THE MCCORD MODEL FOR THE TENSOR PRODUCT OF
A SPACE AND A COMMUTATIVE RING SPECTRUM
NICHOLAS J. KUHN
Abstract. We begin this paper by noting that, in a 1969 paper in
the Transactions, M.C.McCord introduced a construction that can be
interpreted as a model for the categorical tensor product of a based
space and a topological abelian group. This can be adapted to Segal's
very special -spaces - indeed this is roughly what Segal did - and then
to a more modern situation: K R where K is a based space and R is
a unital, augmented, commutative, associative S-algebra.
The model comes with an easy-to-describe filtration. If one lets K =
Sn, and then stabilize with respect to n, one gets a filtered model for
the Topological Andr'e-Quillen Homology of R. When R = 1 1 X,
one arrives at a filtered model for the connective cover of a spectrum X,
constructed from its 0thspace.
Another example comes by letting K be a finite complex, and R
the S-dual of a finite complex Z. Dualizing again, one arrives at
G. Arone's model for the Goodwillie tower of the functor sending Z
to 1 MapT (K, Z).
Applying cohomology with field coefficients, one gets various spec-
tral sequences for deloopings with known E1-terms. A few nontrivial
examples are given.
In an appendix, we describe the construction for unital, commutative,
associative S-algebras not necessarily augmented.
1. Introduction
The point of this paper is to tell a story that begins with a 1969 paper of
M.C.McCord [McC ], and ends with various disparate objects of current in-
terest, e.g. Goodwillie towers, Topological Hochschild Homology, and Topo-
logical Andr'e-Quillen Homology. Line by line, I think most of this story is
known. However, taken as a whole, I think the older work sheds some light
on the newer. Moreover, in this era of tremendous activity in homotopical
algebra of various sorts, it seems important to remind ourselves that the
genesis of many of the most useful ideas lies way back in the literature.
Conceptually, we feature the following categorical notion. Let T be the
category of based topological spaces. If C is a category enriched over T ,
there is the notion of the tensor product of a space K 2 T with an object
____________
Date: January 28, 2002.
2000 Mathematics Subject Classification. Primary 55P43; Secondary 18G55.
This research was partially supported by a grant from the National Science F*
*oundation.
1
2 KUHN
X 2 C: this is an object K X 2 C satisfying
Map C(K X, Y ) = Map T(K, Map C(X, Y ))
for all X, Y 2 C.
We give an overview of the paper.
We consider various topological categories of structured objects. Let Ab
be the category of abelian topological monoids. Let T be G.Segal's category
of -spaces [Se]: functors X : ! T , where is the category of finite based
sets. Let Alg be the category of commutative, associative, augmented S-
algebras, where S is the sphere spectrum.
These categories are closely related. First of all, an abelian topological
monoid A defines in a natural way Ax 2 T . In Segal's terminology, Ax is
an example of a `special' object, and the abelian topological groups define
`very special objects'. A very special -space is roughly the same thing
as an infinite loopspace, and, in this introduction, we will tempt fate and
often identify these two notions. Now we note that if X is either an abelian
topological monoid or an infinite loopspace, then 1 X+ is in the category
Alg.
With C equal to any of these three categories, McCord's construction
yields a functor1
SP 1 : T x C ! C.
His construction generalizes the infinite symmetric product construction
studied by Dold and Thom in the 1950's: with N denoting the natural
numbers, SP 1(K, N) = SP1 (K).
If C is either Ab or Alg, there is an isomorphism
(1.1) SP 1 (K, X) = K X.
See Proposition 2.2(1), and Proposition 4.8. The latter proposition (or at
least its proof) seems to be new.
Equation (1.1) is almost true if C = T . S. Schwede [S], following
Bousfield and Friedlander [BF ], defines a model category structure on T ,
having the very special -spaces as the fibrant objects, and such that
ho(T ) is equivalent to the homotopy category of connective spectra. With
this structure, SP 1 (K, ) preserves fibrant objects and the natural map
K X ! SP1 (K, X) is a weak equivalence.
McCord's interest stemmed from the following fundamental property:
(1.2) for X 2 Ab, SP 1(S1, X) is a classifying space for X.
Suitably interpreted, the same result is true if X is a special object of T .
For R 2 Alg, SP 1(S1, R) is also of interest: SP 1 (S1, R) = S1 R equals
T HH(R; S), the Topological Hochschild Homology of R with coefficients in
the bimodule S. See Proposition 7.1; this is deduced from a similar result
due to J. McClure, R. Schwänzl, and R. Vogt [MSV ].
____________
1McCord uses the notation B(A, K) where we use SP1 (K, A); I have borrowed my
notation from [FF , Chapter 9].
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 3
The construction has two other basic properties, both discussed by Mc-
Cord when C = Ab.
Firstly, there are natural isomorphisms
(1.3) SP 1(K ^ L, X) = SP1 (K, SP1 (L, X)).
From this and (1.2) one quickly deduces that SP 1(Sn, X) is an n-fold de-
looping of X for X 2 Ab , or for very special X 2 T . For R 2 Alg,
SP1 (Sn, R) can be interpreted as `higher' Topological Hochshild Homology
of R.
Secondly, SP 1(K, X) comes with a nice increasing filtration. When C =
Ab, one easily sees that there is an equivalence
(1.4) FdSP 1 (K, X)=Fd-1SP 1 (K, X) ' K(d)^ d X^d,
and little variants of this hold when C = T or Alg. Here K(d)denotes the
d-space obtained from the d-fold smash product K^d by collapsing the
the fat diagonal to a point.2
This much of the story will be fleshed out in sections 2, 3, and 4, with
the statement about T HH appearing in x7.
In [B ], it is observed that Alg is Quillen equivalent to the category Alg0,
of commutative, nonunital S-algebras. In x5, we discuss the corresponding
filtered SP1 construction, which again agrees with the tensor product. This
reduced construction is `smaller' than what is done in Alg, and the category
E of finite sets and epimorphisms replaces .
