*
Homotopy theory of C algebras
Paul Arne Ostvaer *
December 15, 2008
Abstract
In this work we construct from ground up a homotopy theory of C*algebr*
*as.
This is achieved in parallel with the development of classical homotopy th*
*eory by
first introducing an unstable model structure and second a stable model st*
*ructure.
The theory makes use of a full fledged import of homotopy theoretic techni*
*ques
into the subject of C*algebras.
The spaces in C*homotopy theory are certain hybrids of functors repres*
*ented
by C*algebras and spaces studied in classical homotopy theory. In particu*
*lar, we
employ both the topological circle and the C*algebra circle of complexva*
*lued
continuous functions on the real numbers which vanish at infinity. By usin*
*g the
inner workings of the theory, we may stabilize the spaces by forming spect*
*ra and
bispectra with respect to either one of these circles or their tensor prod*
*uct. These
stabilized spaces or spectra are the objects of study in stable C*homotop*
*y theory.
The stable homotopy category of C*algebras gives rise to invariants su*
*ch as
stable homotopy groups and bigraded cohomology and homology theories. We
work out examples related to the emerging subject of noncommutative motives
and zeta functions of C*algebras. In addition, we employ homotopy theory *
*to
define a new type of Ktheory of C*algebras.
________________________________
*Department of Mathematics, University of Oslo, Norway.
MSC Primary: 46L99, 55P99
1
Contents
1 Introduction 3
2 Preliminaries 8
2.1 C *spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . . . 8
2.2 G  C*spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . . 18
2.3 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . *
*. . . . . 19
3 Unstable C*homotopy theory 28
3.1 Pointwise model structures . . . . . . . . . . . . . . . . . . . . . *
*. . . . 28
3.2 Exact model structures . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . 39
3.3 Matrix invariant model structures . . . . . . . . . . . . . . . . . .*
* . . . 50
3.4 Homotopy invariant model structures . . . . . . . . . . . . . . . . . *
*. . 55
3.5 Pointed model structures . . . . . . . . . . . . . . . . . . . . . . . *
*. . . . 69
3.6 Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *
*. . . . . 76
4 Stable C*homotopy theory 79
4.1 C *spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . . . 79
4.2 Bispectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *
*. . . . . . 90
4.3 Triangulated structure . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . 94
4.4 Brown representability . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . 100
4.5 C *symmetric spectra . . . . . . . . . . . . . . . . . . . . . . . . *
*. . . . . 101
4.6 C *functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . . . 116
5 Invariants 123
5.1 Cohomology and homology theories . . . . . . . . . . . . . . . . . . .*
* . 123
5.2 KKtheory and the EilenbergMacLane spectrum . . . . . . . . . . . . .*
* 125
5.3 HLtheory and the EilenbergMacLane spectrum . . . . . . . . . . . . .*
* 129
5.4 The ChernConnesKaroubi character . . . . . . . . . . . . . . . . . *
*. . 130
5.5 Ktheory of C*algebras . . . . . . . . . . . . . . . . . . . . . . . *
*. . . . . 131
5.6 Zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . . 141
6 The slice filtration 1*
*51
2
1 Introduction
In this work we present some techniques and results which lead to new invariant*
*s of
C *algebras. The fundamental organizational principle of C*homotopy theory in*
*fers
there exists a homotopy theory of C*algebras determined by short exact sequenc*
*es,
matrix invariance and by complexvalued functions on the topological unit inter*
*val.
We shall make this precise by constructing model structures on certain spaces w*
*hich
are build up of C*algebras in much the same way as every natural number acquir*
*es
a prime factorization. Our approach combines a new take on C*algebras dictated*
* by
category theory and the recently perfected homotopy theory of cubical sets. The*
* idea
of combining C*algebras and cubical sets into a category of "cubical C*spaces*
*" may
perhaps be perceived as quite abstract on a first encounter. However, these spa*
*ces
arise naturally from a homotopy theoretic viewpoint. We observe next the failur*
*e of
a more straightforward topological approach.
By employing the classical homotopy lifting property formulated in [73] for *
*maps
between C*algebras one naturally arrives at the notion of a C*algebra cofibra*
*tion.
The definition is rigged such that under the GelfandNaimark correspondence a m*
*ap
between locally compact Hausdorff spaces X ! Y is a topological cofibration if *
*and
only if the induced map C0(Y) ! C0(X) is a C*algebra cofibration. Now a standa*
*rd
argument shows every *homomorphism factors as the composition of an injective
homotopy equivalence and a C*algebra cofibration. This might suggest to willi*
*ng
minds that there exists a bona fide model structure on C*algebras with fibrati*
*ons the
C *algebra cofibrations and weak equivalences the homotopy equivalences. In th*
*is
aspiring model structure every C*algebra is fibrant and for a suitable tensor *
*product
the suspension functor = C0(R)  acquires a left adjoint. Thus for every diag*
*ram of
C *algebras indexed by some ordinal ~ the suspension functor induces a homot*
*opy
equivalence Q Q
i2~Ai____//_i2~ Ai.
But this map is clearly not a homotopy equivalence; for example, applying K1 to
the countable constant diagram with value the complex numbers yields an injecti*
*ve
Q
map with image the subgroup of bounded sequences in N Z. The categories of
cubical and simplicial C*spaces introduced in this work offer alternate approa*
*ches to
a homotopy theory employing constructions which are out of reach in the traditi*
*onal
confines of C*algebra theory, as in e.g. [8] and [69].
3
One of the main goals of C*homotopy theory is to provide a modern framework
for cohomology and homology theories of C*algebras. In effect, we introduce t*
*he
stable homotopy category SH *of C*algebras by stabilizing the model structure *
*on
cubical C*spaces with respect to the tensor product C S1 C0(R) of the cubi*
*cal
circle S1 and the nonunital C*algebra C0(R). Combining these two circles allow*
* us to
define a bigraded cohomology theory
i j
Ep,q(A) SH * 1CA, Spq C0(Rq) E (1)
and a bigraded homology theory
i j
Ep,q(A) SH * 1CSpq C0(Rq), A E . (2)
Here, A is a C*algebra considered as a discrete cubical C*space and E is a C**
*spectrum
in the stable homotopy category SH *. Kasparov's KKtheory and ConnesHigson's
Etheory, suitably extended to cubical C*spaces, give rise to examples of Csp*
*ectra.
The precise definitions of the tensor product and the suspension functor 1Cw*
*ill be
explained in the main body of the text. An allied theory of noncommutative moti*
*ves
rooted in algebraic geometry is partially responsible for the choice of bigradi*
*ng, see
[62]. Inserting the sphere spectrum 1CC into the formula (2) yields a theory *
*of
bigraded stable homotopy groups which receives a canonical map from the classic*
*al
homotopy groups of spheres. An intriguing problem, which will not be attempted *
*in
this paper, is to compute the commutative endomorphism ring of 1CC.
The spaces in C*homotopy theory are convenient generalizations of C*algebr*
*as.
A C*space is build out of C*algebras considered as representable setvalued f*
*unctors.
In unstable C*homotopy theory we work with cubical C*spaces X. By definition,*
* for
every C*algebra A we now get homotopically meaningful objects in form of cubic*
*al
sets X(A). Using the homotopy theory of cubical sets, which models the classic*
*al
homotopy theory of topological spaces, declare a map X ! Y of cubical C*spaces*
* to
be a pointwise weak equivalence if X(A) ! Y(A) is a weak equivalence of cubical*
* sets.
This is a useful but at the same time an extremely coarse notion of weak equiva*
*lence in
our setting. In order to introduce a much finer notion of weak equivalence refl*
*ecting
the data of short exact sequences, matrix invariance and homotopy equivalence of
C *algebras, we shall localize the pointwise model structure with respect to a*
* set of
maps in the category *Spc of cubical C*spaces. We define the unstable homotopy
category H of C*algebras as the homotopy category of the localized model struc*
*ture.
4
The homotopy category is universal in the sense that the localized model str*
*ucture
gives the initial example of a left Quillen functor *Spc ! M to some model cate*
*gory
M with the property that every member of the localizing set of maps derives to *
*an
isomorphism in the homotopy category of M.
Section 3 details the constructions and some basic properties of the unstabl*
*e model
structures in C*homotopy theory. Moreover, for the noble purpose of stabilizin*
*g, we
note there exist entirely analogous model structures for pointed cubical C *sp*
*aces
and a pointed unstable homotopy category H*of C*algebras. As in topology, every
pointed cubical C*space gives rise to homotopy groups indexed by the nonnegat*
*ive
integers. These invariants determine precisely when a map is an isomorphism in *
*H*.
We use the theory of representations of C*algebras to interpret Kasparov's KK*
*groups
as maps in H*. Due to the current setup of Ktheory, in this setting it is conv*
*enient to
work with simplicial rather than cubical C*spaces. However, this distinction m*
*akes
no difference since the corresponding homotopy categories are equivalent.
Section 4 introduces the "spaces" employed in stable C*homotopy theory, nam*
*ely
spectra in the sense of algebraic topology with respect to the suspension coord*
*inate C.
These are sequences X0, X1, . .o.f pointed cubical C*spaces equipped with stru*
*cture
maps C Xn ! Xn+1for every n 0. We show there exists a stable model structure
on spectra and define SH *as the associated homotopy category. There is a techn*
*ically
superior category of C*symmetric spectra with a closed symmetric monoidal prod*
*uct.
The importance of this category is not emphasized in full in this paper, but on*
*e would
expect that it will play a central role in further developments of the subject.*
* Its stable
homotopy category is equivalent to SH *.
Using the pointed model structure we define a new type of Ktheory of C*alg*
*ebras.
We note that it contains the Ktheory of the cubical sphere spectrum or Waldhau*
*sen's
Atheory of a point as a retract. This observation brings our C*homotopy theor*
*y in
contact with manifold theory. More generally, working relative to some C*algeb*
*ra A,
we construct a Ktheory spectrum K(A) whose homotopy groups cannot be extracted
from the ordinary K0 and K1groups of A. It would of course be of considerable
interest to explicate some of the Kgroups arising from this construction, even*
* for the
trivial C*algebra. We show that the pointed (unstable) model structure and the*
* stable
model structure on S1spectra of spaces in unstable C*homotopy theory give ris*
*e to
equivalent Ktheories. This is closely related to the triangulated structure on*
* SH *.
5
In [62] we construct a closely related theory of noncommutative motives. On
the level of C *symmetric spectra this corresponds to twisting EilenbergMacLa*
*ne
spectra in ordinary stable homotopy theory by the KKtheory of tensor products
of C0(R). This example relates to Khomology and Ktheory and is discussed in
Section 5.2. The parallel theory of EilenbergMacLane spectra twisted by local *
*cyclic
homology is sketched in Section 5.3. As it turns out there is a ChernConnesKa*
*roubi
character, with highly structured multiplicative properties, between the C*sym*
*metric
spectra build from EilenbergMacLane spectra twisted by KKtheory and local cyc*
*lic
homology. This material is covered in Section 5.4. An alternate take on motives*
* has
been worked out earlier by ConnesConsaniMarcolli in [14].
Related to the setup of motives we introduce for a C*space E its zeta funct*
*ion iE(t)
taking values in the formal power series over a certain Grothendieck ring. The *
*precise
definition of zeta functions in this setting is given in Section 5.6. In the sa*
*me section
we provide some motivation by noting an analogy with zeta functions defined for
algebraic varieties. As in the algebrogeometric situation a key construction i*
*s that
of symmetric powers. It turns out that iE(t) satisfies a functional equation in*
*volving
Euler characteristics O(E) and O+(E) and the determinant det(E) provided E is "*
*finite
dimensional" in some sense. If DE denotes the dual of E then the functional equ*
*ation
reads as follows:
iDE (t1) = (1)O+(E)det(E)tO(E)iE(t)
In the last section we show there exists a filtration of the stable homotopy*
* category
SH *by full triangulated subcategories:
. . . 1CSH*,eff SH *,eff 1CSH*,eff . . .
Here, placed in degree zero is the socalled effective stable C *homotopy cate*
*gory
comprising all suspension spectra. The above is a filtration of the stable C*h*
*omotopy
category in the sense that the smallest triangulated subcategory containing qC*
*SH*,eff
for every integer q coincides with SH *. In order to make this construction wor*
*k we use
the fact that SH *is a compactly generated triangulated category. The filtratio*
*n points
toward a whole host open problems reminiscent of contemporary research in motiv*
*ic
homotopy theory [79]. A first important problem in this direction is to identif*
*y the
zero slice of the sphere spectrum.
6
The results described in the above extend to C*algebras equipped with a str*
*ongly
continuous representation of a locally compact group by C*algebra automorphism*
*s.
This fact along with the potential applications have been a constant inspiratio*
*n for
us. Much work remains to develop the full strength of the equivariant setup.
Our overall aim in this paper is to formulate, by analogy with classical hom*
*otopy
theory, a first thorough conceptual introduction to C*homotopy theory. A next *
*step
is to indulge in the oodles of open computational questions this paper leaves b*
*ehind.
Some of these emerging questions should be resolved by making difficult things *
*easy
as a consequence of the setup, while others will require considerable handson *
*efforts.
Acknowledgments. Thanks go to the the members of the operator algebra and
geometry/topology groups in Oslo for interest in this work. We are grateful to *
*Clark
Barwick, George Elliott, Nigel Higson, Rick Jardine, Andr'e Joyal, Max Karoubi,*
* Jack
Morava, Sergey Neshveyev, Oliver R"ondigs and Claude Schochet for inspiring cor*
*re
spondence and discussions. We extend our gratitude to Michael Joachim for expla*
*in
ing his joint work with Mark Johnson on a model category structure for sequenti*
*ally
complete locally multiplicatively convex C *algebras with respect to some infi*
*nite
ordinal number [42]. The two viewpoints turned out to be wildly different. Aside
from the fact that we are not working with the same underlying categories, one *
*of the
main differences is that the model structure in [42] is right proper, since eve*
*ry object
is fibrant, but it is not known to be left proper. Hence it is not suitable fo*
*dder for
stabilization in terms of today's (left) Bousfield localization machinery. One*
* of the
main points in our work is that fibrancy is a special property; in fact, it gov*
*erns the
whole theory, while left properness is required for defining the stable C*homo*
*topy
category.
7
2 Preliminaries
2.1 C *spaces
Let C* Alg denote the category of separable C*algebras and *homomorphisms. It
is an essentially small category with small skeleton the set of C*algebras whi*
*ch are
operators on a fixed separable Hilbert space of countably infinite dimension. I*
*n what
follows, all C*algebras are objects of C*  Alg so that commutative C*algebra*
*s can
be identified with pointed compact metric spaces via GelfandNaimark duality. L*
*et
K denote the C*algebra of compact operators on a separable, infinite dimension*
*al
Hilbert space, e.g. the space `2 of square summable sequences.
The object of main interest in this section is obtained from C*Alg via embe*
*ddings
C * Alg ____//_C* Spc___//_*C * Spc.
A C *space is a setvalued functor on C * Alg. Let C * Spc denote the categ*
*ory
of C*spaces and natural transformations. By the Yoneda lemma there exists a f*
*ull
and faithful contravariant embedding of C * Alg into C * Spc which preserves
limits. This entails in particular natural bijections C * Alg(A, B) = C * Sp*
*c(B, A)
for all C *algebras A, B. Since, as above, the context will always clearly in*
*dicate
the meaning we shall throughout identify every C*algebra with its corresponding
representable C*space. Note that every set determines a constant C*space. A p*
*ointed
C *space consists of a C*space X together with a map of C*spaces from the tr*
*ivial
C *algebra to X. We let C*  Spc0 denote the category of pointed C*spaces. Th*
*ere
exists a functor C*  Spc ! C * Spc0 obtained by taking pushouts of diagrams of
the form X ; ! 0; it is left adjoint to the forgetful functor. Observe that e*
*very C*
algebra is canonically pointed. The category *C * Spc of cubical C*spaces con*
*sists
of possibly void collections of C *spaces Xn for all n 0 together with face *
*maps
dffi:Xn ! Xn1, 1 i n, ff = 0, 1 (corresponding to the 2n faces of dimensio*
*n n  1 in
a standard geometrical ncube), and degeneracy maps si:Xn1! Xn where 1 i n
subject to the cubical identities dffidfij= dfij1dffifor i < j, sisj= sj+1sifo*
*r i j and
8
>>>s dffi < j
>>< j1 i
dffisj= >>id i = j
>>>
:sjdffi1i > j.
8
A map of cubical C *spaces is a collection of maps of C *spaces Xn ! Yn for a*
*ll
n 0 which commute with the face and degeneracy maps. An alternate description
uses the box category * of abstract hypercubes representing the combinatorics of
power sets of finite ordered sets [41, x3]. The box category * is the subcatego*
*ry of
the category of poset maps 1n ! 1m which is generated by the face and degeneracy
maps. Here, 1n = 1xn = {(ffl1, . .,.ffln)ffli= 0, 1} is the nfold hypercube. *
*As a poset 1n is
isomorphic to the power set of {0, 1, . .,.n}. The category *Set of cubical set*
*s consists
of functors *op ! Set and natural transformations. With these definitions we may
identify *C * Spc with the functor category [C * Alg, *Set] of cubical setva*
*lued
functors on C*Alg . Note that every cubical set defines a constant cubical C**
*space by
extending degreewise the correspondence between sets and C*spaces. A particula*
*rly
important example is the standard ncell defined by *n *(, 1n). Moreover, ev*
*ery
C *algebra defines a representable C*space which can be viewed as a discrete *
*cubical
C *space. The category *C * Spc0 of pointed cubical C*spaces is defined usin*
*g the
exact same script as above. Hence it can be identified with the functor categor*
*y of
pointed cubical setvalued functors on C* Alg.
We shall also have occasion to work with the simplicial category of finite*
* ordinals
[n] = {0 < 1 < . .<.n} for n 0 and orderpreserving maps. The category C * *
*Spc
of simplicial C*spaces consists of C*spaces Xn for all n 0 together with fa*
*ce maps
di:Xn ! Xn1, 1 i n, and degeneracy maps si:Xn1! Xn, 1 i n, subject to
the simplicial identities didj= dj1difor i < j, sisj= sj+1sifor i j and
8
>>>s di < j
>>< j1 i
disj= >>id i = j, j + 1
>>>
:sjdi1 i > j + 1.
Let C*Algdenote a suitable monoidal product on C* Alg with unit the compl*
*ex
numbers. Later we shall specialize to the symmetric monoidal maximal and minimal
tensor products, but for now it is not important to choose a specific monoidal *
*product.
