial rings, and
instead of an equivalence of categories we obtain a Quillen equivalence of mode*
*l categories
Sp(T )conn ' T s-mod:
T sis the endomorphism Gamma-ring of the free T -algebra on one generator (see *
*4.6 for the precise
meaning). The homotopy groups of T sare isomorphic to the ring of stable homoto*
*py operations
of T -algebras. The two rings arising from the theory T are closely related. Th*
*ere is a 1-connected
map of Gamma-rings T s-! HT abwhich governs the relationship between stable hom*
*otopy and
homology of T -algebras. The simplicial ring T abcan be obtained from the Gamma*
*-ring T sby
applying the functor L left adjoint to the Eilenberg-MacLane functor (see Theor*
*em 5.2).
4.1 Spectra of T -algebras. To define spectra, we need to recall the definition*
* of the suspension
of a T -algebra X. Since T -algebras form a simplicial model category, the prod*
*uct X T S1 with
the simplicial circle is defined. The suspension of X is then obtained as the c*
*ofibre, in the category
of T -algebras, of the map X -! X T S1 induced by the inclusion of the unique v*
*ertex into S1.
X is a bar construction with respect to the coproduct of T -algebras. This mea*
*ns that it is
the geometric realization of the simplicial T -algebra which in simplicial degr*
*ee k consists of the
coproduct of k copies of X. The suspension functor has an adjoint which is def*
*ined dually. The
loop functor commutes with the forgetful functor, i.e., the underlying simplici*
*al set of X is the
simplicial set of pointed maps of S1 into the underlying simplicial set of X. *
*and are a Quillen
adjoint functor pair. The total derived functors of and are the suspension an*
*d loop functors
on the homotopy category of T -algebras [32, I.2]. The simplicial structure thu*
*s provides liftings
of these functors from the homotopy category to functors on the actual model ca*
*tegory. One has
to remember that X can have the `wrong' homotopy type if X is not cofibrant jus*
*t as Y can
have the `wrong' homotopy type if Y is not fibrant.
For our purposes, the naive definition of a spectrum suffices. The following is*
* an elaboration on
the construction of [9, x2].
Definition 4.2A spectrum X of T -algebras is a collection of T -algebras Xn; n *
* 0; and T -
algebra homomorphisms Xn -! Xn+1. A morphism of spectra f : X -! Y is a collect*
*ion of
maps fn: Xn -! Yn such that all the diagrams
fn
Xn ______Ynw
| |
| |
| |
| |
|u |u
Xn+1 _____Yn+1wfn+1
commute. We denote the category of spectra by Sp(T ).
15
If T is the theory of sets, a T -algebra is just a simplicial set and the above*
* definition reduces to
the notion of a spectrum as in [9, x2]. In general, a spectrum of T -algebras i*
*s the same thing as
a model of T in the category of spectra of [9]. We call a map of spectra of T -*
*algebras a weak
equivalence (resp. fibration) if it is a stable weak equivalence (resp. stable *
*fibration) as a map of
spectra of simplicial sets. We call a map a cofibration if it has the left lift*
*ing property with respect
to all acyclic fibrations of spectra of T -algebras.
Theorem 4.3 If T is a simplicial theory, then the category Sp(T ) of spectra o*
*f T -algebras is a
closed simplicial model category.
Proof: We apply Lemma A.2 to lift the stable model category structure of spectr*
*a of simplicial
sets to the category Sp(T ). The adjoint functor pair to use consists of the fo*
*rgetful and the free
T -algebra functor, applied dimensionwise to spectra. All limits, colimits as w*
*ell as tensors and
cotensor with simplicial sets are inherited from the category of T -algebras an*
*d they are defined
dimensionwise for spectra of T -algebras. The category Sp(T ) is locally finite*
*ly presentable (Lemma
A.1) and the model category of spectra of simplicial sets is cofibrantly genera*
*ted (Lemma A.3)
.So it remains to describe a functor Q that provides fibrant replacements. In [*
*9, x2] Bousfield and
Friedlander use a functor Q given by
(QX)n = colimiiSing|Xn+i| :
(Kan's functor Ex1 can be substituted for the geometric realization of the sing*
*ular complex). A
priori, the functor Q is defined for spectra of simplicial sets; but this choic*
*e of Q is simplicial and
preserves finite products, so it passes to the category of spectra of T -algebr*
*as. For every spectrum
X, QX is an -spectrum and degreewise a fibrant simplicial set. It is thus fibra*
*nt in the stable
model category of spectra by [9, A.7]. So the lifting lemma A.2 applies. *
* __|_|
Now we can state the main theorem of this section, saying that the (connective)*
* stable homotopy
theory of T -algebras is equivalent to the homotopy theory of modules over a ce*
*rtain Gamma-ring
T s. The fact that we only get connective spectra of T -algebras stems from the*
* fact that -spaces
only represent connective spectra.
Theorem 4.4 To a simplicial theory T there is functorially associated a Gamma-*
*ring T s. The
ring ss*T sis isomorphic to the ring of stable homotopy operations of T -algebr*
*as. There is a Quillen
adjoint functor pair
T s-mod _____wu_____Sp(T )
whose total derived functors are inverse equivalences between the homotopy cate*
*gory of T s-modules
and the homotopy category of connective spectra of T -algebras,
Ho (T s-mod) ~= Ho (Sp(T ))conn:
4.5 The Gamma-ring T s. The -space underlying the Gamma-ring T sis defined as t*
*he com-
posite of the free T -algebra functor, restricted to the category op, with the *
*forgetful functor
from T -algebras to pointed simplicial sets. The composite of the free T -algeb*
*ra functor with the
forgetful functor is a triple on the category of simplicial sets. As we pointed*
* out in 2.3, this means
that T scomes with associative and unital maps
T sO T s-! T s and S -! T s
making it a monoid in the category of -spaces with respect to the composition p*
*roduct O. Com-
position with the assembly map (see 1.8)
T s^T s-! T sO T s-! T s
gives T sa multiplication with respect to the smash product. Since the assembly*
* map is associative
and unital, T sbecomes a Gamma-ring.
16
4.6 A generalization: endomorphism Gamma-rings. The construction of the Gamma-r*
*ing
sis a special case of a more general construction of endomorphism Gamma-rings*
*. As input we
T
can use any pointed category C which has finite coproducts. Then C is tensored *
*over the category
op, i.e., the assignment
X^ n+ = X_q_:_:q:X-z____"
n
is the object function of a functor ^ : C x op -! C. As a consequence, the cate*
*gory C is also
enriched over the category of -sets. This means that for any two objects X and *
*Y of C there is
a homomorphism -set HOM(X; Y ) defined by
HOM (X; Y )(n+) = C(X; Y ^n+) :
Furthermore there is a unit morphism S -! HOM (X; X) induced by the identity o*
*f X and
associative and unital composition pairings
HOM (Y; Z) ^HOM (X; Y ) ----! HOM (X; Z):
The composition pairing is induced by the universal property of the smash produ*
*ct of -sets from
the maps
C(Y; Z ^n+) ^C(X; Y ^m+)----! C(X; Z ^n+^ m+)
f ^ g 7! (f ^idm+) O g :
In particular, for every object X, the endomorphism -set HOM(X; X) becomes a Ga*
*mma-ring.
If the category C is also enriched over the category of simplicial sets (as is *
*the case for algebras
over a simplicial theory), then the enrichment over -sets extends to one over -*
*spaces. This basic
observation gives a rich supply of (endomorphism) Gamma-rings. For a simplicial*
* theory T the
category of T -algebras is pointed, simplicially enriched and has coproducts. T*
*he value at n+ of
the endomorphism Gamma-ring of the free T -algebra on one generator is given by
HOM (F T(1+); F T(1+))(n+) ~= Hom T-alg(F T(1+); F T(n+))
which is naturally isomorphic to the value of the Gamma-ring T sat n+. Since th*
*e isomorphism
also preserves the multiplications we obtain the following
Lemma 4.7 Let T be a simplicial theory. Then the Gamma-ring T sis isomorphic t*
*o the endo-
morphism Gamma-ring of the free T -algebra on one generator.