The quotients of the filtration, and the use of E, may look vaguely familiar
to readers of [Ar ], and this is what we explain in x6. There is a contravariant
functor from T to Alg0 sending a based space Z to the ring spectrum R =
D(Z), where D denotes the S-dual, and the multiplication on R is induced
by the diagonal on Z. We note that there is a natural map in Alg0
K D(Z) ! D(MapT (K, Z))
and reinterprete the main convergence theorem of [Ar ] as saying that the
adjoint of this,
1 MapT (K, Z) ! D(K D(Z)),
is an equivalence if both Z and K are finite dimensional complexes, and the
dimension of K is less than the connectivity of Z. See Theorem 6.6. Since
K D(Z) has a nice increasing filtration, the S-dual is a tower of fibrations.
This tower is visibly equivalent to the tower found by Arone, and is thus
the Goodwillie tower associated to the functor sending a space X to the
spectrum 1 MapT (K, Z).
In x7, we discuss the following construction. Given R 2 Alg, nice prop-
erties of SP 1(K, R) as a functor of K allow us to define a filtered spectrum
T AQ(R) by
T AQ(R) = hocolimn!1 nSP 1 (Sn, R).
____________
2This notation, which the author likes, comes from [Ar].
4 KUHN
This is one of various equivalent definitions of the Topological Andr'e-Quillen
spectrum of R. When R = 1 X+ , with X an infinite loopspace, T AQ(R)
is the connective delooping of X. When R = D(Z+ ), the filtered spectrum
T AQ(R) is related to constructions studied by the author in [K3 ].
Using (1.4), one can identify the quotients of the filtration of T AQ(R):
(1.5) FdT AQ(R)=Fd-1T AQ(R) ' Kd ^h d (R=S)^d,
where Kd is the dth partition complex which arose in the work of Arone and
Mahowald on the Goodwillie tower of the identity [AM ].
In x8 we note that applying ordinary cohomology with field coefficients
F to the filtered spectra SP 1(Sn, R) and T AQ(R) yield spectral sequences
converging to H*(Sn R; F) and H*(T AQ(R); F), and having E1 terms
isomorphic to known functors of H*(R; F): see Theorem 8.1. These spectral
sequences appear to be unexplored, even in the case when R = 1 X+ , with
X an infinite loop space, so that, e.g., the T AQ(R) spectral sequence is
calculating the cohomology of a connective spectrum from knowledge of the
cohomology of its 0th-space. As examples, we use results of ours from [K3 ]
to explain how the spectral sequence works, when F has characteristic 2, in
the cases R = 1 (Z=2+ ), 1 (S1+), and, most interestingly, D(S1+).
In the Appendix, we note how our models need to be slightly tweaked
when one considers commutative unital S-algebras not necessarily aug-
mented.
Very influential to me in my understanding of the older work surveyed
in this paper was Chapter of 9 of the unpublished book The actions of the
classical small categories of topology by Bill and Ed Floyd [FF ]. Writing
this book was Ed's project in the late 1980's, after returning to ordinary
academic life after finishing a term as provost of the University of Virginia.
I have also benefited from conversations with Mike Mandell, Bill Dwyer, and
Greg Arone.
Versions of this work were presented at talks at the Johns Hopkins topol-
ogy conference in the spring of 2000, and at the Union College topology and
category theory conference of fall 2001.
2. McCord's construction
Let K be a based space with basepoint *, and let A be an abelian topo-
logical monoid.
Imagine using (1.1) to guide the construction SP 1 (K, A). As a first
experiment, suppose A = N. In this case (1.1) tells us that, for all A 2 Ab,
we should have
Map Ab(SP 1 (K, N), A) = Map T(K, Map Ab(N, A)).
But, since N is the free abelian (topological) monoid on one generator, we
can identify Map Ab (N, A) with A. Thus we are asking that SP 1 (K, N)
satisfy
Map Ab(SP 1 (K, N), A) = Map T(K, A).
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 5
In other words, SP 1(K, N) should be the free topological abelian monoid
generated by K, a.k.a. SP 1 (K).
Note that elements in SP 1 (K) are words of the form kn11. .k.ndd, with
ki2 K and ni2 N. This suggests a definition.
Definition 2.1. Let SP 1 (K, A) be the abelian topological monoid with
generators ka with k 2 K and a 2 A subject to the relations:
(i) *a = * for all a 2 A,
(ii) k0 = * for all k 2 K,
(iii) ka1ka2 = ka1+a2 for all a1, a2 2 A.
Viewing ka as an element`in K xA, SP1 (K, A) is topologized as the evident
quotient space of 1d=0(K x A)d.
Note that the abelian topological monoid satisfying only the relations of
type (i) and (ii) is SP1 (K ^ A). If B is another abelian topological monoid,
a monoid map SP 1(K ^ A) ! B corresponds to a map of topologial spaces
OE : K ^ A ! B which itself corresponds to a map K ! MapT (A, B).
This latter map takes values in Map Ab(A, B) exactly when OE(k, a1 + a2) =
OE(k, a1)OE(k, a2). Thus we see that the quotient of SP 1(K ^ A) having type
(iii) relations imposed satisfies the universal property of K A. (Compare
with [FF , p.164].) We have checked the first part of the next proposition.
Proposition 2.2. There are the following natural identifications in Ab .
(1) SP 1(K, A) = K A.
(2) SP 1(S0, A) = A.
(3) SP 1(K ^ L, A) = SP1 (K, SP1 (L, A)).
(4) SP 1(K _ L, A) = SP1 (K, A) x SP1 (L, A).
The last three parts of this proposition follow formally from statement
(1). For example, (3) follows by manipulating adjunctions:
Map Ab(SP 1 (K ^ L, A), B)= Map T(K ^ L, Map Ab(A, B))
= Map T(K, Map T(L, Map Ab(A, B)))
= Map T(K, Map Ab(SP 1 (L, A), B))
= Map Ab(SP 1 (K, SP1 (L, A)), B).
For statement (4), one needs to also note that the coproduct in Ab is the
product. (In the next section, we will see that there are also reasonable
direct proofs of (3) and (4).)
6 KUHN
In the introduction to [McC ], McCord comments that SP 1( , A) äh s a
tendancy to convert cofibrations . . . to quasifibrations", and proves this in
various cases [McC , Thm.8.8]. Note that statement (4) of the last proposi-
tion nicely illustrates his statement.