In x2.3 we recall the monoidal product *Setin Jardine's closed symmetric monoi*
*dal
structure on cubical sets [41, x3]. We shall outline an extension of these dat*
*a to a
closed monoidal structure on *C * Spc following the work of Day [17]. The exte*
*rnal
monoidal product of two cubical C *spaces X, Y :C*  Alg ! *Set is defined by
setting
Xe Y *SetO (X x Y).
9
Next we introduce the monoidal product X Y of X and Y by taking the left Kan
extension of C*Algalong Xe Y or universal filler in the diagram:
eY
C * Alg x C* Alg X___//_*Set66m
m m
C*Alg m m
fflfflmm
C * Alg
Thus the *Setvalues of the monoidal product are given by the formulas
X Y(A) colim X(A1) *SetY(A2).
A1 C*AlgA2!A
The colimit is indexed on the category with objects ff: A1 C*AlgA2 ! A and ma*
*ps
pairs of maps (OE, _): (A1, A2) ! (A01, A02) such that ff0(_ OE) = ff. By fu*
*nctoriality
of colimits it follows that X Y is a cubical C*space. When couched as a coe*
*nd,
the tensor product is a weighted average of all of the handicrafted external te*
*nsor
products Xe Y *SetO (X x Y) in the sense that
Z A1,A22C*Algi j
X Y(A) = X(A1) *SetY(A2) *SetC* Alg(A1 C*AlgA2, A).
Since the tensor product is defined by a left Kan extension, it is characterize*
*d by the
universal property
*C * Spc(X Y, Z) = [C * Alg, *C * Spc](Xe Y, Z O C*Alg).
The bijection shows that maps between cubical C*spaces X Y ! Z are uniquely
determined by maps of cubical sets X(A) *SetY(B) ! Z(A C*AlgB) which are
natural in A and B. Note also that the tensor product of representable C *spa*
*ces
A B is represented by the monoidal product A C*AlgB and for cubical sets K,*
* L,
K L = K *SetL, i.e. (C *Alg , C*Alg) ! (*C *Spc , ) and (*, *Set) ! (*C **
*Spc , )
are monoidal functors in the strong sense that both of the monoidal structures *
*are
preserved to within coherent isomorphisms. According to our standing hypothesis,
the C*algebra C (the complex numbers) represents the unit for the monoidal pro*
*duct
.
If eZ is a cubical setvalued functor on C * Alg x C*  Alg and Y is a cubi*
*cal
C *space, define the external function object ]Hom_(Y, eZ) by
i j
H]om_(Y, eZ)(A) *C * Spc Y, eZ(A, ) .
10
Then for every cubical C*space X there is a bijection
i j
*C * Spc X, ]Hom_(Y, eZ) = [C * Alg x C* Alg, *Set](Xe Y, eZ).
A pair of cubical C*spaces Y and Z acquires an internal hom object
Hom__(Y, Z) ]Hom_(Y, Z O C*Alg).
Using the characterization of the monoidal product it follows that
Z O___//_Hom_(Y, Z)
determines a right adjoint of the functor
X O___//_X Y.
Observe that *C * Spc equipped with and Hom__ becomes a closed symmetric
monoidal category provided the monoidal product C*Algis symmetric, which we
may assume.
According to the adjunction the natural evaluation map Hom__(Y, Z) Y ! Z
determines an exponential law
i j
*C * Spc(X Y, Z) = *C * Spc X, Hom__(Y, Z) .
Using these data, standard arguments imply there exist natural isomorphisms
i j
Hom__(X Y, Z) = Hom__X, Hom__(Y, Z)
i j
*C * Spc(Y, Z) = *C * Spc(C Y, Z) = *C * Spc C, Hom__(Y, Z)
and
Hom__(C, Z) = Z.
In what follows we introduce a cubical set tensor and cotensor structure on *
**C *
Spc . This structure will greatly simplify the setup of the left localization *
*theory of
model structures on cubical C*spaces. If X and Y are cubical C*spaces and K i*
*s a
cubical set, define the tensor X K by
X K(A) X(A) *SetK (3)
11
and the cotensor YK in terms of the ordinary cubical function complex
i j
YK(A) hom *SetK, Y(A) . (4)
The cubical function complex hom *C*Spc(X, Y) of X and Y is defined by setting
hom *C*Spc(X, Y)n *C * Spc(X *n, Y).
By the Yoneda lemma there exists a natural isomorphism of cubical sets
hom *C*Spc(A, Y) = Y(A). (5)
Using these definitions one verifies easily that *C * Spc is enriched in cubic*
*al sets
*Set. Moreover, there are natural isomorphisms of cubical sets
i j
hom *C*Spc(X K, Y) = hom *SetK, hom *C*Spc(X, Y) = hom *C*Spc(X, YK).
In particular, taking 0cells we obtain the natural isomorphisms
i j
*C * Spc(X K, Y) = *Set K, hom *C*Spc(X, Y) = *C * Spc(X, YK). (6)
It is useful to note that the cubical function complex is the global section*
*s of the
internal hom object, and more generally that
i j
Hom__(X, Y)(A) = hom *C*SpcX, Y( A) .
In effect, note that according to the Yoneda lemma and the exponential law for *
*cubical
C *spaces, we have
i j
Hom__(X, Y)(A)= hom *C*SpcA, Hom__(X, Y)
i j
= hom *C*Spc(X A), Y .
Hence, since the Yoneda embedding of (C * Alg)op into *C * Spc is monoidal, we
have
Hom__(B, Y) = Y( B).
The above allows to conclude there are natural isomorphisms
i j
Hom__(X, Y)(A)= hom *C*Spc(X A), Y
i j
= hom *C*SpcX, Hom__(A, Y)
i j
= hom *C*SpcX, Y( A) .
12
In particular, the above entails natural isomorphisms
Hom__(B, Y)(A) = Y(A B). (7)
There exist entirely analogous constructs for pointed cubical C*spaces and *
*pointed
cubical sets. In short, there exists a closed monoidal category (*C *Spc 0, ,*
* Hom__) and
all the identifications above hold in the pointed context. Similarly, there are*
* closed
monoidal categories ( C * Spc, , Hom__) and ( C * Spc0, , Hom__) of simplic*
*ial and
pointed simplicial C*spaces constructed by the same method. Here we consider t*
*he
categories of simplicial sets Set and pointed simplicial sets Set*with their *
*standard
monoidal products.
Next we recall some sizerelated concepts which are also formulated in [35, *
*x2.1.1].
One of the lessons of the next sections is that these issues matter when dealin*
*g with
model structures on cubical C*spaces. Although the following results are state*
*d for
cubical C*spaces, all results hold in the pointed category *C * Spc0 as well.
Let ~ be an ordinal, i.e. the partially ordered set of all ordinals < ~. A ~*
*sequence
or transfinite sequence indexed by ~ in *C * Spcis a functor F: ~ ! *C * Spcw*
*hich
is continuous at every limit ordinal fi < ~ in the sense that there is a natura*
*lly induced
isomorphism colimff>>ffifffflj < i
>>< i1 j
ffljffiffi= >>id j = i
>>>
:ffiffifflj1j > i.
Denote by C(*otop) the standard cubical C*algebra
oo___ oo___oo_
n O___//_C(*ntop):oCoC(I1)_oo_oC(I2)o_oo_oo_.o.o._oo_ (8)
oo___oo_
comprising continuous complexvalued functions on the topological standard ncu*
*be.
Its cubical structure is induced in the evident way by the coface and codegener*
*acy
maps of *otopgiven above.
For legibility we shall use the same notation C(*otop) for the naturally ind*
*uced
cocubical C*space
i j
C* Alg C(*otop),  :C * Alg___//_(*Set)op.
The singular functor
Singo*:*C * Spc_____//*C * Spc
is an endofunctor of cubical C*spaces. Its value at a C*space X is by definit*
*ion given
as the internal hom object
i j
Singo*(X) Hom__C(*otop), X .
The cubical structure of Singo*(X) is obtained from the cocubical structure of *
*C(*otop).
15
Plainly this functor extends to an endofunctor of *C * Spc by taking the di*
*agonal
of the bicubical C*space
i j i j
(m, n)O___//_Hom_C(*mtop), Xn = Xn C(*mtop)  .
In particular, the singular functor specializes to a functor from C*spaces
Singo*:C*  Spc_____//*C * Spc.
Its left adjoint is the geometric realization functor
 . : *C * Spc__//C* Spc.
If X is a cubical C*space, then its geometric realization X is the coend of *
*the functor
* x *op ! C* Spc given by (1m, 1n) 7! C(*mtop) Xn. Hence there is a coequali*
*zer in
C * Spc
` m ____//_` n R*op
:1m!1nC(*top) Xn____//_1nC(*top) Xn___//_X C(*ntop) Xn.
The two parallel maps in the coequalizer associated with the maps : 1m ! 1n in*
* the
box category * are gotten from the natural maps
`
C(*mtop) Xn__//_C(*ntop) _Xn_//_nC(*ntop) Xn
and `
C(*mtop) Xn___//_C(*mtop) _Xm_//_nC(*ntop) Xn.
Example 2.8: For every cubical C*space X there is a monomorphism X ! Singo*(X).
In ncells it is given by the canonical map
i j
Xn ____//_Hom_C(*ntop), Xn .
Example 2.9: For n 0 there are natural isomorphisms
i j
Singn*(X)(A) = X A C(*ntop)
and
X C(*ntop) = X C(*ntop).
16
Remark 2.10: The cognoscenti of homotopy theory will notice the formal similari*
*ties
between  .  and the geometric realization functors of Milnor from semisimpli*
*cial
complexes to CWcomplexes [58] and of MorelVoevodsky from simplicial sheaves
to sheaves on some site [59]. Note that using the same script we obtain a geome*
*tric
realization functor for every cocubical C*algebra. The standard cocubical C*s*
*pace
meshes well with the monoidal products we shall consider in the sense that C(*n*
*top)
and C(*1top) . . .C(*1top) are isomorphic as C*algebras, and hence as C*spa*
*ces.
Remark 2.11: Note that n 7! C(*ntop) defines a functor * ! C * Spc *C * Spc.
Since the category *C * Spc is cocomplete this functor has an enriched symmetr*
*ic
monoidal left Kan extension *Set ! *C * Spc which commutes with colimits and
sends *o to C(*otop).
The next result is reminiscent of [59, Lemma 3.10] and [40, Lemma B.1.3].
Lemma 2.12: The geometric realization functor  . : *C * Spc ! C * Spc prese*
*rves
monomorphisms.
Proof.For i < j and n 2 the cosimplicial identities imply there are pullback *
*diagrams:
j1 n1
C(*n2top)d//_C(*top)
di di
fflffl j fflffl
C(*n1top)d//_C(*ntop)
Hence @*n is isomorphic to the union @C(*ntop) of the images di:C(*n1top) ! *
*C(*ntop),
and @*n ! *n is a monomorphism for n 2. And therefore the lemma is equiva*
*lent
to the fact that C(*otop) is augmented, i.e. the two maps @*1 *1 induce an in*
*jection
`
C(*0top) C(*0top) ! C(*1top). *
* *
Remark 2.13: Denote by otopthe topological standard cosimplicial set and by C(*
* otop)
the simplicial C*algebra n 7! C( ntop) of continuous complexvalued functions *
*on ntop
which vanish at infinity. As in the cubical setting, the corresponding cosimpl*
*icial
C *space C( otop) defines a singular functor Singo and a geometric realization*
* functor
 . : C * Spc ! C*  Spc. We note that C( otop) does not mesh well with mono*
*idal
products in the sense that C( ntop) , C( 1top) . . .C( 1top). The other prope*
*rties of the
cubical singular and geometric realization functors in the above hold simplicia*
*lly.
17
2.2 G  C *spaces
Let G be a locally compact group. In this section we indicate the steps require*
*d to
extend the results in the previous section to GC *algebras. Recall that a GC*
* *algebra
is a C*algebra equipped with a strongly continuous representation of G by C*a*
*lgebra
automorphisms. There is a corresponding category G  C* Algcomprised of G  C*
algebras and G equivariant *homomorphisms. Since every C *algebra acquires a
trivial G action, there is an evident functor C * Alg ! G  C*  Alg. It giv*
*es a
unique G  C*algebra structure to C because the identity is its only automorph*
*ism.
Denote by GC*Alga symmetric monoidal tensor product on G  C* Alg with unit
C. To provide examples, note that if C*Algdenotes the maximal or minimal tens*
*or
product on C * Alg, then A C*AlgB inherits two strongly continuous G actions
and hence the structure of a (G x G)  C*algebra for all objects A, B 2 G  C**
* Alg.
Thus A C*AlgB becomes a G  C*algebra by restricting the (G x G)action to t*
*he
diagonal. For both choices of a tensor product on C* Alg this construction fur*
*nishes
symmetric monoidal structures on GC *Alg with unit the complex numbers turning
C * Alg ! G  C* Alg into a symmetric monoidal functor.
With the above as background we obtain embeddings
G  C* Alg ____//_G C* Spc____//_*G  C* Spc
by running the same tape as for C* Alg. The following properties can be establ*
*ished
using the same arguments as in the previous section.
o *G C *Spc is a closed symmetric monoidal category with symmetric monoidal
product X G Y, internal hom object Hom__G(X, Y) and cubical function comp*
*lex
hom *GC*Spc(X, Y) for cubical G  C*spaces X and Y. The unit is represe*
*ntable
by the complex numbers.
o *G  C* Spc is enriched in cubical sets.
o *G  C* Spc is locally presentable.
i j
o Hom__G S1 C0(R),  is finitely presentable.
o There exists a Gequivariant singular functor
SingG,o*:*G  C* Spc____//_*G  C* Spc.
18
The categories of pointed cubical G  C*spaces, simplicial G  C*spaces and p*
*ointed
simplicial G  C*spaces acquire the same formal properties as *G  C* Spc.
2.3 Model categories
In order to introduce C *homotopy theory properly we follow Quillen's ideas for
axiomatizing categories in which we can "do homotopy theory." A striking beauty*
* of
the axioms for a model structure is that algebraic categories such as chain com*
*plexes
also admit natural model structures, as well as the suggestive geometric exampl*
*es of
topological spaces and simplicial sets. The axioms for a stable homotopy catego*
*ry, or
even for a triangulated category, are often so cumbersome to check that the bes*
*t way
to construct such structures is as the homotopy category of some model structur*
*e.
The standard references for this material include [25], [28], [33], [35] and [6*
*5].
Definition 2.14: A model category is a category M equipped with three classes of
*
* ~
maps called weak equivalences, cofibrations and fibrations which are denoted by*
* ! ,
and respectively. Maps which are both cofibrations and weak equivalences are
~
called acyclic cofibrations and denoted by ; acyclic fibrations are defined s*
*imilarly
~
and denoted by . The following axioms are required [35, Definition 1.1.4]:
CM 1: M is bicomplete.
CM 2: (Saturation or twooutofthree axiom) If f :X ! Y and g: Y ! W are maps
in M and any two of f , g, and gf are weak equivalences, then so is the *
*third.
CM 3: (Retract axiom) Every retract of a weak equivalence (respectively cofibr*
*ation,
fibration) is a weak equivalence (respectively cofibration, fibration).
CM 4: (Lifting axiom) Suppose there is a commutative square in M:
X _____//Z>>
fflffl"
p " " q
fflfflfflfflfflffl"
Y ____//_W
Then the indicated lifting Y ! Z exists if either p or q is a weak equiv*
*alence.
CM 5: (Factorization axiom) Every map X ! W may be functorially factored in two
~ ~
ways, as X Y W and as X Z W.
19
If every square as in CM 4 has a lifting Y ! Z, then X ! Y is said to have t*
*he left
lifting property with respect to Z ! W. The right lifting property is defined s*
*imilarly.
When M is a model category, one may formally invert the weak equivalences to ob*
*tain
the homotopy category Ho (M) of M [65, I.1]. A model category is called pointed*
* if the
initial object and terminal object are the same. The homotopy category of any p*
*ointed
model category acquires a suspension functor denoted by . It turns out that Ho*
* (M)
is a pretriangulated category in a natural way [35, x7.1]. When the suspension*
* is an
equivalence, M is called a stable model category, and in this case Ho (M) becom*
*es a
triangulated category [35, x7.1]. We will give examples of such model structure*
*s later
in this text.
A Quillen map of model categories M ! N consists of a pair of adjoint functo*
*rs
____//_
L: M oo___N :R
where the left adjoint L preserves cofibrations and trivial cofibrations, or eq*
*uivalently
that R preserves fibrations and trivial fibrations. Every Quillen map induces a*
*djoint
total derived functors between the homotopy categories [65, I.4]. The map is a *
*Quillen
equivalence if and only if the total derived functors are adjoint equivalences *
*of the
homotopy categories.
For the definition of a cofibrantly generated model category M with generati*
*ng
cofibrations I and generating acyclic cofibrations J and related terminology we*
* refer
to [35, x2.1]. The definition entails that in order to check whether a map in M*
* is an
acyclic fibration or fibration it suffices to test the right lifting property w*
*ith respect to I
respectively J. In addition, the domains of I are small relative to Icell and *
*likewise for
J and Jcell. It turns out the (co)domains of I and J often have additional pro*
*perties.
Next we first recall [36, Definition 4.1].
Definition 2.15: A cofibrantly generated model category is called finitely gene*
*rated
if the domains and codomains of I and J are finitely presentable, and almost fi*
*nitely
generated if the domains and codomains of I are finitely presentable and there *
*exists
a set of trivial cofibrations J0with finitely presentable domains and codomains*
* such
that a map with fibrant codomain is a fibration if and only if it has the right*
* lifting
property with respect to J0, i.e. the map is contained in J0inj.
In what follows we will use the notion of a weakly finitely generated model
structure introduced in [22, Definition 3.4].
20
Definition 2.16: A cofibrantly generated model category is called weakly finite*
*ly
generated if the domains and the codomains of I are finitely presentable, the d*
*omains
of the maps in J are small, and if there exists a subset J0 of J of maps with f*
*initely
presentable domains and codomains such that a map with fibrant codomain is a
fibration if and only if it is contained in J0inj.
Lemma 2.17: ([22, Lemma 3.5]). In weakly finitely generated model categories, t*
*he classes
of acyclic fibrations, fibrations with fibrant codomains, fibrant objects, and *
*weak equivalences
are closed under filtered colimits.