Warning: The homomorphism -spaces HOM(X; Y ) just defined are usually not adjoi*
*nt to any
kind of smash product pairing between C and the category of -spaces. For exampl*
*e HOM(X; Y )
usually does not preserve limits in the second variable (although it does take *
*colimits in the
first variable to limits). If C = GS is the category of -spaces, then the homom*
*orphism -space
HOM(X; Y ) is different from the internal hom -space Hom(X; Y ) which is adjoin*
*t to the smash
product. Indeed the former is made up from homomorphisms into -spaces of the fo*
*rm Y ^n+,
whereas the latter uses maps into -spaces of the form Yn+^-. The natural maps *
*of -spaces
Y ^n+ -! Yn+^- induce a natural map HOM(X; Y ) -!Hom(X; Y ).
4.8 Comparison of the stable categories. We now proceed to compare the two cat*
*egories
Sp(T ) and T s-mod. This will be done through an intermediate category GS(T ), *
*the category of
pointed functors op-! T -alg, alias the models of T in the category of -spaces.*
* We obtain three
model categories with Quillen adjoint functor pairs
_*___ (-)(S)_
T s-mod u_____w GS(T ) u_____ wSp(T )
* (S;-)
The left pair is a Quillen equivalence, the right is a Quillen pair which passe*
*s to an equivalence of
the homotopy category of GS(T ) with that of connective spectra of T -algebras.
17
Step 1. We first establish the stable model category structure for GS(T ). We*
* call a map in
GS (T ) a weak equivalence (resp. fibration) if and only if it is a stable equi*
*valence (resp. stable
Q-fibration) of underlying -spaces. A map is called a cofibration if it has the*
* left lifting property
with respect to all acyclic fibrations.
Theorem 4.9 With these notions of fibrations, cofibrations and weak equivalenc*
*es, the category
GS (T ) becomes a closed simplicial model category. If X -! Y is a cofibration *
*in GS(T ), then for
every pointed simplicial set K, the map X(K) -! Y (K) is a cofibration of T -al*
*gebras.
Proof: We want to apply Lemma A.2 to lift the stable Q-model category structure*
* from -spaces
to GS(T ). The adjoint functor pair to use consists of the forgetful and the fr*
*ee T -algebra functor,
applied objectwise to -objects. As a functor category into a complete and cocom*
*plete simplicially
enriched category, GS(T ) has all limits and colimits as well as tensors and co*
*tensor with simplicial
sets. The category GS(T ) is locally finitely presentable, see Lemma A.1. The c*
*rucial ingredient is
the stably fibrant replacement functor Q. One possible choice of such Q is give*
*n by
(QX)(n+) = colimiiSing|X(Si^n+)| :
Again we can use Ex1 [21] instead of geometric realization and singular complex*
*. A priori, the
functor Q is only defined on the category of -spaces. However, Q is a simplicia*
*l functor and it
preserves finite products, so it passes to an endofunctor on the category GS(T *
*). There is a natural
stable equivalence X -! QX, and QX is pointwise fibrant and very special, so it*
* is fibrant in the
stable Q-model category structure. Thus the lifting lemma A.2 applies.
To get the statement about cofibrations we first consider generating cofibratio*
*ns. These are of
the form X = T sO A -! T sO B = Y for A -! B a cofibration of -spaces. Then th*
*e map
X(K) -! Y (K) is obtained from a cofibration of simplicial sets by application *
*of the free T -
algebra functor, so it is a cofibration of T -algebras. The general case follow*
*s by the small object
argument [37, Lemma A1] since the property in question is preserved under cobas*
*e change, trans-
finite composition and retract. *
* __|_|
Step 2. The comparison of the category Sp(T ) of spectra of T -algebras with th*
*e category GS(T )
of -objects of T -algebras follows easily from the work of Bousfield and Friedl*
*ander. In [9, x5],
they show that the functor X 7! X(S) from -spaces to spectra has a right adjoin*
*t (S; -). Both
functors are simplicial and the left adjoint preserves finite products, so they*
* pass to adjoint functors
between the categories GS(T ) and Sp(T ) (see Lemma 2.5). Since (stable) weak e*
*quivalences and
fibrations in GS(T ) and Sp(T ) are defined on underlying -spaces or spectra re*
*spectively, the
functors (-)(S) and (S; -) still form a Quillen pair. We have to note here that*
* the stable Q-
model category structure for -spaces has more fibrations than the stable Bousfi*
*eld-Friedlander
model category structure. If X is a connective fibrant spectrum of T -algebras,*
* then the adjunction
map (S; X)(S) -~!X is a stable equivalence. So if A is a cofibrant object in GS*
*(T ), then a map
A(S) -! X is a stable equivalence if and only if the adjoint map A -! (S; X) is*
*. Thus the
Quillen adjoint functor pair passes to an equivalence between the homotopy cate*
*gory of GS(T )
and the homotopy category of connective spectra of T -algebras.
Step 3. In this last step we want to construct a Quillen equivalence between th*
*e category GS(T )
and the category of T s-modules. The associative and unital assembly map X ^Y -*
*! X O Y of
1.8 gives a morphism
: T s^- -! T sO -
of triples on the category of -spaces. An algebra over the triple T s^- is not*
*hing but a T s-
module, an algebra over the triple (T sO -) is a -object of T -algebras. Pullin*
*g back along the
18
triple morphism gives a functor * : GS(T ) -! T s-mod. This functor has a left*
* adjoint *
* preserves fibrations and weak equivalen*
*ces since these are
(see [24, Cor. 1]). The right adjoint
defined everywhere on underlying -spaces; so the functors form a Quillen pair. *
*Since the right
adjoint in fact detects and preserves all weak equivalences, it suffices to show
Lemma 4.10 For a cofibrant T s-module A, the unit map A -! **A of the adjuncti*
*on is a
stable equivalence.
Proof: We assume first that the cofibrant T s-module A is induced, i.e., it is *
*of the form A = T s^Y
for some cofibrant -space Y . Pushforward along a map of triples takes free obj*
*ects to free objects,
so in this case the map in question is the assembly map
A = T s^Y -! T sO Y = **A ;
which is a weak equivalence by [25, Prop. 5.23]. Now we assume that the cofibra*
*nt T s-module A
can be written as the pushout of a diagram of T s-modules
A0 u_____K v_____Lw:
in which K -! L is a cofibration and such that Lemma 4.10 holds for the cofibra*
*nt T s-module A0
and the quotient module L=K ~=A=A0. We claim that then the lemma also holds for*
* A. Indeed,
cofibrations of T s-modules are injective and cofibres of T s-modules are calcu*
*lated on underlying
-spaces [37, Thm. 2.2]. So the cofibre sequence of T s-modules A0 -! A -! A=A0*
* gives rise
to a long exact sequence of homotopy groups by [37, Lemma 1.3]. As a left adjoi*
*nt in a Quillen
functor pair * preserves pushout, cofibres and cofibrations. Hence the five le*
*mma gives the
desired conclusion once we know that the cofibre sequence in GS(T )
*A0----! *A ----! *(A=A0)
also gives rise to a long exact sequence of homotopy groups. If we evaluate at *
*the simplicial n-
sphere Sn, we obtain a cofibre sequence of (n - 1)-connected cofibrant T -algeb*
*ras (by Theorem
4.9). By Theorem 3.6, we obtain a long exact sequence of homotopy groups in a s*
*table range, and
we let n go to infinity.