In particular, when applied to the cofibration S0 ,! I ! S1, his ob-
servations suggest that SP 1(I, A) ! SP 1(S1, A) is a quasifibration with
homotopy fiber A = SP1 (S0, A). One has
Proposition 2.3. If A has a nondegenerate base point, then SP 1(S1, A) is
a classifying space for A.
Combined with statement (3) of the previous proposition, this implies
Corollary 2.4. In this case, SP 1(Sn, A) is an n-fold classifying space of
A.
With such a mild point set hypothesis, this proposition and corollary
occur as [FF , Cor.9.16].
We end this section by noting that SP 1(K, A) is filtered by letting the
dth`filtration, which we denote FdSP 1 (K, A), be the evident quotient of
d r ^d
r=0(K x A) . Let i : (K, d) ,! K denote the inclusion of the fat
diagonal into the d-fold smash product. Under reasonable conditions, e.g.
if K is a based C.W. complex, the inclusion i will be a d-equivariant
cofibration, and we let K(d)= K^d= (K, d). Assuming the basepoint in A
is also nondegenerate, the inclusion Fd-1SP 1 (K, A) ,! FdSP 1 (K, A) will
be a cofibration, and there is a homeomorphism
(2.1) FdSP 1 (K, A)=Fd-1SP 1 (K, A) ' K(d)^ d A^d.
Statements similar to this appear in [McC , x6].
Remark 2.5. One should note that specializing the filtration on SP 1(K, A)
to SP1 (K, N) = SP1 (K) does not yield the standard filtration on SP1 (K).
For example, in this paper an element k2 2 SP 1(K) would be in filtration
1.
3. -spaces and Segal's theorem
Our first goal in this section is to rewrite our construction SP 1(K, A) in
a way allowing for generalization.
We begin by defining more precisely the category .
Definition 3.1. Let be the category with objects the based finite sets
0 = ;+ and n = {1, 2, . .,.n}+ , n 1, and with all based functions as
morphisms. Note that 0 is both an initial and terminal object.
We remark that, unfortunately, it was the opposite of this category that
was called in [Se], and the literature is strewn with inconsistent notation.
A -space is then defined to be a covariant functor X : ! T that
is `based' in the sense that it sends 0 to the one point space. These are
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 7
the objects of a category T having the natural transformations as mor-
phisms. This is a category enriched over T : the set of morphisms between
two -spaces, Map (X, X0), has a natural topology. Similarly, a based con-
travariant functor Y : op ! T willobepcalled a op-space, and these are
objects in a topological category T .
Example 3.2. If A is an abelian topological monoid, there is an associated
-space Ax defined as follows. Firstly, Ax (n) = An. Then, givenQff : n !
m, the ith component of ff* : An ! Am sends (a1, . .,.an) to aj, with the
product running over j such that ff(j) = i. This product is interpreted to
be the unit of A if there are no such j.
Example 3.3. Since, for any K 2 T , Kn = MapT (n, K), there is the
evident op-space Kx with Kx (n) = Kn.
Note that theoconstructionspin these last examples embed Ab into T ,
and T into T , as a full subcategories.
We now recall the coend construction. If X is a -space and Y is a
op-space, we let Y ^ X 2 T denote the quotient space
`
Y (n) ^ X(n)=(~),
n
where ff*(y) ^ x ~ y ^ ff*(x) generates the equivalence relation.
It is useful to observe that, because X(0) = * = Y (0), Y ^ X 2 T =
Y x X, where Y x X is the quotient space
a
Y (n) x X(n)=(~),
n
where ff*(y) ^ x ~ y ^ ff*(x) generates the equivalence relation.
By inspection, one observes
Lemma 3.4. SP 1 (K, A) = Kx ^ Ax .
This suggests a generalization of our construction.
Definition 3.5. Given K 2 T and X 2 T , let SP 1(K, X) = Kx ^ X.
A couple of remarks are now in order.
Firstly, a Yoneda's lemma type argument shows that
SP 1 (n, X) = X(n).
Thus, as a functor of K, SP 1(K, X) extends X to T ; more precisely, this
is the left Kan extension [MacL , Chap.X].
Secondly, we can extend our construction to
SP 1 : T x T ! T
by first letting Xn(m) = X(nm ), and then by defining
SP 1 (K, X)(n) = SP1 (K, Xn).
8 KUHN
It is natural to wonder if SP1 (K, X) can be then be interpreted as a tensor
product in T . Alas, this is not the case: the simple minded construction
K ^ X defined by (K ^ X)(n) = K ^ X(n) is easily seen to play this role3.
Since SP1 (K, X) is not the tensor product in T , formal arguments used
to prove the last three statements in Proposition 2.2 don't apply. However,
we still can prove suitable versions of these.
First of all, the two remarks above (along with the observation that 1 =
S0) combine to show that SP 1(S0, X) = X.
Less obvious are the other two. Statement (3) of Proposition 2.2 is un-
changed in our greater generality.
Proposition 3.6. SP 1 (K ^ L, X) = SP1 (K, SP1 (L, X)).
To generalize statement (4), let a : x ! be the functor sending
(m, n) to m + n . Pulling back by a defines a* : T ! T x .
Proposition 3.7. SP 1 (K _ L, X) = (Kx x Lx ) x x a*(X).
Proposition 2.2(4) follows from this, once one observes that
a*(Ax ) = Ax x Ax ,
so that
SP 1(K _ L, Ax )= (Kx x Lx ) x x (Ax x Ax )
= (Kx x Ax ) x (Lx x Ax )
= SP1 (K, Ax ) x SP1 (L, Ax ).
Let m :op xop ! obepthe functorosendingp(m,on)ptoomnp 4. We let
a* : T x ! T and m* : T x ! T respectively denote the
left adjoints to pulling back by a and m. We have two fundamental lemmas.
Lemma 3.8. m*(Kx x Lx ) = (K ^ L)x .
Lemma 3.9. a*(Kx x Lx ) = (K _ L)x .