Remark 2.18: Lemma 2.17 implies that in weakly finitely generated model categor*
*ies,
the homotopy colimit of a filtered diagram maps by a weak equivalence to the co*
*limit
of the diagram. This follows because the homotopy colimit is the total left der*
*ived
functor of the colimit and filtered colimits preserves weak equivalences.
Two fundamental examples of model structures are the standard model structur*
*es
on the functor categories of simplicial sets Set [ op, Set] constructed by Q*
*uillen
[65] and of cubical sets *Set [*op, Set] constructed independently by Cisinsk*
*i [12]
and Jardine [41]. The box category * has objects 10 = {0} and 1n = {0, 1}nfor e*
*very n 1.
The maps in * are generated by two distinct types of maps which are subject to *
*the
dual of the cubical relations, and defined as follows. For n 1, 1 i n and*
* ff = 0, 1
define the coface map ffii,ffn:1n1 ! 1n by (ffl1, . .,.ffln1) 7! (ffl1, . .,.*
*ffli1, ff, ffli, . .,.ffln1).
And for n 0 and 1 i n + 1 the codegeneracy map oein:1n+1 ! 1n is defined
by (ffl1, . .,.ffln+1) 7! (ffl1, . .,.ffli1, ffli+1, . .,.ffln+1). Recall that*
* a map f in *Set is a weak
equivalence if applying the triangulation functor yields a weak equivalence f *
* of
simplicial sets. A cofibration of cubical sets is a monomorphism. The Kan fibra*
*tions
are forced by the right lifting property with respect to all acyclic monomorphi*
*sms.
Theorem 2.19: ([12],[41]) The weak equivalences, cofibrations and Kan fibration*
*s define a
cofibrantly generated and proper model structure on *Set for which the triangul*
*ation functor
is a Quillen equivalence.
Example 2.20: The cubical set @*n is the subobject of the standard ncell *n ge*
*nerated
by all faces dffi: *n1! *n. It follows there is a coequalizer diagram of cubic*
*al sets
` ____n2//_` n1
0 i00.i n
28
It follows that every map between cubical C*spaces acquires a factorization th*
*rough
some sequential I*C*Spccell (respectively J*C*Spccell) composed with a poin*
*twise
acyclic projective fibration (respectively projective fibration).
For example, to prove the claim for J*C*Spc, assume X ! Y is a projective f*
*ibration
and consider commutative diagrams of cubical C*spaces of the form:
A un(ff,i)!X
??? ??
?y ??y (12)
A *n ! Y
By the Yoneda lemma (5) and (6), such diagrams are in onetoone correspondence
with commutative diagrams of cubical sets of the form:
un(ff,i)!hom*C*Spc(A, X)! X(A)
??? ?? ??
?y ??y ??y (13)
*n ! hom *C*Spc(A, Y)! Y(A)
The assumption implies there exists a lifting *n ! X(A) in (13), which means th*
*ere is
a lifting A *n ! X in (12). Now suppose that X ! Y is a map in *C * Spc. Def*
*ine
the pushout Y0by combining all commutative diagrams of the following form, where
n 0, ff = 0, 1 and A runs through the isomorphism classes of C*algebras:
` n
A un(ff,i)!X A u(ff,i)!X
??? ?? ?? ??
?y ??y ??y ??y (14)
` n 0
A *n ! Y A * ! Y
Then X ! Y0 is a pointwise acyclic projective cofibration because the induced m*
*ap
of coproducts in (14) is so and the evaluation functor EvA preserves pushouts a*
*nd
acyclic cofibrations. By iterating this construction countably many times we ob*
*tain a
diagram where the horizontal maps are pointwise acyclic projective cofibrations:
X ____//_Y0___//_Y1___//_._._.//_Yn__//_. . . (15)
Letting Y1 denote the colimit of (15) yields a factorization of the original map
X ____//_Y1___//_Y. (16)
29
It is straightforward to show that X ! Y1 is a pointwise acyclic projective cof*
*ibration
and the map Y1 ! Y acquires the right lifting property with respect to the set *
*J*C*Spc.
This small object argument is due to Quillen [65] and shows that every map of c*
*ubical
C *spaces factors into a pointwise acyclic cofibration composed with a project*
*ive
fibration (16). The remaining arguments constituting a proof of Theorem 3.2 are*
* of
a similar flavor, and one shows easily that the pointwise projective model is w*
*eakly
finitely generated. We deduce the following result.
Corollary 3.3: Pointwise weak equivalences, projective fibrant objects, pointwi*
*se acyclic
projective fibrations and projective fibrations are closed under filtered colim*
*its.
Recall that X ! Y is a projective cofibration if and only if it is a relativ*
*e I*C*Spccell
complex or a retract thereof. Hence projective cofibrations are pointwise cofib*
*rations,
a.k.a. monomorphisms or injective cofibrations of cubical C*spaces.
Lemma 3.4: Every projective cofibration is a monomorphism.
Lemma 3.5: The pointwise projective model is a proper model structure.
Proof.This is a routine check using properness of the model structure on cubica*
*l sets
and Lemma 3.4. *
If A, B are C*algebras and K, L are cubical sets there is a natural isomorp*
*hism
(A K) (B L) = (A C*AlgB) (K *SetL).
This follows since for every cubical C*space X there are natural isomorphisms
i j i j
*C * Spc (A K) (B L), X= *Set K *SetL, X(A C*AlgB)
i j
= *C * Spc (A C*AlgB) (K *SetL), X .
Lemma 3.6: If A is a C*algebra and K is a cubical sets, then A K is a projec*
*tive cofibrant
cubical C*space. In particular, every C*algebra is projective cofibrant.
Proof.By definition, A : *Set ! *C *Spc is a left Quillen functor for the poi*
*ntwise
projective model structure on *C * Spc and every cubical set is cofibrant. We *
*note
the assertion for C*algebras follows by contemplating the set I*C*Spcof gener*
*ating
projective cofibrations. *
* *
30
Since every map between discrete cubical sets is a Kan fibration we get:
Lemma 3.7: Every C*algebra is projective fibrant.
In addition to the Quillen adjunction gotten by evaluating cubical C *space*
*s at
a fixed C*algebra, the constant diagram functor from *Set to *C * Spc has as *
*left
adjoint the colimit functor, which can be derived. Each evaluation functor pas*
*ses
directly to a functor between the corresponding homotopy categories.
Next we observe that the projective model is compatible with the enrichment *
*of
cubical C*spaces in cubical sets introduced in x2.1.
Lemma 3.8: The pointwise projective model is a cubical model structure.
Proof.If i: X Y is a projective cofibration of cubical C*spaces and p: K L*
* is a
cofibration of cubical sets, we claim the naturally induced pushout map
`
X L X KY K ____//_Y L
is a projective cofibration, and a pointwise acyclic projective cofibration if *
*in addition
either i is a pointwise weak equivalence or p is a weak equivalence. In effect*
*, the
model structure on *Set is cubical and, as was noted above, evaluating projecti*
*ve
cofibrations produce cofibrations or monomorphisms of cubical sets. Since colim*
*its
of cubical C*spaces, in particular all pushouts, are formed pointwise the asse*
*rtions
follow. *
One shows easily using the cotensor structure that Lemma 3.8 is equivalent t*
*o the
following result:
Corollary 3.9: The following statements hold and are equivalent.
o If j: U V is a projective cofibration and k: Z W is a projective fibra*
*tion, then
the pullback map
hom *C*Spc(V, Z)___//_hom*C*Spc(V, W) xhom*C*Spc(U,W)hom*C*Spc(U, Z)
is a Kan fibration of cubical sets which is a weak equivalence if in addit*
*ion either j or k
is pointwise acyclic.
31
o If p: K L is a cofibration of cubical sets and k: Z W is a projective *
*fibration,
then
ZL ____//_ZK xWK WL
is a projective fibration which is pointwise acyclic if either k is pointw*
*ise acyclic or p is
acyclic.
Recall that every C *algebra, in particular the complex numbers, determines*
* a
projective cofibrant representable cubical C*space. Thus the next result verif*
*ies that
the projective model is a monoidal model structure.
Lemma 3.10: The following statements hold and are equivalent.
o If i: X Y and j: U V are projective cofibrations, then
`
X V X U Y U ____//_Y V
is a projective cofibration which is pointwise acyclic if either i or j is.
o If j: U V is a projective cofibration and k: Z W is a projective fibra*
*tion, then
the pullback map
Hom__(V, Z) ____//_Hom_(V, W) xHom_(U,W)Hom_(U, Z)
is a projective fibration which is pointwise acyclic if either j or k is.
Remark 3.11: Lemma 3.10 shows that the pointwise projective model structures on
*C * Spc is monoidal [75, Definition 3.1]. Now since the complex numbers C is a
projective cofibrant cubical C*space, it follows that induces a monoidal pro*
*duct on
the associated homotopy category.
Remark 3.12: We note that since limits are defined pointwise, the second part of
Lemma 3.10 implies the first part of Corollary 3.9 by evaluating internal hom o*
*bjects
at the complex numbers.
Proof.To prove the first statement we may assume
i = A (@*m *m) and j = B (@*n *n).
32
Projective cofibrations are retracts of relative I*C*Spccell complexes so if *
*j is a retract
of a transfinite composition of cobase changes of maps in I*C*Spcthe pushout p*
*roduct
map of i and j is a retract of a transfinite composition of cobase changes of p*
*ushout
product maps between i and members of I*C*Spc. Thus, by analyzing the generati*
*ng
projective cofibrations, we may assume the pushout product map in question is of
the form `
(A B) (*m @*n @*m @*n@*m *n___//_*m+n).
This is a projective cofibration since (A B)  is a left Quillen functor for th*
*e pointwise
projective model structure and the map of cubical sets in question is a cofibra*
*tion.
The remaining claim is proved analogously: We may assume i = A (@*m *m)
and j = B (un(ff,i) *n), so that the pushout product map is of the form
`
(A B) (*m un(ff,i)@*m un(ff,i)@*m__/*n/*m+n).
This map is a pointwise acyclic projective cofibration by the previous argument.
In order to prove the second part, note that by adjointness there is a onet*
*oone
correspondence between the following types of commutative diagrams:
`
X V X U Y U ! Z X ! Hom__(V, Z)
??? ?? ??? ???
?y ??y $ ?y ?y
Y V ! W Y ! Hom__(V, W) xHom_(U,W)Hom_(U, Z)
Hence the lifting properties relating projective cofibrations and projective fi*
*brations
combined with the first part finish the proof. *
* *
Lemma 3.13: Suppose Z is a projective cofibrant cubical C*space. Then
Z : *C * Spc ____//_*C * Spc
preserves the classes of projective cofibrations, acyclic projective cofibratio*
*ns and pointwise
weak equivalences between projective cofibrant cubical C*spaces.
Proof.This follows because the pointwise projective model structure is monoidal*
*. *
Lemma 3.14: The monoid axiom holds in the pointwise projective model structure.
33
Proof.We need to check that (*C * Spc J*C*Spc)cell consists of weak equiva*
*lences
[75, Definition 3.3]. Since the monoid axiom holds for cubical sets and colimi*
*ts in
*C * Spc are defined pointwise, it suffices to show that (*C * Spc J*C*Spc*
*)(B) is
contained in *Set J*Setfor every C*algebra B. This follows from the equaliti*
*es
i i jj i j
X A (un(ff,i) *n) =(B)(X A) (un(ff,i) *n) (B)
= (X A)(B) (un(ff,i) *n).
*
Remark 3.15: Lemma 3.14 combined with the work of SchwedeShipley [75] implies
that modules over a monoid in *C * Spc inherit a module structure, where the
fibrations and weak equivalences of modules over the monoid are just the module
maps that are fibrations and weak equivalences in the underlying model structur*
*e. In
the refined model structures on *C *Spc the same result holds for cofibrant mo*
*noids
by reference to [34].
Next we turn to the injective model structure on *C * Spc. Let ~ be the fi*
*rst
infinite cardinal number greater than the cardinality of the set of maps of C**
* Spc. If
!, as usual, denotes the cardinal of the continuum, define the cardinal number *
*fl by
fl ~!~!.
Definition 3.16: Let I~*C*Spcdenote the set of maps X ! Y such that X(A) ! Y(A*
*) is
a cofibration of cubical sets of cardinality less than ~ for all A. Likewise, l*
*et Jfl*C*Spc
denote the set of maps X ! Y such that X(A) ! Y(A) is an acyclic cofibration of
cubical sets of cardinality less than fl for all A.
The next result follows by standard tricks of the model category trade.
Proposition 3.17: Let q: Z ! W be a map of cubical C*spaces.
o If q has the right lifting property with respect to every map in I~*C*Spc*
*, then q has the
right lifting property with respect to all maps X ! Y for which X(A) ! Y(A*
*) is a
cofibration of cubical sets for all A.
o If q has the right lifting property with respect to every map in Jfl*C*Sp*
*c, then q has the
right lifting property with respect to all maps X ! Y for which X(A) ! Y(A*
*) is an
acyclic cofibration of cubical sets for all A.
34
Note that X ! Y is a monomorphism if and only if X(A) ! Y(A) is a cofibration
of cubical sets for all A.
From Proposition 3.17 we obtain immediately the next result.
Corollary 3.18: Let p: X ! Y be a map of cubical C*spaces.
o The map p is an I~*C*Spccofibration, i.e. has the left lifting property *
*with respect to every
map with the right lifting property with respect to I~*C*Spc, if and only*
* if X(A) ! Y(A)
is a cofibration of cubical sets for all A.
o The map p is a Jfl*C*Spccofibration, i.e. has the left lifting property *
*with respect to every
map with the right lifting property with respect to Jfl*C*Spc, if and onl*
*y if X(A) ! Y(A)
is an acyclic cofibration of cubical sets for all A.
Definition 3.19: A map X ! Y of cubical C *spaces is an injective fibration if*
* it
has the right lifting property with respect to all maps which are simultaneousl*
*y a
monomorphism and a pointwise weak equivalence.
We refer to the following as the pointwise injective model structure.
Theorem 3.20: The classes of monomorphisms a.k.a. injective cofibrations, injec*
*tive fibrations
and pointwise weak equivalences determine a combinatorial model structure on *C*
* * Spc.
Lemma 3.21: The following hold for the pointwise injective model structure.
o It is a proper model structure.
o It is a cubical model structure.
o Every cubical C*space is cofibrant.
o The identity functor on *C * Spc yields a Quillen equivalence with the po*
*intwise
projective model structure.
By adopting the same definitions to simplicial C*spaces we get a combinator*
*ial
projective model structure on C * Spc. Using standard notations for boundaries
and horns in Set, a set of generating cofibrations is given by
I C*Spc {A (@ n n)}n 0
35
and a set of generating acyclic cofibrations by
J C*Spc {A ( ni n)}n>00.i n
The projective model structure on C * Spc acquires the same additional proper*
*ties
as the projective model structure on cubical C*spaces.
From Theorem 2.19 we deduce that the corresponding homotopy categories are
equivalent:
Lemma 3.22: There is a naturally induced Quillen equivalence between projective*
* model
structures:
*C * Spc ____//_oCo*_Spc
There is also a pointwise injective model structure on C * Spc where again*
* the
cofibrations and the weak equivalences are defined pointwise. It defines a simp*
*licial
model structure, and acquires the same formal properties as the pointwise injec*
*tive
model structure on *C *Spc . In particular, the identity on C *Spc defines a*
* Quillen
equivalence between the pointwise injective and projective model structures.
At last in this section we shall verify the rather technical cellular model *
*category
conditions for our examples at hand.
Proposition 3.23: The pointwise injective and projective model structures on *C*
* * Spc
and C * Spc and their pointed versions are cellular model structures.
Lemma 3.24: Every cubical or simplicial C*space X is small.
Proof.Suppose X a cubical C*space (the following argument applies also to simp*
*licial
C *spaces). Let ~ be the regular successor cardinal of the set
a
Xn(A).
A2C*Alg,n 0
For every regular cardinal ~ ~ and ~sequence F in *C * Spc we claim there i*
*s a
naturally induced isomorphism
colimff<~*C * Spc(X, Fff)__//_*C * Spc(X, colimff<~Fff). (17)
36
Injectivity of (17) follows by taking sections and using the fact that every cu*
*bical
(simplicial) set is small, cf. [35, Lemma 3.1.1]. The restriction of X ! colim*
*ffFffin
*C * Spc to any cell of X factors through Ffffor some ff < ~. Regularity of ~ *
*and
the fact that there are less than ~ < ~ cells in X implies the restriction to a*
*ny cell of X
factors through Ffffor some ff < ~. Hence the map in (17) is surjective. *
* *
Lemma 3.25: The domains and codomains of the generating cofibrations I of the i*
*njective
model structures on *C * Spc and C * Spc are compact relative to I.
Proof.Let ~ be the regular successor cardinal of the cardinal of the set
a a
Xn(A) t Yn(A).
X!Y2I A2C*Alg,n 0
If Z is a domain or codomain of I we will show that Z is ~compact relative to
I. Suppose f :X ! Y is an Icell complex and Z ! Y a map. Note that since
all monomorphisms are Icells for the injective model structure, ; ! X is an I*
*cell
complex which when combined with any presentation of f yields a presentation of
Y as an Icell complex with X as a subcomplex. Provided the claim holds for the
induced Icell complex ; ! Y then Z factors through a subcomplex X0 of size less
than ~. The union of X and X0is a subcomplex of the same size as X0and Z ! Y
factors through it. This would imply that Z is ~compact relative to I. It rema*
*ins to
consider Icell complexes of the form f :; ! Y.
Consider an Icell presentation of f :; ! Y with presentation ordinal ~
; = F0____//_F1___//._._.//_Ffi<~_//_... .
We use transfinite induction to show that every cell of Y is contained in a sub*
*complex
of size less than ~. The induction starts for the presentation ordinal 0 which *
*produces
a subcomplex of size 1. Suppose efiis a cell with presentation ordinal fi < ~. *
*Using
induction on fi and regularity of ~ one shows that the attaching map hefihas im*
*age
contained in a subcomplex Y0of size less than ~. The subcomplex obtained from Y0
by attaching efivia hefiyields the desired subcomplex. Since the cardinality o*
*f Z is
bounded by ~, the image of Z ! Y is contained in less than ~ cells of Y. Every *
*such
cell is in turn contained in a subcomplex of Y of size less than ~ by the argum*
*ent
above. Taking the union of these subcomplexes yields a subcomplex Y0of Y of size
less than the regular cardinal ~. Clearly Z ! Y factors through Y0. *
* *
37
Lemma 3.26: The cofibrations in the injective model structures on *C *Spc and *
* C *Spc
are effective monomorphisms.