The previous two paragraphs together show that the conclusion of Lemma 4.10 hol*
*ds for all T s-
modules which can be obtained from the trivial module by finitely many cobase c*
*hanges along
cofibrations between modules that are induced from -spaces. Then it also holds*
* for modules
which are filtered direct limits of such modules. But an arbitrary cofibrant T *
*s-module is a retract
of one of this sort by the small object argument [37, Lemma A1]. *
* __|_|
4.11 Stable homotopy operations. Interpreting ss*T sin terms of stable homotopy*
* operations
is a standard representability argument. We will be brief since this result wil*
*l not be used in the
rest of this paper. A homotopy operation of T -algebras is a natural transforma*
*tion ssn -! ssm of
functors T -alg-!(sets)for some n; m. Homotopy operations can be composed if th*
*e source of one
is the target of the other, and they form a category with objects the natural n*
*umbers. The functor
ssn is represented by F T(Sn) in the homotopy category of T -algebras, i.e., ss*
*nX ~=[F T(Sn); X]
(the right hand side denotes maps in the homotopy category). Consequently, the*
* category of
homotopy operations is isomorphic the opposite of the full subcategory of Ho (T*
* -alg)generated by
the F T(Sn). In particular, homotopy operations ssn -! ssm are in bijective cor*
*respondence with
elements of [F T(Sm ); F T(Sn)]. Homotopy operations can be suspended, i.e., if*
* o : ssn -! ssm is
one, one defines o : ssn+1 -! ssm+1 on a T -algebra X as the composite
o|X|
ssn+1|X| ~= ssn|X| ----! ssm |X| ~= ssm+1|X| :
19
The suspension of operations corresponds to the suspension
: [F T(Sm ); F T(Sn)] -! [F T(Sm+1 ); F T(Sn+1)]
in the homotopy category of T -algebras.
A stable homotopy operation of degree n is represented by a sequence (oi)ii0 of*
* homotopy oper-
ations oi: ssi- ! ssi+n with the property that oi+1= oi. Two such sequences def*
*ine the same
stable operation if the components eventually agree, i.e., if almost all compon*
*ents are equal. Sta-
ble homotopy operations can always be composed, the degrees add under compositi*
*on, and they
form a graded ring. The natural isomorphisms of ssn+i|F T(Si)| with the sets of*
* homotopy classes
[F T(Sn+i); F T(Si)] assemble into an isomorphism of ssn T s= colimissn+i|F T(S*
*i)| with the colimit
of the sets [F T(Sn+i); F T(Si)] over suspension; but the elements of colimi[F *
*T(Sn+i); F T(Si)] are
nothing but the stable homotopy operations of degree n. The fact that the isomo*
*rphism between
ss*T sand stable homotopy operations is multiplicative follows from the fact bo*
*th products are
(suitable kinds of) composition products, see Lemma 4.7.
5 Stable homotopy versus homology
We have seen that a simplicial theory T gives rise to two "rings" and a multipl*
*icative map between
them. There is the simplicial ring T abwhose modules are the abelian group obje*
*cts in the cate-
gory of T -algebras. And there is the Gamma-ring T swhose modules are (Quillen *
*equivalent to)
connective spectra of T -algebras. Abelianization induces a Gamma-ring map T s-*
*! HT ab. This
map encodes the relationship between stable homotopy and homology in the homoto*
*py theory of
T -algebras. In this section we will show that the map T s-! HT abis 1-connecte*
*d and we will
establish Hurewicz and Whitehead Theorems for T -algebras as well as universal *
*coefficient and
Atiyah-Hirzebruch spectral sequences.
5.1 Quillen homology. We recall Quillen's definition of homology as the left de*
*rived functor of
abelianization [32, II.5]. By definition, the abelianization functor
-ab: T -alg-! Ab(T )
is the left adjoint to the forgetful functor. If X is a T -algebra, one chooses*
* a cofibrant replace-
ment Xc -~!X and defines the homology of X to be the homotopy of the abelianiza*
*tion of the
replacement:
H *X = ss*(Xcab) :
The unit of the adjunction is a map of T -algebras X -! Xab which we refer to a*
*s the Hurewicz
map. More generally there is (co-)homology of a T -algebra with coefficients. *
* If M is a right
simplicial T ab-module, then the homology of X with coefficients in M is define*
*d as the homotopy
of the tensor product
H*(X; M) = ss*(M TabXcab) :
If N is a left simplicial T ab-module, then the cohomology groups of X with coe*
*fficients in N are
defined as the homotopy classes of T ab-module maps
H *(X; N) = [Xcab; *N]Tab-mod:
Note that unless the simplicial ring T abis commutative, homology and cohomolog*
*y need different
kinds of coefficients. In the case of the theory of sets this notion of homolog*
*y specializes to singular
homology of simplicial sets. For commutative rings it specializes to Andre-Quil*
*len homology [31,
Sec. 4].
20
Given a T -algebra X, we need a model for its suspension spectrum as a T s-modu*
*le. There is
sas the endomorphism Gamma-rin*
*g of the free
a slick definition using the interpretation of T
T -algebra on one generator as in Lemma 4.7. In the notation of 4.6 (with C = T*
* -alg), we can set
1 X = HOM (F T(1+); X) as a -space, and with T s-module structure given by the *
*composition
action of T s~=HOM (F T(1+); F T(1+)). An equivalent description of 1 X is as *
*follows. First
define a -object of T -algebras g1 X by
(g1 X) (n+) = X ^n+ = X_q_._.q.X-z____";
n
the coproduct being taken in the category of T -algebras. As a functor T -alg-!*
* GS(T ), g1 is
left adjoint to evaluation at 1+. Note that when g1 X is extended (by direct li*
*mit and degreewise
application) to a functor from simplicial sets to T -algebras, then we get (g1 *
*X) (K) = X ^K
(this smash product refers to the enrichment of the category of T -algebras ove*
*r pointed simplicial
sets). Hence the spectrum associated to g1 X is isomorphic to the suspension sp*
*ectrum of X as a
T -algebra, which justifies the name. Recall from 4.8 that -objects in T -alg c*
*an be pulled back to
T s-modules via a functor *. This means that the underlying -space of the -T -a*
*lgebra g1 X
is endowed with a left T s-action via the assembly map (1.8)
T s^g1 X -! T sO g1X -! 1 X :
The T s-module *^1 X is isomorphic to 1 X. So the two suspension spectrum objec*
*ts 1 X
and g1 X have the same underlying -spaces, but one is viewed as a T s-module, t*
*he other one as
an object of GS(T ).
Recall from [37, Lemma 1.2] that the left adjoint functor L to the Eilenberg-Ma*
*cLane functor H
is strong symmetric monoidal and preserves finite products. In particular, it t*
*akes Gamma-rings
to simplicial rings. Furthermore, the functors L and H pass to an adjoint funct*
*or pair between
the category GS(T ) of -objects of T -algebras and the category Ab(T ) of abeli*
*an group objects
of T -algebras by Lemma 2.5. In the following theorem we combine these formal p*
*roperties with
some homotopical input to obtain information on the relationship between stable*
* homotopy and
homology for T -algebras.
Theorem 5.2 For a T -algebra X, the object L(g1 X) 2 Ab(T ) is naturally isomo*
*rphic to Xab.
The map of Gamma-rings T s-! HT abinduces an isomorphism on ss0 and an epimorph*
*ism on ss1
and its adjoint is an isomorphism of simplicial rings L(T s) ~=T ab. In particu*
*lar, if T is a discrete
theory, then T ab~=ss0T s. For a right T ab-module M and a T -algebra X, there *
*is a natural map
of -spaces
HM ^LTs1 X ----! H(M TabXab)
which is a stable equivalence whenever X is cofibrant.
Proof: The forgetful functor from abelian group objects factors as a composite
+
Ab(T ) ---H--! GS(T ) -eval.-at-1---!T -alg:
Since L : GS(T ) -! Ab(T ) is left adjoint to H and g1 is left adjoint to evalu*
*ation at 1+, their
composite is a left adjoint to the forgetful functor. The adjoint of the map T*
* s-! HT abis a
homomorphism of simplicial rings so it suffices to show that it is an isomorphi*
*sm in Ab(T ). But
the -space T sunderlies the suspension spectrum of the free T -algebra on one g*
*enerator, so L(T s)
is isomorphic to the free abelian group object on one generator T abby what we *
*already proved.
The fact that T s-! HT abinduces an isomorphism on ss0 and an epimorphism on ss*
*1 then follows
from [37, Lemma 1.2].