Assuming these for the moment, Proposition 3.6 and Proposition 3.7 fol-
low. For example, using Lemma 3.8, we have identifications
SP 1 (K, SP1 (L, X))= Kx x (Lx x X*)
= (Kx x Lx ) x x m*(X)
= m*(Kx x Lx ) x X
= (K ^ L)x x X
= SP1 (K ^ L, X),
and Proposition 3.6 follows. The proof of Proposition 3.7 is similar.
____________
3One can then formally deduce that, given A, B 2 Ab, the natural map K ^ Ax !
SP1 (K, Ax) induces a homeomorphism
Map (SP1 (K, Ax), Bx ) = Map (K ^ Ax, Bx ).
4More precisely, m is the smash product followed by the lexicographic identi*
*fication of
m ^ n with mn .
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 9
Proof of Lemma 3.8. Given a opx op-space Y , m*(Y ) is explicitly given
by
m*(Y )(c) = colimY
c#m
where c # m is the category with objects all triples (a, b, fl) with fl : c !
ab, and morphisms given by pairs (ff : a ! a0, fi : b ! b0) making an
appropriate diagram commute.
One such triple is (c, c, ), where : c ! cc is the diagonal, and there
is a canonical morphism from this triple to any other triple (a, b, fl) given
by the two components of fl : c ! ab . It follows that the canonical map
Y (c, c) ! colimc#mY is a quotient map.
Now we specialize to Y = Kx xLx . Both colimc#mKx xLx and (K ^L)c
are quotients of Kc x Lc, thus we just need to verify that each maps to the
other, as quotient spaces of Kc x Lc.
To construct a map from the former to the latter, we observe that that
the smash product construction,
MapT (A, K) x MapT (B, L) ! MapT (A ^ B, K ^ L),
specializes to give natural maps
^ : Ka ^ Lb ! (K ^ L)ab.
Thus, associated to a triple (a, b, fl), there is a canonical map
*
Ka ^ Lb ^-!(K ^ L)ab fl-!(K ^ L)c,
and these induce the needed map colimc#mKx x Lx ! (K ^ L)c.
To construct a map in the other direction, we observe that (K ^ L)c is
the quotient of Kc x Lc obtained by collapsing to a point the subspace
{(k1, . .,.kc, l1, . .,.lc) | for all i, either ki= * or li=}*.
One checks easily that this subspace is precisely the union of the images of
maps
ff* x fi* : Ka x Lb ! Kc x Lc
such that the composite c -! cc -fffi-!abis the constant map 0. Thus the
subspace maps to the basepoint in colimc#m Kx x Lx , i.e. the projection
Kc x Lc ! colimc#mKx x Lx factors through (K ^ L)c.
Sketch proof of Lemma 3.9.This follows easily from the observation that
there are canonical decompositions
`
(K _ L)c = Ka _ Lb,
with the wedge running over bijections fl : c ! (a + b) which are order
preserving when restricted to fl-1 ((a + b) - a) and fl-1 ((a + b) - b).
10 KUHN
Remarks 3.10. The two lemmas include the statements that wedge and
smash are the left Kan extensions to T xT of the composites x a-! ,! T
and x m-! ,! T .
We suspect that Lemma 3.9 has been observed by others. Lemma 3.8
seems less familiar. (Compare our proof of Proposition 3.6 to the proof of
[Se, Lemma 3.7].) We note that T x T ^-!T x -m*-!T is the smash
product of [L ].
The category of -spaces admits products in the obvious way: if X and
Y are -spaces, one lets (X x Y )(n) = X(n) x Y (n). We have
Proposition 3.11. SP 1 (K, X x Y ) = SP1 (K, X) x SP1 (K, Y ).
To prove this, we first note that X xY = *(X xY ), where * : T x !
T is induced by the diagonal : ! x . Thus we have identifications
SP 1 (K, X x Y )= Kx x *(X x Y )
= *(Kx ) x x (X x Y )
= (Kx x Kx ) x x (X x Y )
= (Kx x X) x (Kx x Y )
= SP1 (K, X) x SP1 (K, Y ),
where we have used the next lemma.
Lemma 3.12. *(Kx ) = Kx x Kx .
Proof.This can be proved in various ways. Perhaps the slickest proof is
to first note that is right adjoint to a. That * = a* formally follows.
Finally, it is evident that a*(Kx ) = Kx x Kx .
It remains, in this section, to discuss how the SP 1 construction interacts
with the homotopy theory of -spaces.
Define ßs*(X) = colimn!1 ß*+n(SP 1 (Sn, X)). The colimit here arises
from maps S1 ^ SP1 (Sn-1 , X) ! SP1 (Sn, X) which themselves are special
cases (K = S1 and Y = SP1 (Sn-1 , X)) of the natural transformation
K ^ Y ! SP1 (K, Y ).
If we define weak equivalences to be maps f : X ! Y with ßs*(f) an iso-
morphism, then Bousfield and Friedlander [BF ], following Segal [Se], showed
that that the localized category T [weq-1 ] is equivalent to the homotopy
category of connective spectra. Even more, this equivalence is induced by
a Quillen equivalence between appropriate model categories. Schwede [S]
modifies the cofibration and fibrations slightly. All these authors work with
simplicial sets rather than topological spaces, but [S, Appendix B] allows
for some translation into our setting.
The upshot is roughly the following. Cofibrations are maps f : X ! Y
where Y is obtained from X by successively attaching appropriate sorts of
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 11
free -spaces. Fibrant objects agree with Segal's notion of a very special
-space, where X is very special means that each map
(3.1) X(a + b) ! X(a) x X(b)
is a weak equivalence of spaces, and also
(3.2) the monoid ß0(X(1)) is a group.
Proposition 3.13. If K is a C.W. complex, then SP 1 (K, ) preserves
cofibrations and acyclic cofibrations.
Proposition 3.14. If K is a C.W. complex, and X is cofibrant, then the
natural map
K ^ X ! SP1 (K, X)
is a weak equivalence.
Proposition 3.15. If K is a C.W. complex, and X is cofibrant and very
special, then SP 1(K, X) is again very special.
Theorem 3.16. If X is cofibrant and very special, then SP 1( , X) takes
cofibration sequences of C.W. complexes to a homotopy fibration sequence.