Proof.Note that X ! Y is an effective monomorphism if and only if Xn(A) ! Yn(A)
is an effective monomorphism of sets for all A 2 C * Alg and n 0. This follo*
*ws
since all limits and colimits are formed pointwise. To conclude, use that for s*
*ets the
class of effective monomorphisms coincides with the class of injective maps. *
* *
According to Proposition 3.23 the pointwise model structures on *C * Spc ca*
*n be
localized using the theory of cellular model structures [33].
38
3.2 Exact model structures
Clearly the pointwise injective and projective model structures involves too ma*
*ny
homotopy types of C*algebras. We shall remedy the situation slightly by introd*
*ucing
the exact model structures. This involves a strengthening of the fibrancy condi*
*tion
by now requiring that every fibrant cubical C*space turns certain types of sho*
*rt exact
sequences in C * Alg into homotopy fiber sequences of cubical sets. First we *
*fix
some standard conventions concerning exact sequences and monoidal structures on
C * Alg [16]. Corresponding to the maximal tensor product we consider all sho*
*rt
exact sequences of C*algebras
0 ____//_A___//_E__//_B___//_0. (18)
That is, the image of the injection A ! E is a closed 2sided ideal, the compos*
*ition
A ! B is trivial and the induced map E=A ! B is an isomorphism of C*algebras.
Corresponding to the minimal tensor product we restrict to completely positive *
*split
short exact sequences. Recall that f :A ! B is positive if f (a) 2 B+ for every*
* positive
element a 2 A+, i.e.ia =ja*and oeA(a) [0, 1), and f is completely positive if*
* Mn(A) !
Mn(B), (aij) 7! f (aij) , is positive for all n. With these conventions, the *
*monoidal
product is flat in the sense that (18) remains exact when tensored with any C**
*algebra.
Example 3.27: The short exact sequence 0 ! C0(R) ! C(I) ! C C ! 0 is not split
since [0, 1] does not map continuously onto {0, 1}, while 0 ! K ! T ! C(S1) ! 0
where the shift generator of the Toeplitz algebra T is send to the unitary gene*
*rator of
C(S1) acquires a completely positive splitting by sending f 2 C(S1) to the oper*
*ator Tf
on the Hardy space H2 L2(S1); here Tf(g) ss(f g), where ss: L2(S1) ! H2 den*
*otes
the orthogonal projection. Note that the splitting is not a map of algebras.
Example 3.28: A deformation of B into A is a continuous field of C*algebras ov*
*er a
halfopen interval [0, "), locally trivial over (0, "), whose fibers are all is*
*omorphic to
A except for the fiber over 0 which is B. Every such deformation gives rise to *
*a short
exact sequence of C*algebras
i j
0_____//C0 (0, "),_A_//_E___//_B__//_0. (19)
When A is nuclear, so that the maximal and minimal tensor products with A coinc*
*ide,
then (19) has a completely positive splitting. In particular, this holds whenev*
*er A is
finite dimensional, commutative or of type I [9].
39
Remark 3.29: Kasparov's KKtheory [47], [48] and the Etheory of ConnesHigson
[15] are the universal bivariant theories corresponding to the minimal and maxi*
*mal
tensor products respectively, cf. [16].
The set of exact squares consists of all diagrams
A ! E
?? ??
E ??y ??y (20)
0 ! B
obtained from a short exact sequence of C*algebras as in (18), and the degener*
*ate
square with only one entry A = 0 in the upper left hand corner. Note that exact*
*ness
of the sequence (18) implies the exact square in (20) is both a pullback and a *
*pushout
diagram in C*  Alg [63]. The exact model structures will be rigged such that e*
*xact
squares turn into homotopy cartesian squares when viewed in *C * Spc.
A cubical C*space Z is called flasque if it takes every exact square to a h*
*omotopy
pullback square. In detail, we require that Z(0) is contractible and applying *
*Z to
every exact square E obtained from some short exact sequence of C*algebras yie*
*lds
a homotopy cartesian diagram of cubical sets:
A ! E Z(A) ! Z(E)
?? ?? ??? ???
E ??y ??y Z(E) ?y ?y (21)
0 ! B * ! Z(B)
The definition translates easily into the statement that Z is flasque if and on*
*ly if
applying Z to any short exact sequence of C*algebras (18) yields a homotopy fi*
*ber
sequence of cubical sets
Z(A) ____//_Z(E)___//_Z(B).
Recall that Z(E) is a homotopy cartesian diagram if the canonical map from Z(A)
to the homotopy pullback of the diagram * ! Z(B) Z(E) is a weak equivalence.
Right properness of the model structure on *Set implies that if Z(B) and Z(E) a*
*re
fibrant the homotopy pullback and homotopy limit of * ! Z(B) Z(E) are natural*
*ly
weakly equivalent. If Z(B) is contractible, then Z(E) is a homotopy cartesian d*
*iagram
if and only if Z(A) ! Z(E) is a weak equivalence. We denote homotopy fibers of
maps in model categories by hofib.
40
If X: I ! *C * Spc is a small diagram, the homotopy limit of X is the cubic*
*al
C *space defined by
holim X(A) holim EvA O X.
I I
Here EvA O X: I ! *Set and the homotopy limit is formed in cubical sets. Likewi*
*se,
the homotopy colimit of X is the cubical C*space defined using (9) by setting
hocolim X(A) hocolim Ev AO X.
I I
Definition 3.30: A cubical C *space Z is exact projective fibrant if it is pro*
*jective
fibrant and flasque. A map f :X ! Y is an exact projective weak equivalence if *
*for
every exact projective fibrant cubical C*space Z there is a naturally induced *
*weak
equivalence of cubical sets
hom *C*Spc(Qf, Z): hom *C*Spc(QY, Z)__//_hom*C*Spc(QX, Z).
Here Q ! id*C*Spcdenotes a cofibrant replacement functor in the pointwise proj*
*ective
model structure. Exact projective fibrations of cubical C*spaces are maps havi*
*ng the
right lifting property with respect to exact projective acyclic cofibrations.
Remark 3.31: Since the pointwise projective model structure on *C *Spc is a cu*
*bical
model structure according to Lemma 3.8, we may recast the localization machiner*
*y in
[33] based on homotopy function complexes in terms of cubical function complexe*
*s.
We are now ready to introduce the exact projective model structure as the fi*
*rst out
of three types of localizations of the pointwise model structures on cubical C**
*spaces.
Theorem 3.32: The classes of projective cofibrations, exact projective fibratio*
*ns, and exact
projective weak equivalences determine a combinatorial cubical model structure *
*on *C *Spc .
Proof.We show the exact projective model structure arise as the localization of*
* the
pointwise projective model with respect to the maps hocolim(E) ! A, i.e. the se*
*t of
maps hocolim(0 B ! E) ! A and ; ! 0 indexed by exact squares. The localized
model structure exists because the projective model structure is combinatorial *
*and
left proper according to Theorem 3.2 and Lemma 3.5. Since the cofibrations and *
*the
fibrant objects determine the weak equivalences in any model structure, it suff*
*ices to
identify the fibrant objects in the localized model structure with the exact pr*
*ojective
fibrant ones defined in terms of short exact sequences.
41
In effect, note that Z is fibrant in the localized model structure if and on*
*ly if it is
projective fibrant, the cubical set Z(0) is contractible and for all short exac*
*t sequences
(18) the cubical set maps
i j
hom *C*Spc(A, Z)___//_hom*C*Spchocolim(0 B ! E), Z
are weak equivalences. By (5), the latter holds if and only if there exist nat*
*urally
induced weak equivalences of cubical sets
i j i j
Z(A) _____//holimhom*C*Spc(0 B ! E), Z = holim Z(0) ! Z(B) Z(E) .
Put differently, the cubical set Z(0) is contractible and there exist naturally*
* induced
weak equivalences
i j
Z(A) ____//_hofibZ(E) ! Z(B) .
This shows that Z is fibrant in the localized model structure if and only if it*
* is exact
projective fibrant. It follows that the classes of maps in question form part o*
*f a model
structure with the stated properties. *
* *
Remark 3.33: By construction, every exact square gives rise to a homotopy carte*
*sian
square of cubical C*spaces in the exact projective model structure. Taking pus*
*houts
along the exact projective weak equivalence ; ! 0 shows that every cubical C*s*
*pace
is exact projective weakly equivalent to its image in *C * Spc0.
Lemma 3.34: The exact projective model structure is left proper.
Proof.Left properness is preserved under localizations of left proper model str*
*uctures.
Lemma 3.5 shows the pointwise projective model is left proper. *
* *
We are interested in explicating sets of generating acyclic cofibrations for*
* the
exact projective model. In the following we shall use the cubical mapping cylin*
*der
construction to produce a convenient set of generators. In effect, apply the cu*
*bical
mapping cylinder construction cylto exact squares and form the pushouts:
A ! E B ! cyl(B ! E) ! E
?? ?? ??? ??? ???
E ??y ??y ?y ?y ?y
`
0 ! B 0 ! cyl(B ! E) B0 ! A
42
By Theorem 3.32 the exact projective model structure is cubical. Lemmas 2.21 an*
*d 3.6
imply B ! cyl(B ! E) is a projective cofibration between projective cofibrant c*
*ubical
`
C *spaces. Thus s(E) cyl(B ! E) B0 is projective cofibrant [35, Corollary *
*1.11.1].
For the sameireasons,japplying the cubical mapping cylinder to s(E) ! A and set*
*ting
t(E) cyls(E) ! A we get a projective cofibration
cyl(E): s(E)____//t(E). (22)
We claim the map cyl(E) is an exact projective weak equivalence. To wit, since *
*cubical
homotopy equivalences are pointwise weak equivalences, it suffices by Lemma 2.21
to prove that s(E) ! A is an exact projective weak equivalence. The canonical
`
map E B0 ! A is an exact projective weak equivalence and there is a factoring
`
E B0 ! s(E) ! A. Moreover, since E ! cyl(B ! E) is an exact acyclic projective
`
cofibration, so is E B0 ! s(E).
Let JE*C*Spcdenote the set of maps J*C*Spc[ Jcyl(E)*C*Spcwhere Jcyl(E)*C**
*Spcconsists of all
pushout product maps
`
s(E) *n s(E) @*nt(E) @*n_//_t(E) *n.
Proposition 3.35: A cubical C*space is exact projective fibrant if and only if*
* it has the right
lifting property with respect to the set JE*C*Spc.
Remark 3.36: Theorem 3.32 shows the members of Jcyl(E)*C*Spcare exact acyclic *
*projective
cofibrations because the exact projective model structure is cubical and the ma*
*p cyl(E)
in (22) is an exact acyclic projective cofibration.
Proof.Note that a projective fibration X ! Y has the right lifting property with
respect to JE*C*Spcif and only if it has the right lifting property with respe*
*ct to Jcyl(E)*C*Spc.
By adjointness, the latter holds if and only if X(0) ! Y(0) is a weak equivalen*
*ce of
cubical sets and for every exact square E obtained from a short exact sequence *
*of
C *algebras as in (18) there exist liftings in all diagrams of the following f*
*orm:
i j
@*n ! hom *C*Spct(E), X
??? ??
?y ??y
i j i j
*n ! hom *C*Spcs(E), X x i hojm*C*Spct(E), Y
hom*C*Spcs(E),Y
43
In other words there are homotopy cartesian diagrams of cubical sets:
i j i j
hom *C*Spct(E), X! hom *C*Spct(E), Y
??? ??
?y ??y
i j i j
hom *C*Spcs(E), X! hom *C*Spcs(E), Y
An equivalent statement obtained from Yoneda's lemma and the construction of cy*
*l(E)
is to require that there are naturally induced homotopy cartesian diagrams:
X(A) ! Y(A)
??? ??
?y ??y
i j i j
hom *C*Spccyl(B ! E), X xX(B)X(0)! hom *C*Spccyl(B ! E), Y xY(B)Y(0)
In particular, a projective fibrant cubical C *space Z has the right lifting p*
*roperty
with respect to Jcyl(E)*C*Spcif and only if Z(0) is contractible and for every*
* exact square
E obtained from a short exact sequence of C*algebras there is a homotopy carte*
*sian
diagram:
Z(A) ! Z(E)
?? ??
Z(E) ??y ??y
* ! Z(B)
This holds if and only if Z is flasque since the latter diagram coincides with *
*(21). *
Corollary 3.37: The exact projective model is weakly finitely generated.
Proof.Members of JE*C*Spchave finitely presentable domains and codomains. *
* *
Corollary 3.38: The classes of exact projective weak equivalences, exact acycli*
*c projective
fibrations, exact projective fibrations with exact projective fibrant codomains*
*, and all exact
projective fibrant objects are closed under filtered colimits.
Recall the contravariant Yoneda embedding yields a full and faithful embeddi*
*ng
of C* Alg into cubical C*spaces. In the next result we note that no nonisomo*
*rphic
C *algebras become isomorphic in the homotopy category associated with the exa*
*ct
projective model structure. This observation motivates to some extent the matr*
*ix
invariant and homotopy invariant model structures introduced in the next sectio*
*ns.
44
Proposition 3.39: The contravariant Yoneda embedding of C* Alg into *C * Spc *
*yields
a full and faithful embedding of the category of C*algebras into the homotopy *
*category of the
exact projective model structure.
Proof.Every C *algebra is projective cofibrant by Lemma 3.6, projective fibran*
*t by
Lemma 3.7 and also flasque: note that Z(E) is a pullback of discrete cubical se*
*ts for
every C*algebra Z so the assertion follows from [28, II Remark 8.17] since * !*
* Z(B)
is a fibration of cubical sets. This shows that every C *algebra is exact pro*
*jective
fibrant. Thus [35, Theorem 1.2.10] implies there is a bijection between maps in*
* the
exact projective homotopy category, say Ho (*C * Spc)(B, A), and homotopy clas*
*ses
of maps [B, A]. Since the exact projective model structure is cubical, maps B !*
* A are
homotopic if and only if there exists a cubical homotopy B *1 ! A by an argum*
*ent
analogous to the proof of [28, II Lemma 3.5] which shows that B *1 is a cylin*
*der
object for B in *C * Spc. Using the Yoneda embedding and the fact that C*alge*
*bras
are discrete cubical C*spaces, so that all homotopies are constant, we get bij*
*ections
Ho (*C * Spc)(B, A) = C* Spc(B, A) = C* Alg(A, B). *
* *
In the next result we note there exists an explicitly constructed fibrant re*
*placement
functor for the exact projective model structure.
E *
Proposition 3.40: There exists a natural transformation id! ExJ*CSpcsuch that *
*for every
E *
cubical C*space X the map X ! ExJ*CSpcX is an exact projective weak equivalen*
*ce with an
exact projective fibrant codomain.
Proof.Use Quillen's small object argument with respect to the maps A (un(ff,i*
*) *n)
`
and s(E) *n s(E) @*nt(E) @*n ! t(E) *n. See [25, x7] and x3.1 for detai*
*ls. *
The fibrant replacement functor gives a way of testing whether certain maps *
*are
exact projective fibrations:
Corollary 3.41: Suppose f :X ! Y is a pointwise projective fibration and Y exac*
*t projective
fibrant. Then f is an exact projective fibration if and only if the diagram
E *
X ! Ex J*CSpcX
?? ??
f??y ??yExJE*C*Spcf
E *
Y ! ExJ*CSpcY
45
is homotopy cartesian in the pointwise projective model structure.
Proof.Follows from [5, Proposition 2.32] and Proposition 3.40. *
* *
In the following we show that the exact projective model structure is monoid*
*al.
This is a highly desirable property from a model categorical viewpoint. It turn*
*s out
our standard conventions concerning short exact sequences of C*algebras is exa*
*ctly
the input we need in order to prove monoidalness.
Lemma 3.42: If X is projective cofibrant and Z is exact projective fibrant, the*
*n Hom__(X, Z)
is exact projective fibrant.
i j
Proof.Lemma 3.10(ii) shows it suffices to check Hom__A (@*n *n), Z has the*
* right
lifting property with respect to Jcyl(E)*C*Spc. By adjointness, if suffices to*
* check that for
every exact square E the pushout product map of
`
jE s(E) *m s(E) @*mt(E) _@*m_//t(E) *m
and A (@*n *n) is a composition of pushouts of maps in Jcyl(E)*C*Spc. Thi*
*s follows
because there is an isomorphism jE A = jE A where E A denotes the exact squ*
*are
obtained by tensoring with A, and the pushout product map of @*m *m and @*n *
* *n
is a monomorphism of cubical sets formed by attaching cells. *
* *
Proposition 3.43: The exact projective model structure is monoidal.
Proof.Suppose X ! Y is an exact acyclic projective cofibration and moreover that
Z is exact projective fibrant. There is an induced commutative diagram of cubi*
*cal
function complexes:
i j i j
hom *C*SpcY, Hom__(A *n, Z) ! hom *C*SpcX, Hom__(A *n, Z)
??? ??
?y ??y (23)
i j i j
hom *C*SpcY, Hom__(A @*n, Z)! hom *C*SpcX, Hom__(A @*n, Z)
Lemma 3.42 implies the horizontal maps in (23) are weak equivalences. Thus (23)*
* is
a homotopy cartesian diagram. *
Next we record the analog of Lemma 3.13 in the exact projective model struct*
*ure.
46
Lemma 3.44: Suppose Z is a projective cofibrant cubical C*space. Then
Z : *C * Spc ____//_*C * Spc
preserves the classes of acyclic projective cofibrations and exact weak equival*
*ences between
projective cofibrant cubical C*spaces.
For reference we include the next result which captures equivalent formulati*
*ons
of the statement that the exact projective model structure is monoidal.
Lemma 3.45: The following statements hold and are equivalent.
o If i: X Y and j: U V are projective cofibrations and either i or j is *
*an exact
projective weak equivalence, then so is
`
X V X U Y U ____//_Y V.
o If j: U V is a projective cofibration and k: Z W is an exact projectiv*
*e fibration,
then the pullback map
Hom__(V, Z) ____//_Hom_(V, W) xHom_(U,W)Hom_(U, Z)
is an exact projective fibration which is exact acyclic if either j or k i*
*s.
o With the same assumptions as in the previous item, the induced map
hom *C*Spc(V, Z)___//_hom*C*Spc(V, W) xhom*C*Spc(U,W)hom*C*Spc(U, Z)
is a Kan fibration which is a weak equivalence of cubical sets if in addit*
*ion either j or k
is exact acyclic.
Next we construct the exact injective model structure on cubical C*spaces.