21
There is a functorial cofibrant replacement (1 X)c -! 1 X in the category of T *
*s-modules.
sprovides a natural stable equivalence of -spac*
*es from
Then [37, Lemma 4.2] with R = T
HM^Ts(1 X)c to H(M TabL((1 X)c)). Since Xab ~=L(g1 X), it remains to show that
the map of cofibrant T ab-modules L((1 X)c) -! L(g1 X) is a weak equivalence. T*
*his follows
if we can show that for any T ab-module W , the induced map on homomorphism spa*
*ces
hom Tab-mod(L(g1 X); W ) ----! homTab-mod(L((1 X)c); W )
is a weak equivalence. By the various adjunctions, this map is isomorphic to th*
*e map
homGS(T)(g1 X; HW ) ----! homGS(T)(*(1 X)c; HW ) ~= homTs-mod((1 X)c; HW )
induced by the stable equivalence of cofibrant objects *(1 X)c -! g1 X (this us*
*es Lemma
4.10). Hence the latter map of homomorphism spaces is a weak equivalence, which*
* finishes the
proof. __*
*|_|
Corollary 5.3 (Hurewicz theorem)Let T be a simplicial theory and X a cofibrant *
*(n- 1)-
connected T -algebra (n 2). Then the Quillen homology of X vanishes below dime*
*nsion n and
the Hurewicz map induces an isomorphism ssnX ~=HnX and an epimorphism ssn+1X -!*
* Hn+1X.
Proof: An application of Theorem 3.6 to the cofibre sequence X -! Cone(X) -! X *
*shows
that the map |X| -! |X| is (2n - 1)-connected. Since n 2, the n-th homotopy gr*
*oup of X is
thus isomorphic to the n-th stable homotopy group and ssn+1X surjects onto the *
*(n + 1)-st stable
homotopy group. By Theorem 5.2, the -space HXabis stably equivalent to HT ab^LT*
*s1 X. So
the Tor spectral sequence for the derived smash product [37, Lemma 3.1] takes t*
*he form
s ab 1
Torss*Tp(ss*T ; ss* X)q =) Hp+q(X) :
Since the map T s-! HT abis 1-connected, this spectral sequence gives the desir*
*ed answer for
the first non-trivial homology groups of X. *
* __|_|
Corollary 5.4 (Whitehead theorem) Let T be a simplicial theory and X -! Y a ma*
*p of
simply connected T -algebras which induces an isomorphism in Quillen homology. *
*Then the map
is a weak equivalence.
Proof: We can assume that X and Y are 1-reduced and cofibrant and that the map *
*is a cofibra-
tion. Then the cofibre Y==X is 1-connected and has vanishing Quillen homology, *
*hence is weakly
contractible by the Hurewicz Theorem 5.3. Again by the Hurewicz Theorem, the ma*
*p X -! Y is
2-connected. An application of Theorem 3.6 to the cofibration X -! Y and induct*
*ion gives that
the map X -! Y is m-connected for all m *
* __|_|
5.5 Spectral sequences relating Quillen homology and stable homotopy. Let X be*
* a
cofibrant T -algebra and M a coefficient module for Quillen homology. By Theor*
*em 5.2, the
homology groups of X with coefficients in M are isomorphic to the stable homoto*
*py groups of the
-space HM ^LTs1 X. So the spectral sequence for the derived smash product [37, *
*Lemma 3.1]
gives a universal coefficient spectral sequence
s s
E2p;q= Torss*Tp(ss*M; ss*X)q =) Hp+q(X; M) :
22
This spectral sequence lies in the first quadrant. If X is (n - 1)-connected, i*
*f we take M = T ab
and if for simplicity we take T to be a discrete theory we can read off the fol*
*lowing six term exact
sequence
(ss1T s sssn+1X) (ss2T s sssnX)----!sssn+2X ----! H n+2X ----!
----! ss1T s sssnX----! sssn+1X ----! H n+1X ----! 0
which refines the part of the Hurewicz theorem that claims the surjectivity of *
*the Hurewicz map
in dimension n + 1.
If W is a right T s-module, then the homotopy groups of the derived*
* smash product
W*X = ss*(W ^LTs1 X) are a generalized homology theory in the T -algebra X. Th*
*e spectral
sequence [37, Lemma 3.1] together with Theorem 5.2 thus gives an Atiyah-Hirzebr*
*uch spectral
sequence
E2p;q= H p(X; ssqW ) =) Wp+q(X) :
6 Relation to theory cohomology
In this section we provide the link between the cohomology and the stable homot*
*opy of an algebraic
theory. In [20, Def. 4.2], M. Jibladze and T. Pirashvili introduce the cohomolo*
*gy of an algebraic
theory as Ext groups in an abelian functor category _ see Remark 6.4 for some b*
*ackground and
explanation about their cohomology theory. The main result of this section, The*
*orem 6.7, says
that the Jibladze-Pirashvili homology groups of a theory T with coefficients in*
* a functor G are
isomorphic to the topological Hochschild homology groups of the Gamma-ring T sw*
*ith coefficients
in a bimodule G!associated to G. This generalizes a theorem of T. Pirashvili an*
*d F. Waldhausen
[30, Thm. 3.2]. We also show that the analogous statement in cohomology holds *
*provided the
coefficient functor G is additive.
6.1 Homological algebra in functor categories. Let C be a small category with z*
*ero object
and R any ring. We denote by F(C; R) the category of covariant pointed functors*
* from C to the
category of left R-modules. This is an abelian category in which exactness is d*
*efined objectwise.
For every object c of C there are functors Pc and Ic defined by
Pc(d) = Re[C(c; d)] and Ic(d) = map *(C(d; c); Rinj) :
Here eR[-] denotes the reduced free R-module on the pointed set of morphisms fr*
*om c to d,
Rinj= Hom Z(R; Q=Z) is the injective cogenerator in the category of left R-modu*
*les and `map*'
denotes the set of pointed maps into Rinjwith the pointwise left R-module struc*
*ture. Because of
the Yoneda-type isomorphisms
Hom F(C;R)(Pc; G) ~= G(c) and Hom F(C;R)(G; Ic) ~= Hom R-mod(G(c); Ri*
*nj) ;
Pcis a projective object and Icis an injective object in the abelian category F*
*(C; R). Furthermore,
the functors Pc form a set of projectives generators and the functors Ic form a*
* set of injectives
cogenerators for F(C; R) when c runs over the objects of C.
23
6.2 (Co-)homology of algebraic theories. We consider a discrete, pointed algebr*
*aic theory
op; T ab) of pointed funct*
*ors from T opto
T . We abbreviate to F(T ) the abelian category F(T
the category of left T ab-modules. We recall from [6, Prop. 3.8.5] that T opis*
* equivalent to the
full subcategory of T -alg given by the finitely generated free T -algebras. A*
*lso the category of
left modules over the ring T abis equivalent to the category Ab(T ) of abelian *
*group objects of
T -algebras. T abis the endomorphism ring of the free abelian group object on *
*one generator,
and by Theorem 5.2 it is isomorphic to the ring ss0T s. The category F(T ) has *
*a special object
Iab, the abelianization functor for T -algebras, restricted to T op. Every (R *
*(T ab)op)-module M
defines a functor M TabIab in F(T op; R). These are precisely the additive func*
*tors, i.e., those
functors which commute with coproducts. The functor R-mod-T ab-! F(T op; R) whi*
*ch sends M
to M TabIabis right adjoint to the functor which sends G 2 F(T op; R) to the R-*
*T ab-bimodule
Gadd = coker(G(2+) (p1)*+(p2)*-r*---------!G(1+)) :
Here the right T ab-action is induced, through the functor G, from the action o*
*f the monoid
homT(1+; 1+) on the free T -algebra on one generator.
In the case where R = T aband G 2 F(T ), the abelian group Gaddthus has a two-s*
*ided action of
the ring T ab. In this case we can equalize the actions and define
QG = Gadd=(tx - xt) ;
i.e., we divide out the subgroup generated by elements of the form tx - xt for *
*x 2 Gaddand
t 2 T ab. Then Q is an additive, right exact functor from F(T ) to the category*
* of abelian groups
and so it has left derived functors LiQ.
Definition 6.3[20, Def. 4.2] Let T be a pointed discrete algebraic theory and G*
* 2 F(T ). The
homology and cohomology of T with coefficients in G are then defined as
H *(T ; G) = (L*Q)(G) and H *(T ; G) = Ext*F(T)(Iab; G) :
Remark 6.4 The notion of (co-)homology of a theory T with coefficients in a fu*
*nctor in the sense
of Jibladze and Pirashvili has to be distinguished from the Quillen (co-)homolo*
*gy of a T -algebra
X with coefficients in an abelian group object which we reviewed in 5.1. If the*
* theory T is fixed,
then Quillen homology provides a homology theory for varying T -algebras. For e*
*xample, Quillen
homology satisfies excision and is homotopy invariant.