In particular, there are weak equivalences of spaces
X(1) ~-! SP 1 (S1, X)(1) ~-! 2SP 1 (S2, X)(1) ~-!. ...
We briefly indicate why the propositions hold. Firstly, under the cofi-
brancy hypotheses, SP 1(K, X) will be nicely filtered, and satisfy
(3.3) FdSP 1 (K, X)=Fd-1SP 1 (K, X) = K(d)^ d (X(d)=Xsing(d))
where Xsing(d) denotes the union of all the images of maps X(c) ! X(d)
with c < d.
It follows then that then K ^ X(1) ! SP1 (K, X) is a weak equivalence
through a stable range, and the first two of the propositions easily can be
deduced.
For the next proposition, we note that, if (3.1) holds, then
Xa+b ! Xa x Xb
is a strict equivalence of -spaces, where a map is a strict equivalence if
evaluating on any n yields a weak equivalence of spaces. Then we have
equivalences
SP 1 (K, X)(a + b)= SP1 (K, Xa+b)
-~!SP 1 (K, X
a x Xb)
= SP1 (K, Xa) x SP1 (K, Xb)
= SP1 (K, X)(a) x SP1 (K, X)(b),
showing that SP 1(K, X) again satisfies (3.1).
For the theorem, see [Se, Prop.3.2] and [BF , Lemma 4.3]. It follows that if
X is cofibrant and very special, then X(1) is canonically weakly equivalent
12 KUHN
to an infinite loop space. Furthermore, for any C.W. complex K, there are
weak equivalences
(3.4) FdSP 1 (K, X)=Fd-1SP 1 (K, X) ' K(d)^ d X(1)^d.
4. Commutative ring spectra
We now show that the results of the previous sections extend nicely to
the world of structured ring spectra.
We work within the category S, the category of S-modules studied in
[EKMM ]. Given K 2 T and X 2 S, 1 K, K ^ X, and Map (K, X) will
denote the usual S-modules5.
Let Alg be the category of unital, commutative, associative, augmented
S-algebras. Thus an object in Alg is an S-module R, together with multi-
plication ~ : R ^ R ! R, unit j : S ! R and counit ffl : R ! S satisfying
the usual identities. Morphisms preserve all structure.
This category is enriched over T : given R, Q 2 Alg, the morphism space
MapAlg (R, Q) is based with basepoint R -ffl!S -''!Q. We also note that the
coproduct in Alg of R and Q is R ^ Q.
As observed in [B , x1], results in [EKMM ] show that Alg has a topological
model category structure in which weak equivalences are morphisms that are
weak equivalences as maps of S-modules6.
We have two important sources of examples.
Example 4.1. If A 2 Ab , then 1 A+ 2 Alg. More generally, if X is an
E1 -space (e.g. an infinite loop space), then 1 X+ is naturally an object
in Alg. (See [M , Ex.IV.1.10] and [EKMM , xII.4].)
Example 4.2. Given a based space Z, let D(Z+ ) denote Map (Z+ , S). This
is an object in Alg : the unit and the counit are respectively induced by
Z+ ! S0 and S0 ! Z+ , and the diagonal : Z ! Z x Z induces the
multiplication
*
D(Z+ ) ^ D(Z+ ) ! D(Z+ ^ Z+ ) --! D(Z+ ).
Given R 2 Alg, we let R^ : ! S denote the functor with R^(n) = R^n
analogous to Example 3.2.
Definition 4.3. Given K 2 T and R 2 Alg, let SP 1(K, R) = Kx ^ R^.
We will momentarily see that SP 1(K, R) is again an object in Alg.
Proofs from x3 extend immediately to prove the next proposition.
Proposition 4.4. There are the following natural identifications.
(1) SP 1(S0, R) = R.
(2) SP 1(K _ L, R) = SP1 (K, R) ^ SP1 (L, R).
____________
5What we are calling Map(K, X) is FS( 1 K, X) in [EKMM ].
6In Basterra's notation, Algis denoted CS=S.
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 13
(3) SP 1(K, R ^ Q) = SP1 (K, R) ^ SP1 (K, Q).
A consequence of this proposition is that SP 1(K, R) takes values in Alg,
with multiplication given by the composite
SP1 (K,~) 1
SP1 (K, R) ^ SP1 (K, R) = SP1 (K, R ^ R) ------! SP (K, R).
We note that this multiplication agrees with the composite
SP1 (r,R) 1
SP1 (K, R) ^ SP1 (K, R) = SP1 (K _ K, R) -------! SP (K, R),
where r : K _ K ! K is the fold map.
With this structure, all the identifications in the last proposition are as
objects in Alg, and we also have the next proposition, whose proof follows
from the arguments of the last section.
Proposition 4.5. SP 1 (K ^ L, R) = SP1 (K, SP1 (L, R)).
Now we check that SP 1(K, R) is the categorical tensor product in Alg.
The following lemmas are easily verified, where we use the following nota-
tion: with C either T or S, and X and Y functors from to C, Map C(X, Y )
denotes the space of natural transformations from X to Y .
Lemma 4.6. For all K, L 2 T , Map T (Kx , Lx ) = MapT (K, L).
Lemma 4.7. For all R, Q 2 Alg, Map S(R^, Q^) = MapAlg (R, Q).
Proposition 4.8. For all K 2 T and R 2 Alg , SP 1 (K, R) is naturally
isomorphic to K R.
Proof.We check that SP1 (K, R) satisfies the universal property of the ten-
sor. Given K 2 T , and R, Q 2 Alg, we have
MapAlg (SP 1 (K, R), Q)= Map S(SP 1 (K, R)^, Q^)
= Map S(SP 1 (K, R^), Q^)
= Map S(Kx ^ m*(R^), Q^)
= Map T(Kx , Map S(m*(R^), Q^))
= Map T(Kx , MapAlg(R^, Q))
= Map T(Kx , MapAlg(R, Q)x )
= Map T(K, MapAlg(R, Q)).
Here m : x ! is multiplication as in the last section.