Definition 3.46: A cubical C*space Z is exact injective fibrant if it is injec*
*tive fibrant
and flasque. A map f :X ! Y is an exact injective weak equivalence if for every*
* exact
injective fibrant cubical C*space Z there is a naturally induced weak equivale*
*nce of
cubical sets
hom *C*Spc(f, Z): hom *C*Spc(Y,_Z)_//_hom*C*Spc(X, Z).
The exact injective fibrations of cubical C*spaces are maps having the right l*
*ifting
property with respect to exact injective acyclic cofibrations.
47
Remark 3.47: Note there is no cofibrant replacement functor involved in the def*
*inition
of exact injective fibrant objects due to the fact that every cubical C*space *
*is cofibrant
in the injective model structure.
The proof of the next result proceeds as the proof of Theorem 3.32 by locali*
*zing
the pointwise injective model structure with respect to the maps hocolim(E) ! A.
Theorem 3.48: The classes of monomorphisms, exact injective fibrations and exac*
*t injective
weak equivalences determine a combinatorial, cubical and left proper model stru*
*cture on
*C * Spc.
Proposition 3.49: The classes of exact injective and projective weak equivalenc*
*es coincide.
Hence the identity functor on *C * Spc is a Quillen equivalence between the ex*
*act injective
and projective model structures.
Proof.If Z is exact injective fibrant then clearly Z is exact projective fibran*
*t. Thus if
f :X ! Y is an exact projective weak equivalence, then map hom *C*Spc(Qf, Z) i*
*s a
weak equivalence a cubical sets. Now Qf maps to f via pointwise weak equivalenc*
*es,
so hom *C*Spc(f, Z) is also a weak equivalence.
If Z is exact projective fibrant there exists a pointwise weak equivalence Z*
* ! W
where W is injective fibrant. It follows that W is flasque. Now if f :X ! Y is *
*an
exact injective weak equivalence, using that the exact projective model structu*
*re is
cubical we get the following diagram with vertical weak equivalences:
hom *C*Spc(QY, Z)_____//hom*C*Spc(QX, Z)
~ ~
fflffl fflffl
hom *C*Spc(QY, W)____//_hom*C*Spc(QX, W)
It remains to note that hom *C*Spc(Qf, W) is a weak equivalence because Qf is *
*an
exact injective weak equivalence. *
Remark 3.50: The proof of Proposition 3.49 applies with small variations to bot*
*h the
matrix invariant and the homotopy invariant model structures on *C * Spc which
will be constructed in Sections 3.3 and 3.4 respectively; details in these proo*
*fs will be
left implicit in the next sections. By Proposition 3.23 we may appeal to locali*
*zations
of left proper cellular model structures for the existence of the exact model s*
*tructures.
The same applies to the matrix invariant and homotopy invariant model structure*
*s.
48
By localizing the pointwise model structures on C * Spc with respect to the
maps hocolim(E) ! A as above, we obtain exact model structures on simplicial C*
spaces. They acquire the same additional properties as the corresponding exact *
*model
structures on cubical C*spaces. As a special case of [33, Theorem 3.3.20] and *
*Lemma
3.22 we deduce that the corresponding homotopy categories are equivalent:
Lemma 3.51: There are naturally induced Quillen equivalences between exact inje*
*ctive and
projective model structures:
*C * Spc ____//_oCo*_Spc
Remark 3.52: In the following we shall introduce the matrix invariant and homot*
*opy
invariant model structures. As above, these model structures furnish two Quill*
*en
equivalences between *C * Spc and C * Spc. This observation will be employed
implicitly in later sections in the proof of representability of Kasparov's KK*
*groups
in the pointed unstable homotopy category and when dealing with the triangulated
structure of the stable homotopy category of C*algebras.
49
3.3 Matrix invariant model structures
In this section we refine the exact projective model structures by imposing a n*
*atural
fibrancy condition determined by the highly noncommutative data of MoritaRieff*
*el
equivalence or matrix invariance. This amounts to the choice of a rankone proj*
*ection
p 2 K such that the corner embedding A ! A K = colimMn(A) given by a 7! a p
becomes a "matrix exact" weak equivalence. To achieve this we shall localize th*
*e exact
model structures with respect to such a rankone projection. With this approach*
* the
results and techniques in the previous section carry over in gross outline. How*
*ever,
there are a couple of technical differences and the exposition tends to emphasi*
*ze these.
As a motivation for what follows, recall that matrix invariance is a natural an*
*d basic
property in the theory of Ktheory of C*algebras [13].
A cubical C *space Z is matrix exact projective fibrant if Z is exact proje*
*ctive
fibrant and for every C*algebra A the induced map of cubical C*spaces
A K ____//_A (24)
given by a rank one projection induces a weak equivalence of cubical sets
Z(A) = hom *C*Spc(A, Z)____//_hom*C*Spc(A K, Z) = Z(A K). (25)
The definition of Z being matrix exact projective fibrant is independent of the*
* choice
of a rank one projection.
Recall Q is a pointwise projective cofibrant replacement functor. A map betw*
*een
cubical C*spaces X ! Y is a matrix exact projective weak equivalence if for ev*
*ery
matrix exact projective fibrant Z there is a naturally induced weak equivalence*
* of
cubical sets
hom *C*Spc(QY, Z)____//_hom*C*Spc(QX, Z). (26)
Example 3.53: The map A K ! A is a matrix exact projective weak equivalence f*
*or
every C*algebra A because representable cubical C*spaces are projective cofib*
*rant.
If A K ! Mn(A) is a matrix exact projective weak equivalence for some n 1, *
*then
so is Mn(A) ! A.
The matrix invariant projective model structure is defined by taking the Bou*
*sfield
localization of the exact projective model structure on *C * Spc with respect *
*to the
set of maps obtained by letting A run through all isomorphism classes of C*alg*
*ebras
in (24). Thus the next result is a consequence of Theorem 3.32 and Lemma 3.34.
50
Theorem 3.54: The classes of matrix exact projective weak equivalences defined *
*by (26),
matrix exact projective fibrations and projective cofibrations form a combinato*
*rial, cubical and
left proper model structure on *C * Spc.
Applying the cubical mapping cylinder to the map in (24) yields a factoring
A K ____//_cylAK__//_A. (27)
Recall the map A K ! cylAKis a projective cofibration and cylAK! A is a cubic*
*al
homotopy equivalence. In particular, cylAKis projective cofibrant. Example 3.53*
* and
saturation imply A K ! cylAKis a matrix exact projective weak equivalence. Si*
*nce
the matrix invariant model structure is cubical, the factoring (27) and the gen*
*erating
cofibrations @*n *n for *Set induce matrix exact acyclic projective cofibrati*
*ons.
Let Jcyl(K)*C*Spcbe the set consisting of the matrix exact acyclic projecti*
*ve cofibrations
` A A
(A K) *n (A K) @*ncylK @*n____//_cylK *n (28)
where A 2 C* Alg and n 0.
Proposition 3.55: Define
JK*C*Spc J*C*Spc[ Jcyl(E)*C*Spc[ Jcyl(K)*C*Spc.
Then a map of cubical C*spaces with a matrix exact projective fibrant codomain*
* has the right
lifting property with respect to JK*C*Spcif and only if it is a matrix exact p*
*rojective fibration.
Proof.Follows from Proposition 3.35 and [33, 3.3.16]. *
* *
We shall use the set Jcyl(K)*C*Spcto prove the following crucial result.
Proposition 3.56: The matrix invariant projective model structure is monoidal.
Proof.If Z is projective cofibrant, Lemmas 3.13 and 3.57 imply the functor
Z : *C * Spc ____//_*C * Spc
preserves matrix exact acyclic projective cofibrations f :X ! Y. In particular*
*, this
result applies to the domains and codomains of the generating projective cofibr*
*ations
51
I*C*Spc. Hence for every C*algebra A and n 0 there is a commutative diagram
where the horizontal maps are matrix exact acyclic projective cofibrations:
(A @*n) X! (A @*n) Y
??? ??
?y ??y
(A *n) X ! (A *n) Y
Thus, by [35, Corollary 1.1.11], the pushout map
` n
(A *n) X____//_(A *n) X (A @*n)(XA @* ) Y
i j
of (A @*n) f along A (@*n *n) X is a matrix exact acyclic projective cofibra*
*tion.
By saturation if follows that the pushout product map of A (@*n *n) and f i*
*s a
matrix exact projective weak equivalence. *
To complete the proof of Proposition 3.56 it remains to prove the next resul*
*t.
Lemma 3.57: If X ! Y is a matrix exact weak equivalence and W is projective cof*
*ibrant,
then the induced map X W ! Y W is a matrix exact projective weak equivalenc*
*e.
Proof.Suppose that Z is matrix exact projective fibrant. We need to show there *
*is an
induced weak equivalence of cubical sets
hom *C*Spc(Y W, Z)____//_hom*C*Spc(X W, Z).
By adjointness the latter identifies with the map
i j i j
hom *C*SpcY, Hom__(W, Z) ____//_hom*C*SpcX, Hom__(W, Z) .
Thus it suffices to show Hom__(W, Z) is matrix exact projective fibrant, see th*
*e next
lemma. *
Lemma 3.58: If X is projective cofibrant and Z is matrix exact projective fibra*
*nt, then the
internal hom Hom__(X, Z) is matrix exact projective fibrant.
Proof.Lemmai3.45(ii) showsjit suffices to check that for every C*algebra B the*
* map
Hom__ B (@*n *n), Z has the right lifting property with respect to Jcyl(K)*
**C*Spc.
52
Using adjointness, if suffices to check that for all C*algebras A and B the*
* pushout
product map of
` A A
(A K) *m (A K) @*mcylK @*m ____//_cylK *m (29)
and B (@*n *n) is a composition of pushouts of maps in JK*C*Spc. This fol*
*lows
using the isomorphism cylAK B = cylAKB, cp. the proof of Lemma 3.42. *
* *
The next result summarizes the monoidal property of the matrix invariant mod*
*el
structure.
Lemma 3.59: The following statements hold and are equivalent.
o If i: X Y and j: U V are projective cofibrations and either i or j is *
*a matrix
exact projective weak equivalence, then so is
`
X V X U Y U ____//_Y V.
o If j: U V is a projective cofibration and k: Z W is a matrix exact pro*
*jective
fibration, then the pullback map
Hom__(V, Z) ____//_Hom_(V, W) xHom_(U,W)Hom_(U, Z)
is a matrix exact projective fibration which is matrix exact acyclic if ei*
*ther j or k is.
o With the same assumptions as in the previous item, the induced map
hom *C*Spc(V, Z)___//_hom*C*Spc(V, W) xhom*C*Spc(U,W)hom*C*Spc(U, Z)
is a Kan fibration which is a weak equivalence of cubical sets if in addit*
*ion either j or k
is matrix exact acyclic.
The matrix invariant injective model structure on cubical C*spaces arises i*
*n an
analogous way by declaring that Z is matrix exact injective fibrant if it is ex*
*act injective
fibrant and Z(A) ! Z(A K) is a weak equivalence for all A. A map X ! Y is a
matrix exact weak equivalence if for every matrix exact injective fibrant Z the*
*re is a
naturally induced weak equivalence of cubical sets
hom *C*Spc(f, Z): hom *C*Spc(Y,_Z)_//_hom*C*Spc(X, Z).
We are ready to formulate the main results concerning the class of matrix ex*
*act
weak equivalences.
53
Theorem 3.60: The classes of monomorphisms, matrix invariant injective fibratio*
*ns and
matrix exact injective weak equivalences determine a combinatorial, cubical and*
* left proper
model structure on *C * Spc.
Proposition 3.61: The classes of matrix invariant injective and projective weak*
* equivalences
coincide. Hence the identity functor on *C *Spc is a Quillen equivalence betwe*
*en the matrix
invariant injective and projective model structures.
Proof.See the proof of Proposition 3.49. *
* *
The category C * Spc acquires matrix invariant model structures. We have:
Lemma 3.62: There are naturally induced Quillen equivalences between matrix inv*
*ariant
injective and projective model structures:
*C * Spc ____//_oCo*_Spc
Remark 3.63: For completeness we note that the Quillen equivalent matrix invari*
*ant
model structures on *C *Spc and C *Spc are examples of cellular model struct*
*ures.
With the matrix invariant model structures in hand we are now ready to const*
*ruct
the last in the series of model structure appearing in unstable C*homotopy the*
*ory.
54
3.4 Homotopy invariant model structures
Let A be a C*algebra, let I = [0, 1] denote the topological unit interval and *
*C(I, A)
the C*algebra of continuous functions from I to A with pointwise operations an*
*d the
supremum norm. At time t, 0 t 1, there is an evaluation map
evAt:C(I, A)____//A.
Recall that *homomorphisms ht:A ! B for t = 0, 1 are homotopic if there exists
a map H :A ! C(I, B) such that evBtO H = ht. The notions of homotopies between
*homomorphisms and contractible C*algebras are defined in terms of C(I) C(I*
*, C)
and the trivial C*algebra exactly as for topological spaces. There is an isomo*
*rphism
of C*algebras C(I, B) C(I) B for the tensor products we consider. These de*
*finitions
correspond under GelfandNaimark duality to the usual topological definitions in
the event that A and B are commutative. It turns out the cone of every C*algeb*
*ra is
contractible and every contractible C*algebra is nonunital. If A is contractib*
*le, then its
unitalization is homotopy equivalent to C. There is a canonical *homomorphism,*
* the
constant function map, from A to C(I, A) sending elements of A to constant func*
*tions.
Composing this map with evAtgives the identity map on A for all t.
Motivated by the notion of homotopies between maps of C*algebras we shall n*
*ow
introduce the homotopy invariant model structures on cubical C*spaces. The main
idea behind these models is to enlarge the class of weak equivalences by formal*
*ly
adding all homotopy equivalences based on employing C(I) as the unit interval.
Indeed, these model structures give rise to the correct unstable homotopy categ*
*ory
in the sense that homotopic C*algebras become isomorphic upon inverting the we*
*ak
equivalences in the homotopy invariant models. Existence of the homotopy invari*
*ant
model structures is shown using localization techniques, as one would expect. We
show there is an abstract characterization of the weak equivalences in the homo*
*topy
invariant model structure and introduce homotopy groups. These invariants give a
way of testing whether a map between C*spaces is a weak equivalence.
A cubical C*space Z is called C*projective fibrant if Z is matrix exact pr*
*ojective
fibrant and for every C*algebra A the canonically induced map of cubical C*sp*
*aces
C(I, A) ! A induces a weak equivalence of cubical sets
i j
hom *C*Spc(A, Z)___//_hom*C*SpcC(I, A), Z . (30)
55
It follows immediately that a matrix exact projective fibrant Z is C*projectiv*
*e fibrant
if and only if for every A and for some 0 t 1 the evaluation map evAtyields*
* a weak
equivalence
i j
hom *C*SpcC(I, A), Z___//_hom*C*Spc(A, Z). (31)
Moreover, note thatithe mapjin (30) is a weak equivalence if and only if the in*
*duced
map ss0Z(A) ! ss0Z C(I, A) is a surjection and for every 0cell x of Z(A) and *
*n 1
there is a similarly induced surjective map of higher homotopy groups
i j ` i j '
ssn Z(A), x_____//ssn Z C(I, A) , x .
Likewise, the map (31) induces surjections on all higher homotopy groups. An
alternate formulation of Z being C*projective fibrant is to require that for 0*
* t 1
there are naturally induced pointwise weak equivalences
i j Z(evCt )
Z() ! Z C(I)  ! Z().
In terms of internal hom objects, yet another equivalent formulation obtained f*
*rom
(7) is that for every C*algebra A evaluating the naturally induced maps
i j
Hom__(A, Z)____//_Hom_C(I, A), Z___//_Hom_(A, Z)
at the complex numbers yield weak equivalences of cubical sets.
Remark 3.64: The notion of an C*projective fibrant cubical C*space depends on*
*ly on
the unit interval C(I) and the matrix exact projective fibrancy condition in th*
*e sense
that it may be checked using any of the evaluation maps or the constant functio*
*n map.
A map X ! Y is a projective C*weak equivalence if for every C*projective f*
*ibrant
Z there is an induced weak equivalence of cubical sets
hom *C*Spc(QY, Z)____//_hom*C*Spc(QX, Z).
Recall that Q is our notation for a cofibrant replacement in the pointwise proj*
*ective
model structure. Every C *algebra A is projective C *weakly equivalent to C(*
*I, A).
All matrix exact projective weak equivalences are examples of projective C *we*
*ak
equivalences since the matrix exact projective model structure is cubical.
56
A map X ! Y is a projective C*fibration if it has the right lifting propert*
*y with
respect to every C*acyclic projective cofibration. The class of projective C**
*fibrations
coincides with the fibrations in the C*projective model structure which we def*
*ine by
localizing the matrix invariant projective model at the set of maps C(I, A) ! A.
Theorem 3.65: The projective cofibrations and projective C*weak equivalences d*
*etermine a
combinatorial, cubical and left proper model structure on *C * Spc.
The unstable C*homotopy category, denoted by H, is defined by inverting the
class of projective C*weak equivalences between cubical C*spaces.
We trust that the notions of C*injective fibrant cubical C*spaces, injecti*
*ve C*weak
equivalences and injective C*fibrations are clear from the above and the defin*
*itions
of the injective model structures constructed in the previous sections. Next we*
* state
two basic results concerning the injective homotopy invariant model structure.
Theorem 3.66: The classes of monomorphisms, injective C*fibrations and injecti*
*ve C*weak
equivalences determine a combinatorial, cubical and left proper model structure*
* on *C *Spc .
Proposition 3.67: The classes of injective and projective C *weak equivalences*
* coincide.
Hence the identity functor on *C * Spc is a Quillen equivalence between the ho*
*motopy
invariant injective and projective model structures.
In the following we write C*weak equivalence rather than injective or proje*
*ctive
C *weak equivalence. We note there exist corresponding homotopy invariant model
structures for simplicial C*spaces, and include the following observation.
Lemma 3.68: There are naturally induced Quillen equivalences between homotopy i*
*nvariant
injective and projective model structures:
*C * Spc ____//_oCo*_Spc
An elementary homotopy between maps ht:X ! Y of cubical C*spaces is a map
H :X C(I) ! Y such that H O (idX evCt) = htfor t = 0, 1. Two maps f and g a*
*re
homotopic if there exists a sequence of maps f = f0, f1, . .,.fn = g such that *
*fi1is
elementary homotopic to fifor 1 i n. And f :X ! Y is a homotopy equivalence*
* if
there exists a map g: Y ! X such that f O g and g O f are homotopic to the resp*
*ective
identity maps.