The Jibladze-Pirashvili cohomology plays the same role for algebraic theories t*
*hat is played by
Hochschild cohomology for algebras over a field, and it generalizes MacLane coh*
*omology [26] for
arbitrary rings. For example in [20, Sec. 4], Jibladze and Pirashvili give int*
*erpretations of the
theory cohomology groups in dimensions 0, 1 and 2 as suitable `center', `outer *
*derivation' and
`singular extension' groups respectively. For the theory of modules over a ring*
* and for an additive
coefficient functor these reduce to the corresponding classical interpretations*
* of the MacLane
cohomology groups. Indeed, by [20, Theorem A] the cohomology groups of the the*
*ory of R-
modules with coefficients in a bimodule are isomorphic to the MacLane cohomolog*
*y groups.
6.5 Topological Hochschild (co-)homology. Let S be a Gamma-ring and M an S-bimo*
*dule.
We choose a cofibrant approximation cS -! S in the model category of Gamma-ring*
*s of [37, Thm.
24
2.5]. For us the topological Hochschild homology groups of S with coefficients *
*in M are defined as
the homotopy groups of the derived smash product of S and M as cS-S-bimodules,
THH n(S; M) = ssn (S ^LcS^SopM) :
This is not the original definition of topological Hochschild homology given by*
* B"okstedt [5]. How-
ever Shipley [39, Sec. 4] shows in the context of symmetric spectra that the tw*
*o definitions are
equivalent; a proof of the analogous statements in the context of Gamma-rings i*
*s similar, but
easier. The topological Hochschild cohomology groups of S with coefficients in *
*M are defined as
the homotopy classes of cS-S-bimodule maps from S to M,
ae n
THH n(S; M) = [S;[-M]cS-SnS; ifnM]0
if n cS-S< 0.
Here refers to the suspension functor in the homotopy category of cS-S-bimodul*
*es.
6.6 The bimodule construction. To a functor G 2 F(T op; R) there is a functoria*
*lly associated
cHR-T s-bimodule G!. This generalizes the construction of [30, Ex. 2.6]. Here c*
*HR is a cofibrant
approximation of HR in the model category of Gamma-rings of [37, Thm. 2.5]. As *
*a -space, G!
is equal to the composite functor
T G forget
op -F---!T op----! R-mod ----! (pt. simpl. sets):
In other words, the value of the -space G!on n+ is the underlying set of the va*
*lue of G on the
free T -algebra on n generators. There is a map HR O G!O T s-! G!given at n+ 2 *
*opby
(HR O G!O T s)(n+) = eR[G(F T(F T(n+)))] ----! G(F T(n+)) = G!(n+) ;
this map is evaluation both inside and outside of G and it uses that G takes va*
*lues in R-modules
and that F Tis a triple. Composition with the assembly map (1.8) and the stabl*
*e equivalence
of Gamma-rings cHR -~! HR gives the bimodule structure cHR ^G!^T s-! G!. For ex*
*ample,
the cHR-T s-bimodule associated to the additive functor M TabIab is the Eilenbe*
*rg-MacLane
module HM.
Theorem 6.7 Let T be a pointed discrete algebraic theory and G 2 F(T ). There*
* is a natural
isomorphism
H *(T ; G) ~= THH *(T s; G!) :
For T ab-bimodules M, the groups THH*(T s; HM) are trivial in negative dimensio*
*ns and for * 0
there is a natural isomorphism
H*(T ; M TabIab) ~= THH *(T s; HM) :
Remark 6.8 A special case of interest is the case when T is the theory of left*
* R-modules for a
ring R. We are then looking at functors G 2 F(R) from the category of finitely *
*generated free
R-modules to all R-modules. In this case the homotopy groups ss*G!are (essentia*
*lly by definition)
the stable derived functors of G in the sense of A. Dold and D. Puppe [13, 8.3]*
*. The homological
case of Theorem 6.7 then specializes to [30, Thm. 3.2]. By a theorem of Jibladz*
*e and Pirashvili [20,
Thm. A] the groups Ext*F(R)(I; M R -) are naturally isomorphic to the MacLane c*
*ohomology
groups H*ML(R; M) introduced in [26]. So the cohomological part of Theorem 6.7*
* implies that
MacLane cohomology coincides with topological Hochschild cohomology.
25
The cohomology of T with coefficients in a non-additive functor can differ from*
* the topological
Hochschild cohomology, see Remark 6.15. To prove the homological part of Theore*
*m 6.7 we use the
same strategy as [30]: we show that topological Hochschild homology has the uni*
*versal properties
of the derived functors of Q. The cohomological part follows from a comparison *
*of the derived
category of the abelian category F(T op; R) with the homotopy category of cHR-T*
* s-bimodules.
We start with three short lemmas.
Lemma 6.9 Let Pn be the projective object of F(T op; R) represented by n+ 2 T *
*op(see 6.1). Then
Pn!is stably equivalent to cHR ^n+^ T sas a cHR-T s-bimodule.
Proof: The -space underlying Pn!is the composite of three other -spaces, Pn!= H*
*R O n O T s.
The cHR-T s-bimodule structure comes through the left and right composition fac*
*tors. Since cHR
is cofibrant as a Gamma-ring, it is also cofibrant as a -space [37, Thm. 2.5]. *
*By [25, Prop. 5.23]
the assembly map
cHR ^n^ T s--~--! HR O n O T s
from the smash to the composition product is thus a stable equivalence. The lem*
*ma follows since
the cofibrant -spaces n and S ^n+ are stably equivalent. *
* __|_|
Recall from [37, Sec. 4] that the functor L which is adjoint to the Eilenberg-M*
*acLane functor H
passes to a functor L : cHR-mod-T s-! R-mod-T ab(using the isomorphism T ab~=L(*
*T s)).
Lemma 6.10 There is a natural isomorphism Gadd~= LG! of functors F(T op; R) -!*
* R-mod-
T ab.
Proof: The evaluation map eZ[G(1+)] -! G(1+) passes to a natural map of R-T ab-*
*bimodules
from LG!to Gadd. By Lemma 6.9 and [37, Lemma 1.2], LPn!is isomorphic to the fre*
*e bimodule
(R T ab)n, so the map is an isomorphism for the projective generators. Since b*
*oth expressions
are right exact in G, the map is an isomorphism in general. *
* __|_|
We denote by sF(T op; R) the category of simplicial objects in F(T op; R) (whic*
*h is the same as the
category of pointed functors from T opto the category of simplicial left R-modu*
*les). The bimodule
construction 6.6 which takes G to G!can be applied dimensionwise to simplicial *
*functors.
Lemma 6.11 The bimodule construction G 7! G!takes short exact sequences of sim*
*plicial func-
tors in sF(T op; R) to homotopy cofibre sequences of cHR-T s-bimodules.
Proof: When the underlying -spaces of the bimodules associated to a short exact*
* sequence are
evaluated at a simplicial sphere Sn, one obtains a short exact sequence of simp*
*licial R-modules
which give rise to a long exact sequence in homotopy. When n tends to infinity,*
* these assemble
into a long exact sequence for the homotopy groups of the cHR-T s-bimodules. *
* __|_|
Proof of the homological part of Theorem 6.7: We show that the functors THH*(T *
*s; (-)!)
have the universal properties of the derived functors of Q. By [37, Lemmas 1.2 *
*and 4.1] we can
identify THH0(T s; G!) as
THH 0(T s; G!) ~= L(T s^cTs^(Ts)opG!) ~= T abT ab(Tab)opLG!:
So by Lemma 6.10, the group THH0(T s; G!) is naturally isomorphic to QG. Short *
*exact sequences
of objects in F(T ) go to homotopy cofibre sequences of bimodules (Lemma 6.11),*
* which become
homotopy cofibre sequences of -spaces after derived smash product with T sover *
*cT s^(T s)op.