As before, SP 1(K, R) is naturally filtered. Let R=S denote the cofiber
of j : S ! R. If K is a C.W. complex, and j is a cofibration, then the
inclusion Fd-1SP 1 (K, R) ,! FdSP 1 (K, R) will be a cofibration, and there
is an isomorphism of S-modules
(4.1) FdSP 1 (K, R)=Fd-1SP 1 (K, R) ' K(d)^ d (R=S)^d.
14 KUHN
5.The reduced model
It is sometimes useful to replace Alg by a slightly different category. Let
Alg0be the category of nonunital, commutative, associative S-algebras (the
category denoted NS in [B ]). Basterra observes that the functor S_ : Alg0!
Alg, that wedges a unit S onto a nonunital algebra, has as right adjoint the
augmentation ideal functor J : Alg ! Alg 0, defined by letting J(R) be
the fiber of R -ffl!S. She then notes that, with a natural topological model
category on Alg0, these adjoint functors form a Quillen pair, and thus induce
adjoint equivalences on the associated homotopy categories.
Example 5.1. If Z is a based space, J(D(Z+ )) = D(Z). The multiplication
on D(Z) is induced by the reduced diagonal : Z ! Z ^ Z.
Our SP 1(K, ) construction has a `reduced' analogue in Alg0.
Definition 5.2. Let E be the category with objects n, for n 1, and
with morphisms from n to m equal to all epimorphisms from {1, . .,.n} to
{1, . .,.m}.
As observed in [Ar ] (see also [AK ]), a based space K defines a functor
K^ : Eop ! T with K^ (n) = K^n . Also, J 2 Alg0 defines J^ : E ! S in
the obvious way.
Definition 5.3. Given K 2 T and J 2 Alg0, let SP 1(K, J) = K^ ^E J^ .
The analogues of all the properties of SP1 (K, R) proved in the last section
hold in our setting, with virtually identical proofs. In particular, SP1 (K, J)
is again an object in Alg0, and it agrees with the categorical tensor product
K J.
From the above comments, one can formally deduce the following isomor-
phism in Alg.
Proposition 5.4. SP 1 (K, J) _ S = SP1 (K, J _ S).
Though we won't show this here, this proposition can also be given a
direct proof, and there are analogues in other contexts. Readers may wish
to compare this result with observations in [P ].
As usual, SP 1 (K, J) is filtered: if Ed denote the full subcategory of E
with objects n for n d, then we let FdSP 1 (K, J) = K^ ^Ed J^ . If K is a
C.W. complex, then the inclusion Fd-1SP 1 (K, J) ,! FdSP 1 (K, J) will be
a cofibration, and there is an isomorphism of S-modules
(5.1) FdSP 1 (K, J)=Fd-1SP 1 (K, J) ' K(d)^ d J^d.
We note that the isomorphism of the last proposition is filtration preserv-
ing.
6. Reinterpretation of Arone's tower for 1 MapT (K, X)
In this section, we let K be a finite C.W. complex.
In [Ar ], G. Arone described a model for the Goodwillie tower of the functor
sending a based space Z to the S-module 1 MapT (K, Z). Here we show
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 15
that, if Z is a finite complex, his tower arises as the S-dual of the filtered
object K D(Z) of the last section.
We recall Arone's construction and some of its properties [Ar ]. For more
detail, see also [AK ].
Definitions 6.1. Let Z be a based space.
(i) Let P K(Z) = Map ES(K^ , Z^ ), the spectrum of natural transformations
from 1 K^ to 1 Z^ .
(ii) Let PdK(Z) = Map EdS(K^ , Z^ ).
(iii) Let (K, Z) : 1 MapT (K, Z) ! P K(Z) be the natural transformation
that sends f : K ! Z to the natural transformation with nth component
equal to 1 f^n : 1 K^n ! 1 Z^n .
The spectrum P K(Z) is the inverse limit of the tower of fibrations
. .!.PdK+1(Z) ! PdK(Z) ! PdK-1(Z) ! . .,.
and the dth fiber is isomorphic to
Map Sd(K(d), Z^d).
Because K(d) is finite, and the d action on this is free away from the
basepoint, this fiber is naturally homotopy equivalent to the homotopy orbit
spectrum
(D(K(d)) ^ Z^d)h d.
From this last description one sees that the tower has the form of a Good-
willie tower, and also that the connectivity of the fibers goes up if the con-
nectivity of Z is greater than the dimension of K. Arone then proves that
this is the Goodwillie tower of 1 MapT (K, Z) by proving
Theorem 6.2. [Ar ] If the connectivity of Z is greater than the dimension
of K, then (K, Z) is a homotopy equivalence.
Now we connect these constructions to K D(Z).
Definitions 6.3. Let Z be a based space.
(i) Let ~ (K, Z) : K D(Z) ! D(MapT (K, Z)) be the map in Alg0 adjoint
to the composite
K eval--!MapT(MapT (K, Z), Z) D-!Map Alg0(D(Z), D(MapT (K, Z)).
(ii) Let (K, Z) : 1 MapT (K, Z) ! D(K D(Z)) be the S-module map
adjoint to ~ (K, Z).
(iii) Let ff(K, Z) : P K(Z) ! D(K D(Z)) be the map of cofiltered S-
modules defined as follows. Let i : Z^ ! D(D(Z)^) be the natural trans-
formation adjoint to D(Z)^ ! D(Z^ ). Now let ff(K, Z) be the composite
induced by i:
Map ES(K^ , Z^ ) i-!Map ES(K^ , D(D(Z)^)) = MapS (K^ ^E D(Z)^, S).
A check of the definitions verifies the next lemma.
16 KUHN
Lemma 6.4. There is a commutative diagram
1 MapT (K,RZ)
(K,Z)ooooo RRRR(K,Z)RR
ooo RRRR
wwoooo ff(K,Z) RR((
P K(Z) __________________________//D(K D(Z))
Lemma 6.5. ff(K, Z) is a homotopy equivalence if Z is a finite complex.
Proof.If Z is finite, then i : Z^n ! D(D(Z)^n) is a an equivalence for all
n. Now the lemma follows by observing that K^ is a cofibrant Eop-space,
or more simply, note that the fibers of the towers will be equivalences, as
K(d)is a free d-complex for all d. (This has been noted before; see e.g.
[AK , McC ].)