57
Remark 3.69: Note that maps between representable cubical C*spaces are homotop*
*ic
if and only if the maps between the corresponding C*algebras are so.
Lemma 3.70: Homotopy equivalences are C*weak equivalences.
Proof.The proof reduces to showing that elementary homotopic maps ht:X ! Y
become isomorphic in the unstable C*homotopy category: If f :X ! Y is a homoto*
*py
equivalence with homotopy inverse g we need to show that f Og and gOf are equal*
* to
the corresponding identity maps in the homotopy category, but the composite maps
are homotopic to the corresponding identity maps. Now for the projective cofibr*
*ant
replacement QX ! X the assertion holds for the maps QX ! QX C(I) induced by
evaluating at t = 0 and t = 1. And hence the same holds for the two composite m*
*aps
QX ! QX C(I) ! X C(I). Composing these maps with the homotopy yields maps
naturally isomorphic to h0 and h1 in the unstable C*homotopy category. *
* *
Remark 3.71: Lemma 3.70 shows that cubical C*spaces represented by homotopy
equivalent C*algebras are C*weakly equivalent.
Lemma 3.72: Suppose f, g: X ! Y are homotopy equivalent maps and Z is a cubical
C *space. Then Hom__(f, Z) and Hom__(g, Z) respectively f Z and g Z are h*
*omotopy
equivalent maps. Thus the internal hom functor Hom__(, Z) and the tensor funct*
*or  Z
preserves homotopy equivalences.
Proof.An elementary homotopy from f to g determines a map of cubical C*spaces
i j
Hom__(Y, Z)____//_Hom_X C(I), Z .
According to the closed symmetric monoidal structure of *C * Spc detailed in x*
*2.1
there exists by adjointness a map
i j
Hom__(Y, Z) ____//_Hom_C(I), Hom__(X, Z) .
By adjointness of the latter map we get the desired elementary homotopy
Hom__(Y, Z) C(I)___//_Hom_(X, Z).
The claims concerning f Z and g Z are clear. *
* *
58
Corollary 3.73: For every cubical C*space X and n 0 the canonical map
i j
X ____//_X C(*ntop) 
is a C*weak equivalence.
Corollary 3.74: The canonical map X ! Singo*(X) is a C*weak equivalence.
Proof.Applying the homotopy colimit functor yields a commutative diagram with
naturally induced vertical pointwise weak equivalences [33, Corollary 18.7.5]:
i j
hocolim Xn ! hocolim Hom__ C(*ntop), Xn
*op? *op
??? ???
y ?y
X ! Singo*(X)
i j
In ncells there is a homotopy equivalence Xn ! Hom__C(*ntop), Xn . The same ma*
*p is
a C*weak equivalence of discrete cubical C*spaces, so the upper horizontal ma*
*p is a
C *weak equivalence on account of Corollary 3.76. *
* *
Lemma 3.75: Suppose X ! Y is a natural transformation of small diagrams I ! *C *
** Spc
such that for every i 2 I the induced map X(i) ! Y(i) is a C*weak equivalence.*
* Then the
induced map
hocolim X ____//_hocolimY
I I
is a C*weak equivalence.
Proof.If Z is a cubical C*space there is a canonical isomorphism of cubical se*
*ts
hom *C*Spc(hocolim X, Z) = holim hom *C*Spc(X, Z).
I Iop
The lemma can also be proven using general properties of homotopy colimits. *
* *
Corollary 3.76: A map of cubical C*spaces which induces C*weak equivalences o*
*f discrete
cubical C*spaces in all cells is a C*weak equivalence.
Lemma 3.77: Suppose f :X ! Y is a map between projective C*fibrant cubical C**
*spaces.
The following statements are equivalent where in the third item we assume X is *
*projective
cofibrant.
59
o f is a C*weak equivalence.
o f is a pointwise weak equivalence.
o f is a cubical homotopy equivalence.
Proof.The equivalence between the first two items follows because the homotopy
invariant model structure is a localization of the pointwise projective model s*
*tructure.
Now if X is projective cofibrant, then X *1 is a cylinder object for X since *
*we are
dealing with a cubical model structure, cf. [28, II Lemma 3.5]. Hence the first*
* item is
equivalent to the third by [35, Theorem 1.2.10]. *
* *
Remark 3.78: We leave the formulation of Lemma 3.77 for maps between injective
C *fibrant cubical C*spaces to the reader. Note that no cofibrancy condition *
*is then
required in the third item since every cubical C *space is cofibrant in the in*
*jective
homotopy invariant model structure.
A map between C*spaces is called a C*weak equivalence if the associated map
of constant cubical C *spaces is a C *weak equivalence, and a cofibration if *
*it is a
monomorphism. Fibration of C*spaces are defined by the right lifting property.
We are ready to state the analog in C*homotopy theory of [40, Theorem B.4] *
*which
can be verified by similar arguments using Lemma 2.12 and Corollary 3.74. Furth*
*er
details are left to the interested reader. In this setting,
hom C*Spc(X, Y)n C* Spc(X *n, Y)
and i j
XK lim Hom__C0(*ntop), X ,
*n!K
where X, Y are C*spaces, and the limit is taken over the cell category of the *
*cubical
set K.
Theorem 3.79: The classes of monomorphisms, fibrations and C*weak equivalence *
*form a
combinatorial, cubical and left proper model category on C* Spc.
The singular and geometric realization functors yield a Quillen equivalence:
 . : *C * Spc_//_C*oSpc:oSingo*_
60
Next we introduce unstable C*homotopy group (functors) ss*nfor integers n *
* 0,
and show that f :X ! Y is a C*weak equivalence if and only if ss*0X ! ss*0Y is*
* a
bijection and for every 0cell x of X and n 1 there is a group object isomorp*
*hism
i j
ss*n(X,_x)__//ss*nY, f (x) . (32)
If n 2, then ss*n(X, x) takes values in abelian groups. To achieve this we re*
*quire the
construction of a fibrant replacement functor in the C*projective model struct*
*ure.
Using the cubical mapping cylinder we may factor the constant function map of
cubical C*spaces
C(I, A)___//_A
into a projective cofibration composed with a cubical homotopy equivalence
i j
C(I, A)____//_cylC(I, A) ! A___//_A. (33)
i j
Observe that cylC(I, A) ! A is finitely presentable projective cofibrant and th*
*e maps
in (33) are C*weak equivalences.
Definition 3.80: Let Jcyl(I)*C*Spcdenote the set of pushout product maps from
a i j
C(I, A) *n cylC(I, A) ! A @*n
C(I,A) @*n
i j
to cylC(I, A) ! A *n indexed by n 0 and A 2 C* Alg.
Lemma 3.81: A matrix exact projective fibration whose codomain is C*projective*
* fibrant is
a C*projective fibration if and only if it has the right lifting property with*
* respect to Jcyl(I)*C*Spc.
Proof.The cubical function complex hom *C*Spc(Z, ) preserves cubical homotopi*
*es,
which are examples of pointwise weak equivalences. Proposition 3.56 and a check
using only the definitions reveal that a matrix exact projective fibrant cubica*
*l C*space
Z is C*projective fibrant if and only if the map Z ! * has the right lifting p*
*roperty
with respect to J**C*Spc. This completes the proof by [33, Proposition 3.3.6].*
* *
Corollary 3.82: The C*projective model is weakly finitely generated by the set
J**C*Spc J*C*Spc[ Jcyl(E)*C*Spc[ Jcyl(K)*C*Spc[ Jcyl(I)**
*C*Spc.
61
Next we employ J**C*Spcin order to explicate a fibrant replacement functor *
*in the
C *projective model structure by means of a routine small object argument as i*
*n the
proof of Proposition 3.40.
Proposition 3.83: There exists a natural transformation
*
id____//_ExJ*C*Spc
of endofunctors of cubical C*spaces such that for every X the map
*
X ____//_ExJ*C*SpcX
is a C*weak equivalence with C*fibrant codomain.
Definition 3.84: Let (X, x) be a pointed cubical C*space. Define the nth C*ho*
*motopy
group
ss*n(X, x): C* Alg__//_Set
by 8 i j
>>>: i J* * j
ssn Ex *CSpcX(A), xn > 0.
A cubical C*space X is nconnected if it is nonempty and ss*i(X, x) is trivial*
* for all
0 i n and x. Note that ss*n(X, x) is a contravariant functor taking values *
*in sets if
n = 0, groups if n = 1 and abelian groups if n 2.
Lemma 3.85: A map (X, x) ! (Y, y) is a C*weak equivalence if and only if for e*
*very integer
n 0 there are naturally induced isomorphisms between C*homotopy groups
ss*n(X, x)__//_ss*n(Y, y).
* *
Proof.Using the properties of the natural transformation id! ExJ*CSpcappearing*
* in
Proposition 3.83 and the Whitehead theorem for localizations of model categorie*
*s, it
follows that X ! Y is a C*weak equivalence if and only if
* * ____//_J* *
ExJ*CSpcX Ex *CSpcY
is a pointwise weak equivalence. *
62
Next we show a characterization of the class of C*weak equivalences.
Proposition 3.86: The class of C*weak equivalences is the smallest class of ma*
*ps *  weq
of cubical C*spaces which satisfies the following properties.
o *  weq is saturated.
o *  weq contains the class of exact projective weak equivalences.
o *  weq contains the elementary matrix invariant weak equivalences
A K ____//_A.
and the elementary C*weak equivalences
C(I, A)____//_A.
o Suppose there is a pushout square of cubical C*spaces where f is in *  w*
*eq:
g
X ! Z
?? ??
f??y ??yh
Y ! W
If g is a projective cofibration, then h is in *  weq. If f is a projecti*
*ve cofibration, then
h is a projective cofibration contained in *  weq.
o Suppose X: I ! *C * Spc is a small filtered diagram such that for every i*
* ! j, the
induced map X(i) ! X(j) is a map in *  weq. Then the induced map
X(i)____//_colimj2i#IX(j)
is in *  weq.
It is clear that the class of C*weak equivalences satisfies the first, seco*
*nd and third
conditions in Proposition 3.86.
The first part of the fourth item holds because the homotopy invariant proje*
*ctive
model structure is left proper, while the second part holds because acyclic cof*
*ibrations
are closed under pushouts in any model structure.
63
If X ! Y is a map of diagrams as in the fifth part, [33, Proposition 17.9.1]*
* implies
there is an induced C*weak equivalence
colim X(i)____//_colimY(i).
i2I i2I
Now consider the small filtered undercategory i # I with objects the maps i *
*! j in
I, and with maps the evident commutative triangles of objects. Applying the abo*
*ve
to X(i) ! X implies the last item. Note that in the formulation of the last ite*
*m we may
replace colimits by homotopy colimits.
The proof of Proposition 3.86 makes use of a functorial fibrant replacement *
*functor
in the homotopy invariant projective model structure. Denote by
id*C*Spc___//_()fC(I)
the fibrant replacement functor obtained by applying the small object argument *
*to
the set of (isomorphism classes of) elementary C*weak equivalences.
With these preliminaries taken care of we are ready to begin the proof.
Proof.(of Proposition 3.86.) Every elementary weak equivalence A ! C(I, A) is
contained in *  weq. We shall prove that any C *weak equivalence X ! Y can
be constructed from elementary weak equivalences using constructions as in the
statement of the proposition.
There is a commutative diagram:
X ! Y
??? ??
?y ??y
XfC(I)! YfC(I)
The vertical maps are constructed out of direct colimits of pushouts of element*
*ary
weak equivalences. The third and fourth items imply that the vertical maps are
contained in *  weq. On the other hand, saturation for C*weak equivalences an*
*d the
defining property of the fibrant replacement functor imply the lower horizontal*
* map is
a projective C*weak equivalence. This implies that it is a matrix exact projec*
*tive weak
64
equivalence. Since *  weq contains all matrix exact projective weak equivalen*
*ces
according to the second and third conditions, the lower horizontal map is conta*
*ined
in *  weq. Thus saturation, or the twooutofthree property, for the class * *
* weq
which holds by the first item implies the C*weak equivalence X ! Y is containe*
*d in
*  weq. *
In order to construct a fibrant replacement functor for the injective homoto*
*py
invariant model structure we shall proceed a bit differently. A flasque cubica*
*l C*
space Z is called quasifibrant ifithe mapsjZ(A) ! Z(A K)  corresponding to
matrix invariance  and Z(A) ! Z C(I, A)  corresponding to homotopy invariance
 are weak equivalences for every C *algebra A. We note that every C *projec*
*tive
fibrant cubical C*space is quasifibrant.
For a C*space X, we set
(Ex cyl(E,K)SingoX)0 SingoX
* *
and form inductively pushout diagrams
` o cyl(E,K)
ffnsff___//_Sing*(Ex Singo*X)n

 
 
` fflffl cfflfflyl(E,K)
ffntff____//_(Ex Singo*X)n+1
indexed by the set ffn of all commutative diagrams
sff____//_Singo*(Ex cyl(E,K)SingoX)n
 *
 
 
fflffl fflffl
tff____________//*
where sff ! tff is member of Jcyl(E)*C*Spc[ Jcyl(K)*C*Spc. There is an induce*
*d map from X to the
colimit Excyl(E,K)SingoX of the sequential diagram of alternating injective acy*
*clic C*weak
*
equivalences according to Example 2.8, Corollary 3.74 and Jcyl(E)*C*Spc[ Jcyl(*
*K)*C*Spcacyclic
cofibrations:
. ._.__//_(Ex cyl(E,K)SingoX)n//_Singo(Ex cyl(E,K)oX)n//_(Ex cyl(E,K)oX)n+1/*
*/_.(.3.4)
* * Sing* Sing*
65
Lemma 3.87: There is an endofunctor Excyl(E,K)Singoof *C * Spc and a natural t*
*ransformation
*
id*C*Spc___//_Excyl(E,K)Singo
*
such that Excyl(E,K)SingoX is quasifibrant for every cubical C*space X and the*
* map
*
cyl(E,K)
X ____//_ExSingo*X
is an injective acyclic C*weak equivalence.
Proof.The natural transformation exists by naturality of the map X ! Ex cyl(E,K*
*)SingoX
*
from X to the colimit of (34), and by [33, Proposition 17.9.1] it is an injecti*
*ve acyclic
C *weak equivalence. To show quasifibrancy, note that homotopy invariance holds
on account of the singular functor and that Excyl(E,K)SingoX has the right lift*
*ing property
*
with respect to Jcyl(E)*C*Spc[ Jcyl(K)*C*Spcsince the domains and codomains o*
*f maps in the
latter set preserve sequential colimits. *
* *
Corollary 3.88: Let
id*C*Spc___//_R
denote a fibrant replacement functor in the pointwise injective model structure*
*. Then
id*C*Spc___//_R Excyl(E,K)Singo
*
is a fibrant replacement functor in the injective homotopy invariant model stru*
*cture.
Proof.For every cubical C*space X the composite map
cyl(E,K) // cyl(E,K)
X ____//_ExSingo*X____R(Ex Singo*X)
is an injective acyclic C*weak equivalence by Lemma 3.87 and the defining prop*
*erty of
R. Moreover, R(Ex cyl(E,K)SingoX) is clearly injective fibrant, matrix invarian*
*t and homotopy
*
invariant. To show that it is flasque, note that for every exact square E the d*
*iagram
Excyl(E,K)Singo(A)!Excyl(E,K)o(E)
*? Sing*
?? ???
Ex cyl(E,K)Singo(E) ?y ?y (35)
*
* ! Excyl(E,K)Singo(B)
*
66
is homotopy cartesian due to Lemma 3.87. Applying the pointwise injective fibra*
*nt
replacement functor R yields a pointwise weak equivalence between (35) and:
R Excyl(E,K)Singo(A)!R Excyl(E,K)o(E)
?* Sing*
?? ???
R Excyl(E,K)Singo(E) ?y ?y (36)
*
* ! R Excyl(E,K)Singo(B)
*
It follows that (36) is homotopy cartesian [33, Proposition 13.3.13]. *
* *
Next we note the homotopy invariant projective model structure is compatible
with the monoidal structure. The proofiis analogousjto the proofiof Proposition*
* 3.56,j
using that for C*algebras A and B, cylC(I, A) ! A B = cylC(I, A B) ! A *
*B .
Proposition 3.89: The homotopy invariant projective model structure on *C * Sp*
*c is
monoidal.
Lemma 3.90: If X is projective cofibrant and Z is C*projective fibrant, then t*
*he internal
hom object Hom__(X, Z) is C*projective fibrant.
Lemma 3.91: If X is a projective cofibrant cubical C*space, then
i j
 X, Hom__(X, )
is a Quillen adjunction for the homotopy invariant projective model structure o*
*n *C * Spc.
The above has the following consequence.
Lemma 3.92: The following statements hold and are equivalent.
o If i: X Y and j: U V are projective cofibrations and either i or j is *
*a C*weak
equivalence, then so is
`
X V X U Y U ____//_Y V.
o If j: U V is a projective cofibration and k: Z W is a projective C*fi*
*bration,
then the pullback map
Hom__(V, Z) ____//_Hom_(V, W) xHom_(U,W)Hom_(U, Z)
is a projective C*fibration which is C*acyclic if either j or k is.
67
o With the same assumptions as in the previous item, the induced map
hom *C*Spc(V, Z)___//_hom*C*Spc(V, W) xhom*C*Spc(U,W)hom*C*Spc(U, Z)
is a Kan fibration which is a weak equivalence of cubical sets if in addit*
*ion either j or k
is C*acyclic.
Lemma 3.93: A map of cubical C*spaces X ! Y is a C*weak equivalence if and on*
*ly if for
every projective C*fibrant Z the induced map of internal hom objects
Hom__(QY, Z) ____//_Hom_(QX, Z) (37)
is a pointwise weak equivalence.
Proof.Lemma 3.90 implies (37) is a map between C*projective fibrant objects. H*
*ence,
by Lemma 3.77, (37) is a C*weak equivalence if and only if for every C*algebr*
*a A the
induced map
Hom__(QY, Z)(A) ____//_Hom_(QX, Z)(A),
or equivalently
i j i j
hom *C*SpcQY, Hom__(A, Z)____//_hom*C*SpcQX, Hom__(A, Z)
is a weak equivalence of cubical sets. Since every C*algebra is projective cof*
*ibrant
according to Lemma 3.6, the internal hom object Hom__(A, Z) is C*projective fi*
*brant
again by Lemma 3.90. *
The next result follows now from [35, Theorem 4.3.2].