So the functors THH*(T s; (-)!) have a connecting homomorphism with respect to *
*which short
exact sequences of objects in F(T ) go to long exact sequences in homology. So *
*it remains to show
that topological Hochschild homology vanishes in positive dimensions for each o*
*f the projective
generators Pn of F(T ). Using Lemma 6.9 we calculate
THH *(T s; Pn!) ~= ss*(T s^LcTs^(Ts)op(HT ab^n+^ T s)) ~= ss*(HT ab^n+) ~=*
* (ss*T ab)n
which is indeed trivial in positive dimensions since T abis a discrete ring. *
* __|_|
26
6.12 Model structures for simplicial functors. The cohomological part of Theore*
*m 6.7 is a
special case of a more general statement about the relationship between the cat*
*egory of simplicial
functor from T opto R-modules and the category of cHR-T s-bimodules. Quillen [3*
*2, II.4 Thm. 4]
provides a standard model category structure on the category sF(T op; R) of sim*
*plicial functors.
The weak equivalences (resp. fibrations) are the maps which are objectwise weak*
* equivalences
(resp. fibrations) of simplicial R-modules. We refer to this model category str*
*ucture as the strict
structure for simplicial functors. By the Dold-Kan theorem the normalized chain*
* complex functor
induces an equivalence of the strict homotopy category of sF(T op; R) with the *
*derived category
D+ (F(T op; R)) of non-negative dimensional chain complexes over the abelian ca*
*tegory F(T op; R).
The bimodule construction takes objectwise weak equivalences of simplicial func*
*tors to stable
equivalences of cHR-T s-bimodules. Lemma 6.11 implies that the induced functor *
*on the level of
homotopy categories
(-)! : D+ (F(T op; R)) ----! Ho(cHR-mod-T s)
is a triangulated functor.
We call a map of simplicial functors F -! G a stable equivalence (resp. stable *
*fibration) if the
associated map of cHR-T s-bimodules F !-! G!is a stable equivalence (resp. stab*
*le fibration). The
stable cofibrations coincide with the strict cofibrations. A simplicial functor*
* G is stably fibrant if
and only if it is homotopy-additive, i.e., if for all X; Y 2 T opthe map F (X) *
* F (Y ) -! F (X q Y )
is a weak equivalence of simplicial R-modules.
Theorem 6.13 The stable notions of fibrations, cofibrations and weak equivalen*
*ces make the cat-
egory sF(T op; R) of simplicial functors into a closed simplicial model categor*
*y. The functor (-)!
is the right adjoint of a Quillen equivalence between the stable model category*
* of simplicial functors
sF(T op; R) and the model category of cHR-T s-bimodules.
Proof: The functor G 7! G!preserves all limits and we first want to see that it*
* actually has a
left adjoint. It suffices to show this for discrete functors in F(T op; R). The*
* category F(T op; R)
is complete, it has a set of cogenerators (see 6.1) and it is well-powered (i.e*
*., every object has
only a set of subobjects). So Freyd's Special Adjoint Functor Theorem (see e.g.*
* [27, V.8, Cor.])
provides a left adjoint (-)!. To obtain the model category structure we apply t*
*he lifting lemma
A.2. The category sF(T op; R) of simplicial functors is complete, cocomplete, s*
*implicially enriched
and locally finitely presentable (Lemma A.1). The model category structure of c*
*HR-T s-bimodules
is cofibrantly generated. It remains to find a stably fibrant replacement funct*
*or Q for the category
sF(T op; R). We first note that a simplicial functor G 2 sF(T op; R) can be ext*
*ended to a functor
from the category of T -algebras to simplicial R-modules by the coend construct*
*ion
Z k+2Top
G(X) = Xk^ G(k+ ) for X 2 T -alg.
Then the functor Q is given by
(QG)(k+ ) = colimnnG(nF T(k+ )) :
The underlying -space of (QG)!is a stably fibrant replacement on the -space und*
*erlying G!,
so Q in fact has the properties needed to apply the lifting lemma A.2. We conc*
*lude that the
stable notions of cofibrations, fibrations and weak equivalences make the categ*
*ory sF(T op; R) of
simplicial functors into a closed simplicial model category.
By definition the right adjoint (-)!preserves and detects weak equivalences and*
* fibrations. So
it remains to show that for every cofibrant cHR-T s-bimodule A the unit map A -*
*! (A!)!is a
27
stable equivalence. This is very similar to Lemma 4.10. We first consider the c*
*ase when A is one
n^ K) ^T sf*
*or some pointed
of the generating bimodules, i.e., when it is of the form A = cHR ^(
simplicial set K. Then A!~=Pn eZ[K] and the unit map
A = cHR ^(n^ K) ^T s----! HR O (n^ K) O T s= (A!)!
is the composite of the assembly map and the map induced by the stable equivale*
*nce cHR -! HR.
The assembly map is a stable equivalence when all except possibly one of the fa*
*ctors are cofibrant
[25, Prop. 5.23]. Since cHR is cofibrant as a Gamma-ring, it is also cofibrant *
*as a -space [37,
Thm. 2.5]. Hence the map A -! (A!)!is a stable equivalence if A is a generating*
* bimodule.
An arbitrary cofibrant cHR-T s-bimodule is obtained from the trivial bimodule b*
*y iterated pushouts
along cofibrations between bimodule of the above form, transfinite composition *
*and retract. So
the rest of the argument is exactly as in Lemma 4.10. We only have to observe t*
*hat the functor
(-)!takes cofibre sequences of cHR-T s-bimodules to cofibre sequences of simpli*
*cial functors. But
cofibre sequences in sF(T op; R) are in particular short exact sequences which *
*posses long exact
sequences in homotopy by Lemma 6.11. *
* __|_|
If we take R = T aband F = Iab in the following corollary, the left hands side*
* becomes
H*(T ; M TabIab) and the right hand side becomes [HT ab; HM]*cHTab-Ts. Change o*
*f rings gives
an isomorphism
[HT ab; HM]*cHTab-Ts~=[T s; HM]*cTs-Ts= THH *(T s; HM) ;
so the cohomological part of Theorem 6.7 is a special case of
Corollary 6.14Let F be any functor in F(T op; R) and M an R-T ab-bimodule. Then*
* the groups
[nF !; HM]cHR-Tsare trivial for n > 0 and the functor (-)!induces natural isomo*
*rphisms
ExtnF(Top;R)(F; M TabIab) ~= [F !; nHM]cHR-Ts:
Proof: For an arbitrary cHR-T s-bimodule W , the group [W; HM]cHR-Tsis isomorph*
*ic to the
group of R-T ab-bimodule homomorphism from ss0W to M. Since ss0(nF !) is trivia*
*l for n > 0,
the first claim follows. When we regard functors F and G in F(T op; R) as cons*
*tant simplicial
objects, ExtnF(Top;R)(F; G) is isomorphic to the maps from F to nG in the stric*
*t homotopy
category of simplicial functors sF(T op; R). The functor M TabIabis additive an*
*d the associated
bimodule HM is stably fibrant, so M TabIabis fibrant in the stable model catego*
*ry structure of
simplicial functors. So the maps from F to M TabIabcoincide in the strict and s*
*table homotopy
categories. Since (-)!is the right adjoint of a Quillen equivalence of model ca*
*tegories (Theorem
6.13), the groups of maps from F to n(M TabIab) in the stable homotopy category*
* of simplicial
functors is mapped isomorphically to the group of maps from F !to HM in Ho(cHR-*
*mod-T s). __|_|
Remark 6.15 The map ExtnF(Top;R)(F; G) -! [F !; nG!]cHR-Tsis not bijective for*
* arbitrary
functors in F(T op; R). For example, if we take F to be one of the projective g*
*enerators Pn then
HomF(Top;R)(Pn; G) ~=G(n+), but [Pn!; G!]cHR-Ts~=(ss0G!)n by Lemma 6.9. These t*
*wo expressions
are different unless G is additive. In particular we can take T to be the theor*
*y of pointed sets and
R = T ab= Z. In this case the abelianization functor Iab is isomorphic to the p*
*rojective object
P1, so H0(T ; G) ~=G(1+) and H*(T ; G) is trivial for * 1. On the other hand T*
* sis the sphere
spectrum, so THH*(T s; G!) ~=ss*G!, which can have higher homotopy groups.