Summarizing, we conclude
Theorem 6.6. If both K and Z are finite complexes, and the dimension of
K is less than the connectivity of Z, then
(K, Z) : 1 MapT (K, Z) ! D(K D(Z))
is a weak equivalence, and thus the algebra map
~(K, Z) : K D(Z) ! D(MapT (K, Z))
can be identified as the map from a spectrum to its double dual.
We end this section by noting how the homological version of this discus-
sion would go.
Let F be a field, HF the associated commutative S-algebra, and Alg F
the category of commutative, nonunital HF algebras. Let K F J 2 AlgF
denote the tensor product of a based space K and an J 2 AlgF. As before,
one learns that
K F J = Kx ^E J^ ,
where smash products are taken over HF.
Now let DF(Z) = Map (Z, HF), the HF-module whose homotopy groups
are the cohomology groups of Z with F-coefficients. In this case, the natural
map i : HF ^ Z^n ! DF(DF(Z)^n) is an equivalence for any space Z with
H*(Z; F) of finite type. Reasoning as before, from Arone's theorem one
deduces
Theorem 6.7. If K is a finite complex of dimension less than the connec-
tivity of Z, and H*(Z; F) is of finite type, then the natural map in AlgF,
: K F DF(Z) ! DF(MapT (K, Z)),
is an equivalence.
Remark 6.8. It seems likely that this theorem can be deduced from older
convergence results for the Anderson spectral sequence [An ], and then one
can run our arguments backwords, and deduce Arone's theorem. The novelty
would then be to identify the filtration as Arone did.
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 17
7.Topological Hochschild homology and Topological
Andre-Quillen homology
Let T HH(R; M) denote the Topological Hochschild homology spectrum
associated to a S-algebra R and an R-bimodule M (see e.g. [EKMM ,
Chap.9]). If R is commutative and augmented, then ffl : R ! S makes S
into an R-bimodule. We have
Proposition 7.1. S1 R = T HH(R; S).
Proof.This is a variant of a theorem of J. McClure, R. Schwanzl, and R. Vogt
[MSV ]. They show that if R is a commutative S-algebra, then T HH(R; R)
is the tensor product of R with S1 with the tensor product in the category
of commutative S-algebras. In the appendix, we note that if R is also
augmented, then this would agree with S1+ R 2 Alg. Thus
T HH(R; R) = S1+ R.
Applying R to the pushout square in T
S0 ______//*
| |
| |
fflffl| fflffl|
S1+_____//S1,
yields a pushout square in Alg
R ___________//S
| |
| |
fflffl| fflffl|
S1+ R _____//S1 R,
and we conclude that
S1 R = (S1+ R) ^R S = T HH(R, R) ^R S = T HH(R; S).
Given K 2 T and R 2 Alg, there is a natural map
K ^ R ! K R,
and thus
K ^ (L R) ! K (L R) = (K ^ L) R.
This map is easily seen to be filtration preserving.
Specializing to the case when K = S1, and L = Sn yield filtration pre-
serving maps
(Sn R) ! Sn+1 R,
or, equivalently,
Sn R ! (Sn+1 R).
Definition 7.2. Let T AQ(R) = hocolimn!1 nSn R.
18 KUHN
M. Mandell has shown the author that this definition agrees with other
definitions of Topological Andr'e Quillen Homology in the literature, e.g.
[B ]. In particular T AQ(R) is homotopy equivalent to the cofiber of J(R) ^
J(R) ! J(R)7.
As the next example makes clear, T AQ(R) can be viewed as an `infinite
delooping' of R.
Example 7.3. If X is a connective S-module, T AQ( 1 ( 1 X)+ ) ' X. To
see this, just recall that Sn yields the (n - 1)-connected n-fold delooping
of an infinite loopspace.
Note that T AQ(R) is filtered with
FdT AQ(R)=Fd+1T AQ(R) ' hocolimn!1 -n Sn(d)^ d (R=S)^d.
As in [AM ], let Kd be the unreduced suspension of the classifying space
of the poset of nontrivial partitions of a set with d elements.
Lemma 7.4. [AD ] There is a d-equivariant map
hocolimn!1 -n Sn(d)! Kd
that is a nonequivariant equivalence.
The original short proof of this, due to Arone and Mahowald, appears in
[K3 , Appendix].
Corollary 7.5. There is a homotopy equivalence
FdT AQ(R)=Fd+1T AQ(R) ' ( Kd ^ (R=S)^d)h d.
8. Spectral sequences and examples
Applying homology or cohomology with F-coefficients to our filtered mod-
els for Sn R and T AQ(R) yields highly structured convergent spectral
sequences with E1 terms equal to known functors of H*(R; F). To see why
this is true, we note that there is an explicit equivariant duality map [AK ]
F (Rn, d)+ ^ Sn(d)! Snd,
where F (Rn, d) is the usual configuration space of d distinct points in Rn.
Thus the homology calculations of [CLM ] apply.
To be more precise, let {E*,*r(Sn R; F)} and {E*,*r(T AQ(R); F)} re-
spectively denote the spectral sequences for computing H*(Sn RF) and
H*(T AQ(R); F). Let H~*(R; F) denote the reduced cohomology of R, i.e.
H*(J(R); F).
Theorem 8.1. For R 2 Alg with H*(R; F) of finite type, there are natural
isomorphisms as follows.
____________
7Strictly speaking, R should be replaced by by a fibrant object in Algand th*
*en J(R)
replaced by a cofibrant object in Alg0.
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 19
(1) If F has characteristic 0, then
E*,*1(Sn R; F) = S*( 1-nL( -1H~*(R; F))),
and
E*,*1(T AQ(R); F) = L( -1H~*(R; F))).
(2) If F has characteristic p, then
E*,*1(Sn R; F) = S*(Rn( 1-nLr( -1H~*(R; F)))),
and
E*,*1(T AQ(R); F) = R( Lr( -1H~*(R; F)))).
In this theorem, dV denotes the d-fold shift of a graded vector space V ,
L is the free Lie algebra functor, Lr is the free restricted Lie algebra functo*
*r,
S* is the free commutative algebra functor8, and R and Rn are appropriate
free Dyer-Lashof operation functors.