Corollary 3.94: The total derived adjunction of ( , Hom__) gives a closed symme*
*tric monoidal
structure on the unstable C*homotopy category H. The associativity, commutati*
*vity and
unit isomorphisms are derived from the corresponding isomorphisms in *C * Spc.
Remark 3.95: Comparing with the corresponding derived adjunction obtained from
the closed monoidal structure on C * Spc and the Quillen equivalent homotopy
invariant model structure, we get compatible closed symmetric monoidal structur*
*es
on the unstable C*homotopy category.
68
3.5 Pointed model structures
It is straightforward to show the results for the model structures on *C * Spc*
* have
analogs for the categories *C * Spc0 of pointed cubical C*spaces and C * Sp*
*c0
of pointed simplicial C *spaces. Let H* denote the unstable pointed C *homot*
*opy
category. In this section we identify a set of compact generators for H*, form*
*ulate
Brown representability for H*and compute Kasparov's KKgroups of C*algebras as
maps in H*. To prove this result we use the simplicial category C * Spc0.
The next observation will be used in the context of cubical C*spectra.
Lemma 3.96: Suppose X is projective cofibrant and Z C*projective fibrant in *C*
* * Spc0.
Then the pointed internal hom object Hom__0(X, Z) is C*projective fibrant.
Proof.There is a pullback diagram of cubical C*spaces:
Hom__0(X, Z)____//Hom_(X, Z)

 
 
fflffl fflffl
*__________//Hom_(*, Z)
By monoidalness in the form of Lemma 3.92 the right vertical map is a C*projec*
*tive
fibration. Now use that fibrations pull back to fibrations in every model struc*
*ture. *
Suppose M is a pointed model category. Recall that G is a set of weak genera*
*tors for
Ho (M) if for every nontrivial Y 2 Ho (M) there is an X 2 G such that Ho (M)( n*
*X, Y)
is nontrivial. An object X 2 Ho (M) is called small if, for every set {Xff}ff2~*
*of objects
of Ho (M), there is a naturally induced isomorphism
` `
colim Ho (M)(X, ff2~0Xff)__//_Ho(M)(X, X ).
~0 ~,~0<1 ff2~ff
By [35, Section 7.3] the cofibers of the generating cofibrations in any cofibra*
*ntly
generated model category M form a set of weak generators for Ho (M). However,
it is more subtle to decide whether these weak generators are small in Ho (M). *
*The
argument given in [35, Section 7.4] relies not only on smallness properties of *
*the
domains and codomains of the generating cofibrations of M, but also on detailed
knowledge of the generating trivial cofibrations. For further details we refer *
*to the
proof of the analogous stable result Theorem 4.29.
69
Theorem 3.97: The cofibers of the generating projective cofibrations
{A (@*n *n)+}nA0
form a set of compact generators for the pretriangulated homotopy category H*o*
*f the homotopy
invariant model structure on pointed cubical C*spaces.
Next we formulate Brown representability for contravariant functors from the
pointed homotopy category of C*spaces to pointed sets.
Theorem 3.98: Suppose the contravariant functor F from H*to Set*satisfies the f*
*ollowing
properties.
o F (0) is the onepoint set.
o For every set {Xff} of objects in *C * Spc0 there is a naturally induced *
*bijective map
W Q
F ( Xff)___//_F (Xff).
o For every pointed projective cofibration X ! Y and pushout diagram
X _______//_Y
 
 
fflffl fflffl
Z ____//_Z [X Y
there is a naturally induced surjective map
F (Z [X Y) ____//_F (Z) xF (X)F (Y).
Then there exists a pointed cubical C*space W and a natural isomorphism
H*(, W) = F ().
Proof.Theorem 3.97 and left properness imply the C*projective model structure *
*on
*C * Spc0satisfies the assumptions in Jardine's representability theorem for p*
*ointed
model categories [39]. *
Remark 3.99: Left properness ensures the contravariant pointed set valued funct*
*or
H*(, W) satisfies the conditions in the formulation of Theorem 3.98.
70
Lemma 3.100: If X is a projective cofibrant pointed C*space, then there is a Q*
*uillen map
X : *C * Spc0 ____//_*Co*oSpc0:Hom__(X,_)
of the homotopy invariant model structure.
Lemma 3.101: Suppose X is a projective C*fibrant pointed cubical C*space. The*
*n for every
C *algebra A and integer n 0 there is a natural isomorphism
ssnX(A) = H*(A Sn, X).
Proof.Let ' be the equivalence relation generated by cubical homotopy equivalen*
*ce.
Since S1X is projective C*fibrant by the assumption on X, the Yoneda lemma and
Proposition 2.23 imply there are isomorphisms
ssnX(A) = ssnhom *C*Spc0(A, X)
= *C * Spc0(A, nS1X)= '
= H*(A Sn, X).
*
In the next theorem we use unstable C*homotopy theory to represent Kasparov*
*'s
KKgroups. The proof we give makes extensive use of Ktheoretic techniques which
are couched in simplicial sets; it carries over to the cubical setting in the l*
*ikely event
that the cubical nerve furnishes an equivalent way of constructing Ktheory. Se*
*ction
5.5 gives a fuller review of the Ktheory machinery behind categories with cofi*
*brations
and weak equivalences.
Theorem 3.102: Let F be a C*algebra. The pointed simplicial C*space
FRep:C * Alg ____//_ Set0
defined by i j
FRep(E) K Rep (F, E) =  NhtSoRep (F, E) 
is projective C*fibrant. For n 0 there is a natural isomorphism
KKn1(F, E) = H*(E Sn, FRep).
71
Here Rep (A, B) is the idempotent complete additive category of representati*
*ons
between C*algebras A and B. It is a category with cofibrations the maps which *
*are
split monomorphisms and weak equivalences the isomorphisms. Now passing to
the Ktheory of Rep (A, B) by using a fibrant geometric realization functor we *
*get a
pointed simplicial C*space FRep for every C*algebra F. Next we briefly outlin*
*e the
part of the proof showing FRep is exact projective fibrant: It is projective fi*
*brant by
construction (every simplicial abelianigroup is fibrant).jTo show it is flasque*
* we shall
trade Rep (A, B) for the category Ch bRep (A, B) of bounded chain complexes. *
*The
extra information gained by passing to chain complexesiallowsjus to finish the *
*proof.
The canonical inclusion of Rep (A, B) into Ch bRep (A, B) as chain complexe*
*s of
length one induces an equivalence in Ktheory [78, Theorem 1.11.7]. Thus we may
assume Rep (A, B) acquires a cylinder functor and satisfies the cylinder, exten*
*sion and
saturation axioms. Applying the fibration theorem [81, Theorem 1.6.4] furnishes*
* for
every short exact sequence (18) of C*algebras with a completely positive split*
*ting the
desired homotopy fiber sequence
FRep(A)____//_FRep(E)__//_FRep(B).
The second part of Theorem 3.102 follows by combining the first part with Lemma
3.101 and work of Kandelaki [44].
To prepare ground for the proof of Theorem 3.102 we shall recall some notion*
*s from
[44] and [46]. In particular, we shall consider categories enriched in the symm*
*etric
monoidal category of C*algebras, a.k.a. C*categories. The category of Hilbert*
* spaces
and bounded linear maps is an example. Every unital C*algebra defines a C*cat*
*egory
with one object and with the elements of the algebra as maps.
If B is a C*algebra, then a Hilbert Bmodule H consists of a countably gene*
*rated
right Hilbert module over B equipped with an inner product <  >: H xH ! B. Den*
*ote
by H(B) the additive C*category of Hilbert Bmodules with respect to sums of H*
*ilbert
modules and by K(B) its C*ideal of compact maps. Next we consider pairs (H, ae)
where H 2 H(B) and ae: A ! L(H) is a *homomorphism. Here L(H) is the algebra
of linear operators on H which admit an adjoint with respect to the inner produ*
*ct. A
map (H, ae) ! (H0, ae0) consists of a map f :H ! H0 in H(B) such that f ae(a) *
* ae0(a)f
is in K(B)(H, H0) for all a 2 A. This defines the structure of an additive C*c*
*ategory
inherited from H(B). Let Rep (A, B) denote its universal pseudoabelian C*categ*
*ory.
72
Its objects are triples (H, ae, p) where p: (H, ae) ! (H, ae) satisfies p = p*a*
*nd p2 = p, and
maps (H, ae, p) ! (H0, ae0, p0) consists of maps of pairs f :(H, ae) ! (H0, ae0*
*) as above,
subject to the relation f p = p0f = f . We note that triples are added accordin*
*g to the
formula (H, ae, p) (H0, ae0, p0) (H H0, ae ae0, p p0).
i j
Let Ch bRep (A, B) be the chain complex category of bounded chain complexes
Eb: 0 ! Em ! . .!.En ! 0 in the additive category Rep (A, B). It acquiresithe *
* j
structure of a category with cofibrations and weak equivalences htCh bRep (A, B*
*) in
the sense of Waldhausen [81] with cofibrations the degreewise split monomorphis*
*ms
and weak equivalences theimaps whosejmapping cones are homotopy equivalent to
acyclic complexes in Ch bRep (A, B) .
i j
If f :Eb ! Fb is a map in Ch bRep (A, B) , let T(f ) be the bounded chain co*
*mplex
given
T(f )p Ep Ep1 Fp.
The boundary maps of T(f ) are determined by the matrix:
0 1
BBBdEb id 0 CC
BBB0 d b 0 CCC
B@ E CCA
0 f dFb
There exist natural inclusions of direct summands iEb:Eb T(f ) and iFb:Fb T*
*(f ).
These maps fit into the commutative diagram:
iEb iFb
Eb ____//_DDT(F)Fboo_

DDD ss 
f DD!!Dfflffl
Fb
Here ss is definedidegreewisejby ssp (f, 0, id). Standard chain complex techn*
*iques
imply htCh bRep (A, B) satisfies the cylinder axioms [81, x1.6].
Lemma 3.103: If f :Eb ! Fb is a chain map, then ss is a chain homotopy equivale*
*nce, iEb iFb
is a degreewise split monomorphisms and T(0 ! Fb) = Fb, ss = iFb= idFb.
i j
Moreover, the next lemma shows that htCh b Rep(A, B) satisfies the extension
axiom formulated in [81, x1.2].
73
Lemma 3.104: Suppose
Bb ____//_Eb___//_Ab
  
  
fflfflfflffl fflffl
eBb____//_eEb__//_eAb
i j
is a map of cofibration sequences in Ch bRep (A, B) . If the left and right ver*
*tical maps are
weak equivalences, then so is the middle vertical map.
i j
Next we note that Ch bRep (A, B) satisfies the saturation axiom [81, x1.2].
i j
Lemma 3.105: If f and g are composable maps in Ch bRep (A, B) and two of the m*
*aps f , g
and f g are weak equivalences, then the third map is a weak equivalence.
i j
Suppose F is a C*algebra. Applying the functor Ch bRep (F, ) to a complet*
*ely
positive split short exact sequence 0 ! A ! E ! B ! 0 yields functors
i j i j i j
Ch b Rep(F, A)_____//ChbRep (F, E)____//ChbRep(F, B) .
i j i j
Denote by ehtChbRep (F, E) the category Chb Rep (F, E) with cofibrations degree*
*wise
split monomorphisms and weak equivalences the chainimaps withjmapping cones
homotopy equivalent to acyclic complexes in Ch bRep (F, B) . It inherits a cyl*
*inder
i j
functor from htCh bRep (F, E) .
i j
Lemma 3.106: The category ehtChbRep (F, E) satisfies the extension and saturati*
*on axioms
and acquires a cylinder functor satisfying the cylinder axioms.
i jeht i j
Define htCh b Rep(F, E) to be the full subcategory of htCh b Rep(F, E) wh*
*ose
i j
objects are Eb such that 0 ! Eb is a weak equivalence in ehtChbRep (F, E) . It *
*acquires
the structureiofja category with cofibrations and weak equivalences inherited f*
*rom
htCh b Rep(F, E) . With these definitions there are equivalences
i jeht i j
htCh b Rep (F, E) ' htCh bRep (F, A)
and i j i j
ehtChb Rep(F, E) ' htCh bRep (F, B) .
74
i j
Clearly every weak equivalence in htCh b Rep(F, E) is also a weak equivalence *
*in
i j
ehtChb Rep(F, E) . Thus by [81, Theorem 1.6.4] there is a homotopy cartesian sq*
*uare:
i j i jeht
htCh b Rep (F, A)____//_ehtChbRep(F, E) ' *
 
 
i fflffl j i fflffl j
htChb Rep (F, E)______//_htChbRep(F, B)
This implies FRep is flasque. Theorem 3.102 follows now simply by combining the
isomorphism
ssnFRep(E) = KKn1(F, E)
for n 0 [44, Theorem 1.2] and Lemma 3.101.
The results in [44] employed in the above hold equivariantly. Thus we may in*
*fer:
Theorem 3.107: Let F be a G  C*algebra where G is compact second countable. T*
*hen the
pointed simplicial G  C*space
FRep:G  C* Alg ____//_ Set0
defined by i j
FGRep(E) K G  Rep (F, E) =  NhtSoG  Rep (F, E) 
is projective G  C*fibrant. For n 0 there is a natural isomorphism
G  KKn1(F, E) = G  H*(E Sn, FGRep).
Here, the left hand side denotes the G equivariant Kasparov KKgroups and the *
*right hand
side maps in the unstable pointed Gequivariant C*homotopy category.
75
3.6 Base change
For every C*algebra A the slice category *C * Spc # A consists of cubical C**
*spaces
together with a map to A. Maps in *C * Spc # A are maps in *C * Spc which are
compatible with the given maps to A. We claim *C *Spc # A acquires the exact *
*same
four types of model structures as *C *Spc by defining the relevant homotopical*
* data
via the forgetful functor
*C * Spc # A ____//_*C * Spc.
In the slice category setting the model structures on *C *Spc correspond to th*
*e trivial
C *algebra. More generally, we have the following result.
Lemma 3.108: For any of the pointwise, exact, matrix invariant and homotopy inv*
*ariant
model structures on *C * Spc the slice category *C * Spc # X has a correspon*
*ding
combinatorial and weakly finitely generated left proper model structure where a*
* map f is a
weak equivalence (respectively cofibration, fibration) in *C * Spc # X if and *
*only if f is a
weak equivalence (respectively cofibration, fibration) in *C * Spc.
Proof.The existence of the model structure follows from [33, Theorem 7.6.5]. Si*
*nce
pushouts are formed by taking pushouts of the underlying maps in *C * Spc, it
follows that *C * Spc # X is left proper since *C * Spcis so. With these defi*
*nitions it
is trivial to check that the (acyclic) cofibrations are generated by generating*
* (acyclic)
cofibrations over X. *
Remark 3.109: There is a straightforward analog of Lemma 3.108 for pointed cubi*
*cal
and pointed simplicial C*spaces. We leave the formulation of Brown representab*
*ility
in this setting to the reader.
If f :X ! Y is a map between cubical C*spaces, there is an induced Quillen *
*pair
between the corresponding slice categories:
f!:*C * Spc # X ____//_*Co*oSpc_# Y :f * (38)
The left adjoint is defined by (Z ! X) 7! (Z ! X ! Y) and the right adjoint by
(Z ! Y) 7! (Z xY X ! X). When f is a weak equivalence between fibrant objects,
then the adjunction (38) is a Quillen equivalence, but without the fibrancy con*
*dition
this may fail.
76
For objects X and Y of *C * Spc let X # *C * Spc # Y be the category of ob*
*jects
of *C * Spc under X and over Y in which an object is a diagram X ! Z ! Y of
maps of cubical C*spaces. A map from X ! Z ! Y to X ! W ! Y consists of
a map f :Z ! W such that the obvious diagram commutes. The next result can be
proved using similar arguments as in the proof of Lemma 3.108 and there are dir*
*ect
analogs for pointed cubical and pointed simplicial C*spaces which we leave imp*
*licit.
Lemma 3.110: For any of the pointwise, exact, matrix invariant and homotopy inv*
*ariant
model structures on *C * Spcand for every pair of cubical *C *spaces X and Y *
*the category
X # *C * Spc # Y has a corresponding combinatorial and weakly finitely generat*
*ed left
proper model structure where f is a weak equivalence (respectively cofibration,*
* fibration) in
X # *C * Spc # Y if and only if Z ! W is a weak equivalences (respectively cof*
*ibration,
fibration) in *C * Spc.
The Ktheory of a C *algebra or more generally of a cubical *C *space Z us*
*es
the homotopy theory of the retract category (Z, *C * Spc, Z) with objects trip*
*les
(X, i: Z ! X, r: X ! Z) where ri = id and maps f :(X, i: Z ! X, r: X ! Z) !
(X, j: Z ! Y, s: Y ! Z) respecting the retractions and sections.
We have the following variant of Lemma 3.110.
Lemma 3.111: For any of the pointwise, exact, matrix invariant and homotopy inv*
*ariant
model structures on *C * Spc the retract category (Z, *C * Spc, Z) of a cubic*
*al *C *
space Z has a corresponding combinatorial and weakly finitely generated left pr*
*oper cubical
model structure where f is declared a weak equivalence (respectively cofibratio*
*n, fibration) in
(Z, *C * Spc, Z) if and only if X ! Y is a weak equivalences (respectively cof*
*ibration,
fibration) in *C * Spc.
Since it is perhaps not completely obvious we define the cubical structure of
(Z, *C * Spc, Z). If K is a cubical set the tensor
(X, i: Z ! X, r: X ! Z) K
is defined as the pushout of the diagram
Z Z *0 oo___Z K ____//_X K,
while the cotensor
(X, i: Z ! X, r: X ! Z)K
77
is defined as the pullback of the diagram
0 _____//K oo___K
Z Z* Z X .
The cubical function complex
i j
hom (Z,*C*Spc,Z)(X, i: Z ! X, r: X ! Z), (X0, i0:Z ! X0, r0:X0! Z)
of X and X0is the subcomplex of hom *C*Spc(X, X0) comprising maps which respect
the retraction and section [65, II. 2 Proposition 6].
Remark 3.112: We leave implicit the formulations of the corresponding equivaria*
*nt
results in this section. Several functoriality questions arise when the groups *
*vary.
78
*
4 Stable C homotopy theory
Stable homotopy theory in the now baroque formulation of spectra is bootstrappe*
*d to
represent all generalized homology and cohomology theories for topological spac*
*es.