28
7 Examples
7.1 Sets. In the theory of pointed sets, the algebras are the pointed simplicia*
*l sets and the stable
category is (a model for) the usual stable homotopy category. The associated Ga*
*mma-ring is the
sphere spectrum S, so Theorem 4.4 reduces to [9, Thm. 5.8] saying that the homo*
*topy theory of
-spaces is equivalent to that of connective spectra.
7.2 Simplicial sets with G-action. Let G be a simplicial monoid and consider th*
*e theory of
pointed simplicial sets with pointed G-action. The stable category is the categ*
*ory of spectra with
G-action (i.e., G-objects in the category of spectra in the sense of [9]). The *
*stable equivalences are
equivariant maps which induce isomorphisms of the homotopy groups of underlying*
* spectra. The
associated Gamma-ring is S[G], the monoid ring of G over the sphere spectrum (s*
*ee 1.11). The map
from stable homotopy to homology is represented by the map of monoid rings S[G]*
* -! H(Z[G]).
If G is a simplicial group (not just a simplicial monoid), then the homotopy th*
*eory of pointed
G-simplicial sets is the same as the homotopy theory of retractive spaces over *
*the classifying space
BG. This is well known and can be seen as follows: we let EG denote a universal*
* principal G-
space, i.e., any weakly contractible simplicial set with a free G-action, and w*
*e take the orbit space
of EG by the G-action as our model for the classifying space. Then pullback alo*
*ng the orbit map
EG -! BG is an equivalence of categories between the category of simplicial set*
*s containing BG
as a retract, and the category of (unpointed) G-simplicial sets containing EG a*
*s an equivariant
retract (in both cases the section and retraction are part of the data).
The appropriate model category structure for retractive G-spaces over EG is the*
* one in which
fibrations and weak equivalences are those morphisms that are fibrations and we*
*ak equivalences
of simplicial sets after forgetting the G-action, the retraction and the sectio*
*n. The functor that
collapses the retract EG to a point is then the left adjoint of a Quillen equiv*
*alence between the
category of retractive G-simplicial sets over EG, and the category of pointed G*
*-simplicial sets. So
altogether Theorem 4.4 can be interpreted as saying that the stable homotopy th*
*eory of spaces
retractive over BG is equivalent to the homotopy theory of S[G] modules, or spe*
*ctra with an
action of G. This is exploited by J. Klein and J. Rognes to prove a chain rule *
*for the Calculus of
Functors [22].
7.3 Monoids and groups. The theories of sets, monoids and groups have equivale*
*nt stable
homotopy theories. This follows from the fact (see [29, Thm. 1]) that the free *
*monoid and the
free group generated by a connected simplicial set are weakly equivalent to the*
* loop space on
the suspension of the simplicial set. Since the map from a simplicial set to th*
*e loop space of its
suspension is twice as highly connected as the space itself, the maps of Gamma-*
*rings
S ----! (monoids)s ----! (groups)s
are stable equivalences.
7.4 Nilpotent groups. The lower central series of a group G is a filtration by *
*normal subgroups
rG. These subgroups are defined inductively by 1G = G and rG = [r-1G; G], the s*
*ubgroup
generated by commutators. A group is called nilpotent of class r if r+1G is tri*
*vial. We denote
by Nilr the theory of class r nilpotent groups. We obtain a tower of theories
(groups) -! . . .-! Nilr -! Nilr-1- ! . . .-! Nil1= (abelian groups) :
It follows from a theorem of E. Curtis [12, Thm. 1.4] that the unit map S -!(Ni*
*lr)s is (log2r -1)-
connected. So the associated sequence of Gamma-rings interpolates between S and*
* (Nil1)s = HZ.
29
7.5 p-local groups. Fix a prime number p. By considering p-local nilpotent grou*
*ps we obtain a
Gamma-ring model for the p-local sphere spectrum, together with a `multiplicati*
*ve filtration'. This
provides a different view at the mod p-lower central series spectral sequence o*
*f [8]. A nilpotent
group G is called p-local if for all primes q 6= p the set map x 7! xq is a bij*
*ection of G onto
itself. On the category of nilpotent groups there exists a p-localization funct*
*or G 7! G(p)which
is left adjoint to the inclusion of nilpotent p-local groups [41, Sec. 8]. p-lo*
*calization is exact and
commutes with the terms in the lower central series, i.e.,
(G=rG)(p)~= G(p)=r(G(p)) :
p-local groups of fixed nilpotence class r form a theory which we denote by Nil*
*r(p). By [10, Ch. IV]
the group-theoretic localization map G -! G(p)induces p-localization on homotop*
*y groups for
every simplicial nilpotent group G. This implies that the map of Gamma-rings (N*
*ilr)s -! (Nilr(p))s
is the p-localization map on the associated spectra.
The category of simplicial theories has inverse limits and these are calculated*
* pointwise [6, Prop.
3.11.1]. We denote by Nil^(p)the inverse limit theory of the Nilr(p). If X is a*
* reduced simplicial set,
GX its Kan loop group, then the inverse limit of the simplicial groups (GX=r(GX*
*))(p)is weakly
equivalent to the loop group of the Z(p)-completion of X by [10, Ch. IV Prop. 4*
*.1]. In particular,
the free Nil^(p)-algebra generated by a reduced simplicial set X is a model for*
* the p-localization of
|X|. So the Gamma-ring (Nil^(p))s is a model for the p-local sphere spectrum.
*
* i
We define Ji as the pointwise fibre of the map of Gamma-rings associated to Nil*
*^(p)-! Nilp(p).
Then i
Ji -! (Nil^(p))s -! (Nilp(p))s
is a homotopy fibre sequence of -spaces: when evaluated at any simplicial set i*
*t gives a short
exact sequence of simplicial groups. Since Jiis the fibre of a multiplicative m*
*ap between Gamma-
rings, it inherits a multiplication (but no unit), and it behaves like an ideal*
* of (Nil^(p))s. One can
show that the Ji's even form a multiplicative filtration of (Nil^(p))s, i.e., t*
*he image of Ji^Jj in
(Nil^(p))s under the Gamma-ring multiplication is contained in Ji+j. Altogether*
* we have obtained
a convergent multiplicative filtration on a Gamma-ring model of the p-local sph*
*ere spectrum. This
filtration in turn gives rise to a multiplicative spectral sequence. There is a*
* variant which starts
with the p-lower central series, and gives a multiplicative filtration on the p*
*-completed sphere
spectrum. In that case, the spectral sequence obtained from the filtration is *
*the mod-p lower
central series spectral sequence of [8]. From the E2-term on this spectral sequ*
*ence is the Adams
spectral sequence.
7.6 Infinite loop spaces. In our simplicial set-up, the Barratt-Eccles model [*
*1, 2] gives an
algebraic theory modeling infinite loop spaces. M. G. Barratt and P. J. Eccles *
*define a functor +
from the category of pointed simplicial sets to itself [1, Def. 3.1]. To avoid *
*notational confusion
with the category opof finite pointed sets, we use the notation fl+ for the fun*
*ctor of Barratt and
Eccles. The functor fl+ is degreewise defined and commutes with filtered colimi*
*ts, i.e., it comes
from a -space, and fl+ has the structure of a triple [1, Prop. 3.6]. So fl+ is*
* the free algebra
functor of a simplicial theory. This is in fact the only example of a simplicia*
*l theory which we
consider explicitly and which is not a discrete theory. The algebras over this *
*theory are called
`simplicial set with +-structure' in [1]. For connected pointed simplicial sets*
* X, fl+ X is a model
for 1 1 |X| (this is proved for Kan complexes in [1, Thm. 4.10, 5.4], but since*
* the functor fl+
is a prolonged -space, it preserves weak equivalences of simplicial sets [9, 4.*
*9] so that property
holds for arbitrary X). Every fl+ -algebra X is naturally a simplicial monoid. *
*Barratt and Eccles
show furthermore [2, Thm. A] that if ss0X is a group, then the fl+ -structure p*
*rovides natural
30
infinite deloopings of X. In this sense, the algebraic theory fl+ models infini*
*te loop spaces. The
+ )s arising from the theory fl+ is yet another model for the sph*
*ere spectrum; it
Gamma-ring (fl
has the property that its underlying -space is special.