The author plans to write more about this elsewhere.
We end with three examples. All are nontrivial, and most of what I say
follows immediately from work done in [K3 ]. More detail about the last
example will appear in [K4 ].
Example 8.2. If R = 1 S1+, then T AQ(R) ' HZ. At least when local-
ized at 2, the filtration of T AQ(R) will correspond to the symmetric product
of spheres filtration of HZ9. In particular, the dth associated graded spec-
trum is contractible unless d is a power of 2,
k k+1
F2k = SP 2(S), and F2k=F2k-1= L(k).
Here L(k) is as in [MP ]. The boundary maps of the filtration yields the
complex of spectra
. .!. L(2) ! L(1) ! L(0) ! HZ
occurring in the Whitehead conjecture [K1 ]. The sequence is exact in ho-
motopy, but zero in mod 2 homology: indeed, the spectral sequence for
computing H*( HZ; F2) = A=ASq1 collapses at E1.
Example 8.3. If R = 1 Z=2+ , then T AQ(R) ' HZ=2. The dthassociated
graded spectrum is contractible unless d is a power of 2,
k 0 k
F2k = SP 2(S ), and F2k=F2k-1= M(k).
Here we recall [MP ] that SP 2k(S0) is defined to be the cofiber of the `diag-
onal' : SP 2k-1(S0) ! SP 2k(S0), and M(k) = L(k) _ L(K - 1). As in the
previous example, the boundary maps of the filtration yields the complex of
spectra
. .!.M(2) ! M(1) ! M(0) ! HZ=2
____________
8One has sign conventions of the usual sort.
9This presumably holds integrally: [AD , Thm.1.14] says that the filtration *
*quotients
are correct.
20 KUHN
occurring in the mod 2 Whitehead conjecture [K2 ]. The sequence is exact
in homotopy, but the spectral sequence for computing H*(HZ=2; F2) = A
collapses at E1.
Example 8.4. If R = D(S1+), then T AQ(R) ' -1HQ. There are various
ways to see this; in [K4 ], we will prove that S2 R ' HQ _ S0. Localized
at 2, the dth associated graded spectrum is contractible unless d is a power
of 2, and
k 0
F2k=F2k-1= -1SP 2(S ).
Thus H*(F2k=F2k-1; F2) = -1A=Lk+1, where Lk is the span of all admis-
sible sequences in the Steenrod algebra of length at least k. The boundary
maps of the filtration yields a complex of spectra
. .!. -2SP 4(S0) ! -1SP 2(S0) ! S0
that is exact in cohomology: each map sends the bottom class of the cyclic
A-module to Sq1 applied to the bottom class of the next module.
Appendix A. Augmented versus nonaugmented ring spectra
Let Algu be the category of unital commutative S-algebras, but not nec-
essarily augmented. Thus we have forgetful maps
Alg ! Algu ! Alg0.
Alg uis enriched over Tu, the category of unbased topological spaces, so one
can look for a convenient model for K R with K 2 Tu and R 2 Algu. In this
appendix we describe such a model, and compare this to the construction
in x4.
A.1. K R for unital commutative algebras. Let Su be the category
of S-modules under S, so an object is an S-module map j : S ! X.
__
Definition A.1. Given K 2 Tu and X 2 Su, let K^ X 2 Su be the pushout:
K+ ^ S _____//S ^ S = S
| |
| |
fflffl| fflffl|_
K+ ^ X _______//K^ X.
It is easy to check
Lemma A.2. There is an adjunction
__
Map Su(K^ X, Y ) = Map Tu(K, Map Su(X, Y )).
Let Set be the category of finite sets. Given K 2 Tu, there is an apparent
functor Kx : Setop! Tu. Now note that, as it was in Alg, ^ is the coproduct
in Algu. Then R 2 Algu defines R^ : Set ! Su.
Lemma A.3. For all K, L 2 Tu, Map SetT(Kx , Lx ) = Map Tu(K, L).
Lemma A.4. For all R, Q 2 Algu, Map SetSu(R^, Q^) = Map Algu(R, Q).
A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 21
Definition A.5. Given K 2 Tu and R 2 Algu, let
__ ^
SP1 (K, R) = Kx ^SetR .
As in x4, the lemmas combine to prove the analogue of Proposition 4.8.
Proposition A.6. For all K 2 Tu and R 2 Algu, SP 1(K, R) is again in
Algu and is naturally isomorphic to the categorical tensor product K R.
SP 1(K, R) is filtered in the usual way, and one gets an isomorphism of
S-modules
(A.1) FdSP 1 (K, R)=Fd-1SP 1 (K, R) ' (K+ )(d)^ d (R=S)^d.
We note that (K+ )(d)is just S if d = 0 and Kxd =(fat diagonal) if d > 0.
A.2. The unbased versus the based construction. In this subsection,
we denote the tensor in Alg by and the tensor in Algu by u.
Given R 2 Algu , the product R x S will be in Alg , with augmentation
given by projection, and unit S -! S x S ! R x S. This construction is
right adjoint to the forgetful functor:
Lemma A.7. Given Q 2 Alg and R 2 Algu, there is an adjunction
Map Algu(Q, R) = Map Alg(Q, R x S).
Note that, if Q 2 Alg and R 2 Algu, then Map Algu(Q, R) is based with
basepoint Q ! S ! R.
Proposition A.8. Given K 2 T , Q 2 Alg , and R 2 Alg u, there is an
adjunction isomorphism
Map Algu(K Q, R) = Map T(K, Map Algu(Q, R)).
Proof.We have natural isomorphisms
Map Algu(K Q, R)= Map Alg(K Q, R x S)
= Map T(K, Map Alg(Q, R x S))
= Map T(K, Map Algu(Q, R)).
Corollary A.9. If Q 2 Alg and L 2 Tu, then L+ Q = L u Q.
Proof.Let K = L+ in the proposition, and note that
Map T(L+ , Map Algu(Q, R))= Map Tu(L, Map Algu(Q, R))
= Map Algu(L u Q, R).
22 KUHN
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Department of Mathematics, University of Virginia, Charlottesville, VA
22903
E-mail address: njk4x@virginia.edu