7.7 Modules. Let B be a simplicial ring and consider the theory of simplicial l*
*eft B-modules.
The Gamma-ring obtained from this theory is the Eilenberg-MacLane Gamma-ring HB*
* as defined
in 1.5. The homotopy theory of B-modules remains unchanged under stabilization *
*(cf. [36, Thm.
2.2.2]). Theorem 4.4 thus says that the homotopy theory of simplicial B-modules*
* is equivalent to
the homotopy theory of HB-modules; we recover [37, Thm. 4.4].
7.8 Associative algebras. Let B be a commutative simplicial ring and consider t*
*he theory of
augmented associative B-algebras (alias associative B-algebras without unit). W*
*e claim that the
map from the theory of B-modules to the theory of augmented associative B-algeb*
*ras induces a
weak equivalence on associated Gamma-rings
HB --~--! (Ass. B-alg)s :
The connective stable homotopy theory of augmented associative B-algebras is th*
*us equivalent
to the homotopy theory of simplicial B-modules. This fact could have been prove*
*n without ever
introducing Gamma-rings by the methods of [36, Sec. 3]. To prove the claim we n*
*ote that the free
associative non-unital B-algebra generated by a pointed simplicial set K decomp*
*oses as the direct
sum 1
M
eB[ K^_._.^.K_-z___"]
n=1 n
where eB[-] denotes the reduced free B-module. If K is taken to be a k-dimensio*
*nal sphere, all
homogeneous components of degree 2 are at least (2k - 1)-connected, so the map*
* from the free
B-module on Sk to the free non-unital associative B-algebra on Sk is (2k - 2)-c*
*onnected.
7.9 Commutative algebras. Let B be a commutative simplicial ring and consider t*
*he theory of
augmented commutative B-algebras (alias commutative B-algebras without unit) Co*
*mmutative
simplicial algebras have been the object of much study [31, 14, 17, 18, 36]. Th*
*e homology theory
arising as the derived functor of abelianization in this case is known as Andre*
*-Quillen homology
for commutative rings.
We denote by DB the Gamma-ring arising from the theory of augmented commutative*
* B-algebras.
If B is a Q-algebra, the map HB -! DB induced from the symmetric algebra functo*
*r is a stable
equivalence (cf. [36, Thm. 3.2.3]). In general, the Eilenberg-MacLane spectrum *
*splits off DB, but
the category of commutative augmented B-algebras can have higher stable homotop*
*y operations,
in which case DB is not equivalent to HB.
We claim that as a -space, DB is stably equivalent to HB ^LHZ. In particular, t*
*he homotopy
groups of DB are additively isomorphic to the integral spectrum homology of the*
* Eilenberg-
MacLane spectrum HB. To prove our claim we use that the free commutative B-alge*
*bra without
unit on a pointed set X is additively isomorphic to the reduced free B-module o*
*n the infinite
symmetric product of X. Hence if we let SP denote the -space that sends a poin*
*ted set to
the infinite symmetric product, then DB is isomorphic to the composite -space H*
*B O SP. For
every connected simplicial abelian monoid the map to its group completion is a *
*weak equivalence
[40, Cor. 5.7], so the group completion map SP-! HZ is a stable weak equivalenc*
*e of -spaces.
31
Hence DB is weakly equivalent to the derived smash product of HB and HZ by [25,*
* Prop. 5.23].
LHZ can be constructed as an E -ring spectrum, but the weak equivalence to*
* DB can not be
HB ^ 1
multiplicative in any sense since the ring of homotopy groups of HB ^LHZ is gra*
*ded commutative,
that of DB is generally not.
By 4.11 the ring ss*DB is isomorphic to the ring of stable homotopy operations *
*of commutative
simplicial B-algebras. These operations are also referred to as the stable Cart*
*an-Bousfield-Dwyer
algebra (since these authors calculated the unstable operations for B = Fp, see*
* [11, 7, 14]).
Additively, ss*DB is the direct sum of the stable derived functors, in the sens*
*e of Dold and Puppe
[13, 8.3], of the symmetric power functors on the category of B-modules. In [7,*
* x12], Bousfield
calculates the ring ss*DFp under the name of `stable algebra of the functor alg*
*ebra' of symmetric
powers. For p = 2 ([7, Thm. 12.3]; see also [14]) it is the associative unital*
* F2-algebra with
generators ffifor i 2 subject to the relations
X n X n
ff2m+iff1+m+j = 0 and ff1+2m+iff1+m+j = 0
i+j=n i i+j=n i
for m; n 0. [7, Thm. 12.6] gives a similar but more complicated description fo*
*r odd primes.
If X is an augmented commutative simplicial B-algebra, its stable homotopy is d*
*efined as the
homotopy groups of the suspension spectrum of any cofibrant replacement: sss*X *
*= ss*1 Xc. Then
the stable homotopy and Andre-Quillen homology of X are related by the universa*
*l coefficient and
Atiyah-Hirzebruch spectral sequences of 5.5
Torss*DBp(ss*B; sss*X)q=)HAQp+qX
HAQp(X; ssqDB) =) sssp+qX :
7.10 Divided power and Lie-algebras. In the spirit of the previous two examples*
*, one can
consider other types of algebras over a commutative ring B and study the Gamma-*
*rings they
give rise to. For divided power algebras over Fp, Bousfield [7, Thm. 12.3 and 1*
*2.6] calculates the
graded ring of homotopy groups of the associated Gamma-ring, again under the na*
*me of `stable
derived functors' of the divided power functors. For the case of restricted Lie*
*-algebras over Fp, this
calculation is carried out in [8, 2.4 and 2.4']. The result is known as the -al*
*gebra, and it shows
up as a E1-term of the Adams spectral sequence for the stable homotopy groups o*
*f spheres. The
case of restricted Lie-algebras is closely related to Example 7.5 since the ass*
*ociated graded to the
p-lower central series filtration of a free group is the free restricted Lie-al*
*gebra on the abelianized
group.
A Cofibrantly generated model categories
In [32, p. II 3.4], Quillen formulates his small object argument, which is now *
*a standard device
for producing model category structures. An example of this is the lifting lemm*
*a A.2 below which
we use several times in this paper. After Quillen, various other authors have *
*axiomatized and
generalized the small object argument. We work with the `cofibrantly generated *
*model categories'
of [15]. We have given a review of cofibrantly generated model categories in [*
*37, App. A] and
we will continue to use that terminology. In all the cases we treat in this pap*
*er, category theory
automatically takes care of the smallness conditions. The basic reason is that*
* we are dealing
with suitable functor categories with values in simplicial sets. The relevant c*
*ategory theoretical
32
notion is that of a locally presentable category. The categories we consider ar*
*e even locally finitely
presentable. In general, categories involving actual topological spaces tend n*
*ot to be locally
presentable.
An object K of a category C is called finitely presentable if the hom functor h*
*omC(K; -) preserves
filtered colimits. A set G of objects of a category C is called a set of strong*
* generators if for every
object K and every proper subobject there exists G 2 G and a morphism G -! K wh*
*ich does not
factor through the subobject. A category is called locally finitely presentable*
* if it is cocomplete
and has a set of finitely presentable strong generators.
Lemma A.1 Let T be a simplicial theory. Then the categories T -alg, GS (T ) *
*and Sp(T ) are
locally finitely presentable. If T is a discrete theory and R a ring, then the *
*category sF(T op; R)
of simplicial functors is locally finitely presentable. If S is a Gamma-ring, t*
*hen the category of
S-modules is locally finitely presentable.
Proof: All the above categories are cocomplete. Finitely presentable strong g*
*enerators exist
because objectwise evaluation is representable in all these categories. More p*
*recisely, possible
choices of generators are as follows. In T -alg, we can choose the T -algebras *
*freely generated by
the simplicial standard simplices (i)+. In GS(T ) we can take the objects T sO *
*(n^ (i)+). In
Sp(T ) we take the spectra of T -algebras Fnidefined by
ae
(Fni)j = j-nF T*((i)+) ifijf