We are interested in an analogous theory for cubical C*spaces which captures s*
*uitably
defined cohomology and homology theories in one snap maneuver. The mixing of
C *algebras and cubical sets in *C * Spc allows us to vary the suspension coo*
*rdinate
in a manner which is out of reach in the more confined settings of C* Alg and *
**Set.
Indeed the "circle" C we will be using is the tensor product S1 C0(R) of the s*
*tandard
cubical set model *1=@*1 for the topological circle and the C *algebra of comp*
*lex
valued continuous functions on the real numbers which vanish at infinity. In t*
*he
modern formulation of stable homotopy theory the use of symmetric spectra obvia*
*te
ordinary spectra by solving the problem of finding a monoidal model structure w*
*hich
is Quillen equivalent to the stable model structure. To set up the stable C*ho*
*motopy
theory we consider symmetric spectra of pointed cubical C*spaces with respect *
*to C.
A great deal of the results can be proved by referring to the works of Hovey [3*
*6] and
Jardine [40], a strategy we will follow to a large extend. Another valuable vie*
*wpoint
which offers considerable flexibility and opens up some new subjects to explore*
* is to
consider model structures on enriched functors from the subcategory fp*C * Spc*
* of
finitely presentable cubical C*spaces into *C * Spc. This falls into the real*
*ms of [22].
4.1 C *spectra
We start out by adapting the definition of spectra to our setting.
Definition 4.1: The category SptC of cubical C *spectra consists of sequences E
(En)n 0of pointed cubical C*spaces equipped with structure maps oeEn: CEn ! En*
*+1
where C C  is the suspension functor. A map f :E ! F of cubical C*spectra
consists of compatible maps of pointed cubical C*spaces fn: En ! Fn in the sen*
*se
that the diagrams
oeEn
CEn ____//_En+1
C fn fn+1
fflffloeFfflffln
CFn ____//_Fn+1
commute for all n 0.
79
What follows is a list of examples of cubical C*spectra we will be working *
*with.
Example 4.2: The suspension cubical C*spectrum of a C*space X is given by
1CX {n O___//_C n X}
with structure maps the canonical isomorphisms CC n X ! C (n+1) X. The sphere
spectrum is the suspension cubical C*spectrum 1CC of the complex numbers.
Example 4.3: If E is a cubical C*spectrum and X is a pointed cubical C*space,*
* there
is a cubical C *spectrum E ^ X with nth level En X and structure maps oeEn *
*X.
The suspension E ^ C of E is left adjoint to the Cloops cubical C *spectrum *
*CE
of E defined by setting ( CE)n C(En) = Hom__(C, En) and with structure maps
oenCE: C C(En) ! C(En+1) adjoint to the composite C C(En) C ! C En ! En+1.
Example 4.4: The fake suspension CE of E has nth level C En and structure ma*
*ps
oenCE C oeEn. We note that C :SptC ! Spt Cis left adjoint to the fake Cl*
*oops
functor `Cdefined by `C(E) C(En) and with structure maps adjoint to the ma*
*ps
C(eoeEn): C(En) ! 2C(En+1). It is important to note that the adjoint of the *
*structure
map oenCE differs from C(eoeEn) by a twist of loop factors. In particular, the*
* fake Cloops
functor is not isomorphic to the Cloops functor.
Example 4.5: If X is a pointed cubical C*space, denote by hom SptC(X, E) the c*
*ubical
C *spectrum hom SptC(K, E)n hom *C*Spc0(K, En) with structure maps adjoint *
*to the
composite maps C hom *C*Spc0(K, En) K ! C En ! En+1. With these definiti*
*ons
there is a natural bijection
i j
Spt C(E ^ K, F ) = SptC E, hom SptC(K, F ) .
The function complex hom SptC(E, F ) of cubical C*spectra E and F are defined *
*in level
n as all maps E ^ *n+! F of cubical C*spectra.
Example 4.6: The mth shift E[m] of a cubical C*spectrum E is defined by
8
>>>
:* m + n < 0.
The structure maps are reindexed accordingly.
80
Example 4.7: The layer filtration of E is obtained from the cubical C *spectra*
* LmE
defined by
8
>>>
:C nm Em n > m.
There is a canonical map 1CEm[m] ! LmE and Lm+1E is the pushout of the diagra*
*m:
1C(Em+1)[m  1]oo___1C(C Em)[m  1]____//_LmE
Observe that E and colim LmE are isomorphic.
A map f :E ! F is a level weak equivalence (respectively fibration) if fn: E*
*n ! Fn
is a C*weak equivalence (respectively projective C*fibration). And f is a pro*
*jective
cofibration if f0 and the maps
`
En+1 CEn CFn ____//_Fn+1
are projective cofibrations for all n 0. By the results in [36, x1] we have:
Proposition 4.8: The level weak equivalences, projective cofibrations and level*
* fibrations
furnish a combinatorial, cubical and left proper Quillen equivalent model struc*
*ture on SptC.
Our next objective is to define the stable model structure as a Bousfield lo*
*calization
of the level model structure. The fibrant objects in the localized model struct*
*ure have
been apprehended as spectra since the days of yore. In our setting this amoun*
*ts to
the following definition.
Definition 4.9: A cubical C*spectrum Z is stably fibrant if it is level fibran*
*t and all
the adjoints eoeZn:Zn ! Hom__(C, Zn+1) of its structure maps are C*weak equiva*
*lences.
The stably fibrant cubical C*spectra determine the stable weak equivalences*
* of
cubical C*spectra. Stable fibrations are maps having the right lifting propert*
*y with
respect to all maps which are projective cofibrations and stable weak equivalen*
*ces.
Definition 4.10: A map f :E ! F of cubical C*spectra is a stable weak equivale*
*nce
if for every stably fibrant Z taking a cofibrant replacement Qf :QE ! QF of f i*
*n the
level model structure on SptC yields a weak equivalence of pointed cubical sets
hom SptC(Qf, Z): hom SptC(QF , Z)__//_homSptC(QE, Z).
81
Example 4.11: The map 1CEm[m] ! LmE mentioned in Example 4.7 is a stable weak
equivalence.
By specializing the collection of results in [36, x3] to our setting we have:
Theorem 4.12: The classes of stable weak equivalences and projective cofibratio*
*ns define a
combinatorial, cubical and left proper model structure on SptC. Let SH *denote *
*the associated
stable homotopy category of C*algebras.
Denote by 'E: E ! E `CE[1] the natural map where ( E)n Hom__(C, En+1)
and oenE Hom__(C,eoeEn). The stabilization 1 E of E is the colimit of the di*
*agram:
'E 'E 2'E
E ____//_ E___//_ 2E___//_. . .
By [36, Proposition 4.6] we get the following result because Hom__(C, ) preser*
*ves
sequential colimits according to Example 2.7 and the projective homotopy invari*
*ant
model structure on *C * Spc0 is almost finitely generated.
Lemma 4.13: The stabilization of every level fibrant cubical C*spectrum is sta*
*bly fibrant.
Now let fE: E ! RE be a natural level fibrant replacement for E in SptC, mea*
*ning
that fE a level weak equivalence and projective cofibration and RE is level fib*
*rant.
Lemma 4.13 motivates the next definition.
Definition 4.14: Lete'E:E ! 1 RE be the composite of fE and '1RE:RE ! 1 RE.
We have the following convenient characterization of stable weak equivalences
given by [36, Theorem 4.12] and a corollary which shows that certain stable map*
*s can
be approximated by unstable maps [36, Corollary 4.13].
Theorem 4.15: A map f :E ! F is a stable weak equivalence if and only if the in*
*duced map
e'E(f ): 1 RE ! 1 RF is a level weak equivalence.
Corollary 4.16: If X is a finitely presentable cofibrant cubical C*space and F*
* is level fibrant,
then there is a canonical isomorphism
SH *( 1CX, F ) = colim H*(X, nCFn). (39)
n
In addition, if F is stably fibrant so that all of the transition maps in the d*
*irected system in
(39) are isomorphisms, then there is a canonical isomorphism
SH *( 1CX, F ) = H*(X, F0).
82
Next we characterize an important class of stable fibrations.
Lemma 4.17: A level fibration f :E ! F with a stably fibrant target is a stable*
* fibration if
and only if the diagram
E ____//_ 1 RE
f e'E(f)
fflffl fflffl
F ____//_ 1 RF
is a homotopy pullback in the levelwise model structure on C* SpcC.
Proof.See the proof of Corollary 3.41. *
* *
Lemma 4.18: The loop functor E 7! CE preserves stable weak equivalences betwee*
*n level
fibrant objects.
Proof.If E is level fibrant there is an isomorphism 1 ( CE)n = C( 1 E)n. *
* *
Next we seek an interpretation of stable weak equivalences in terms of bigra*
*ded
stable homotopy groups ssp,qfor integers p, q 2 Z. Suppose E is level fibrant *
*and
consider the sequential diagram:
2CeoeC
. ._.__//EneoeC//_ CEn+1_CeoeC//_ 2CEn+2//_. . .
In Asections, the homotopy group ssp 1 En(A) is isomorphic to the colimit of t*
*he
sequential diagram:
ssp(eoeC)(A) ssp( CeoeC)(A) ssp( 2CeoeC)(A)
. ._.__//_ssp(En)(A)_//_ssp( CEn+1)(A)_//_ssp( 2CEn+2)(A)_////_. . .
Passing to the homotopy category associated with the stable model structure on
SptC # A, we can recast the latter diagram as:
. ._.__//[Sp, EnA]_//_[Sp C, En+1A]//_[Sp C 2, En+2A]//_. . .
In the stable homotopy category SH *Aof pointed cubical C *spaces over A, where
there is no need to impose fibrancy, one obtains from the definition C = S1 C0*
*(R) an
alternate description of the homotopy groups as the colimit of the sequential d*
*iagram:
. ._.__//[Sp, EnA]_//_[Sp+1 C0(R), En+1A]_//_[Sp+2 C0(R2), En+2A]//_. .*
* .
83
Definition 4.19: Let E be a cubical C*spectrum. The degree p and weight q sta*
*ble
homotopy group ssp,qE is defined in Asections by
i j
ssp,qE(A) colim [Sp+n C0(Rq+n), EnA]__//[Sp+n+1 C0(Rq+n+1), En+1A]//_...*
* .
In Asections there are natural isomorphisms
ssp,qE(A) = sspq 1 REq(A). (40)
Lemma 4.20: Define S1() Hom__(S1, ) and C0(R)() Hom__(C0(R), ). For *
*every
cubical C*spectrum E there are isomorphisms
8
>>> qS1p
:ss0 C 1 (RE[p]) p q.
0(R) 0
Proof.There are isomorphisms
colim [Sp+n C0(Rq+n), EnA]= colim [Spq+n C0(Rn), E[q]nA]
= colim [Sn C0(Rn), pqS1RE[q]nA]
if p q, and
colim [Sp+n C0(Rq+n), EnA]= colim [Sn C0(Rqp+n), E[p]nA]
= colim [Sn C0(Rn), qpC0(R)RE[p]nA]
if q p. *
We are ready to give an algebraic characterization of stable weak equivalenc*
*es.
Lemma 4.21: The map E ! F is a stable weak equivalence if and only if there is *
*an induced
isomorphism of bigraded stable homotopy groups
ssp,qE = ssp,qF .
Proof.A stable equivalence between E and F induces for every integer m 2 Z a le*
*vel
weak equivalence 1 RE[m] ! 1 RF [m]. Hence, in all sections, the induced maps
between the bigraded stable homotopy groups of E and F are isomorphisms (40).
Conversely, if ssp,qE ! ssp,qF is an isomorphism of presheaves for p q 0, t*
*hen
again by (40) there is a level weak equivalence 1 RE ! 1 RF . *
* *
84
The usual approach verifies that stable homotopy groups preserve colimits in*
* the
following sense.
Lemma 4.22: Let {Ei}i2Ibe a filtered diagram of cubical C*spectra. Then the na*
*tural map
colimi2Issp,qEi__//_ssp,q(hocolim i2IEi)
is an isomorphism for all p, q 2 Z.
Proof.Without loss of generality we may assume the diagram consists of stably f*
*ibrant
C *spectra. Then the colimit E of the diagram  where En = colimi2IEin is st*
*ably
fibrant: Each En is a filtered colimit of C*projective fibrant pointed cubical*
* C*spaces.
Lemma 3.96 shows that Hom__(C, Ein+1) is C*projective fibrant. It follows that*
* the map
Ein! Hom__(C, Ein+1) is a pointwise weak equivalence. Using that Hom__(C, ) co*
*mmutes
with filtered colimits we conclude that the adjoints of the structure maps
En = colimi2IEin___//_colimi2IHom_(C, Ein+1) = CEn+1
are pointwise weak equivalence, and thus C*weak equivalences.
There is a natural isomorphism between
i j
colimi2Iss0hom *C*Spc0Sp C0(Rpq), Ei
and i j
ss0hom *C*Spc0Sp C0(Rpq), colimi2IEi
since the functor i j
hom *C*Spc0Sp C0(Rpq), 
commutes with filtered colimits. Now suppose there is a stable weak equivalence
Ei ! F ibetween stably fibrant C*spectra for every i 2 I. Then levelwise there*
* are
pointwise weak equivalences Ein! Fniso that the map
colimi2IEin___//_colimi2IFni
is a pointwise weak equivalence, which implies E ! F is a stable weak equivalen*
*ce.
Hence the homotopy colimit of {Ei}i2Imaps by a natural stable weak equivalence *
*to E
and we are done. *
85
The next lemma dealing with the cyclic permutation condition on the circle C*
* is a
key input in stable C*homotopy theory. We shall refer to this lemma when compa*
*ring
C *spectra with C*symmetric spectra. It ensures that the stable C*homotopy c*
*ategory
inherits a symmetric monoidal product.
Lemma 4.23: The circle C satisfies the cyclic permutation condition. That is, t*
*here exists a
homotopy C C C C(I) ! C C C from the cyclic permutation to the identi*
*ty on C 3.
Proof.In *C * Spc0 there is an isomorphism C 3 = S3 C0(R3). Clearly the cyc*
*lic
permutation condition holds for the cubical 3sphere since (321): S3 ! S3, x *
*y z 7!
y z x, has degree one. Using the isomorphism C0(R3) C(I) = C0(R3x I) the *
*claim
for C0(R3) follows by defining a homotopy C0(R3x I) ! C0(R3) in terms of the ma*
*trix:
0 1
BBBt 1  t 0 CC
BBB0 t 1  tCCC
B@ CCA
1  t 0 t
These two observations imply that C satisfies the cyclic permutation condition.*
* *
Lemma 4.24: For every cubical C*space X there is a stable weak equivalence
1CX ____//_ C( 1CX ^ C)___//_ CR( 1CX ^ C).
Proof.Using Lemma 4.23 this follows as in the proof of [40, Lemma 3.14]. *
* *
Theorem 4.25: For every cubical C*spectrum E there is a stable weak equivalence
E ____//_ C(E ^ C)___// CR(E ^ C).
Proof.The idea is to reduce the proof to Lemma 4.24 using the layer filtration.*
* Indeed,
since the shift functor preserves stable weak equivalences, Lemma 4.24 implies *
*there
are stable weak equivalences
1CEn[n]____//_ CR( 1CEn[n] ^ C).
Clearly the functors  ^ C and CR( ^ C) preserve stable weak equivalences and
there is a stable weak equivalence 1CEn[n] ! LnE. To conclude the map in ques*
*tion
is a stable weak equivalence we refer to the next lemma, which follows using the
arguments in [40, Lemma 3.12, pp. 498499]. *
86
Lemma 4.26: If E0 ! E1 ! . . .is a sequential diagram of cubical C*spectraisuc*
*h thatj
Ei! CR(Ei^C) is a stable weak equivalence, then so is colim Ei! CR (colim Ei)*
*^C .
We include the next results for completeness. The proofs can be patched from*
* the
arguments in [40, x3.4].
Corollary 4.27: Let E and F be cubical C*spectra.
o If E is level fibrant the evaluation map CE ^ C ! E is a stable weak equi*
*valence.
o A map E ^ C ! F is a stable weak equivalence if and only if E ! CF ! CRF*
* is
a stable weak equivalence.
o A map E ! F is a stable weak equivalence if and only if E ^ C ! F ^ C is s*
*o.
o There are natural stable weak equivalences between CE, E[1] and E ^ C.
o If E is level fibrant there are natural stable weak equivalences between *
*`CE, E[1] and
CE.
As for the circle C, we may define categories of S1spectra SptS1and C0(R)s*
*pectra
SptC0(R)of pointed cubical C*spaces and construct level and stable model struc*
*tures.
In particular, a map E ! F of cubical S1spectra is a stable weak equivalence i*
*f and
only if 1 RE ! 1 RF is a pointwise level weak equivalence, i.e. all the induc*
*ed
maps ssp 1 RE(A) ! ssp 1 RF (A) of homotopy groups are isomorphisms. Here 1
is defined as above using Hom__(S1, ) and R is a fibrant replacement functor i*
*n the
level model structure on SptS1. The group ssp 1 RE(A) is isomorphic to the coli*
*mit of
the sequential diagram
. ._.__//_[Sp, EnA]_//_[Sp+1, En+1A]//_[Sp+2, En+2A]//_. . . (41)
in the homotopy category H(*C * Spc0 # A). Define sspE in Asections to be the
colimit of (41). Thus E ! F is a stable weak equivalence if and only if there *
*are
induced isomorphisms sspE = sspF for every integer p. Every level fiber sequen*
*ce
F ! E ! E0 can be functorially replaced up to level weak equivalence by a fiber
sequence of level fibrant cubical S1spectra, so that in Asections we get a le*
*vel fiber
sequence RF (A) ! RE(A) ! RE0(A). This implies there is a long exact sequence:
. ._.__//_ssp+1E0_//_sspF__//_sspE__//_sspE0__//. . . (42)
87
Applying the proof of [40, Corollary 3.6] to our setting we get the next result.
Lemma 4.28: Every cofiber sequence E ! E0! E0=E induces a natural long exact se*
*quence:
. ._.__//ssp+1E0=E__//_sspE__//_sspE0_//_sspE0=E__//_. . .
We end this section with a useful result which reduces questions about SH *to
properties of a set of wellbehaved compact generators. It is the stable analo*
*g of
Theorem 3.97. For completeness we indicate the proof following [35, Section 7.4*
*].
Theorem 4.29: The cofibers of the generating projective cofibrations
i j
Frm A (@*n *n)+
form a set of compact generators for the stable homotopy category SH *.
Proof.By [35, Section 7.3] the cofibers form a set of weak generators. It remai*
*ns to
show that the cofibrant and finitely presentable cofibers are compact in SH *.
Suppose E is cofibrant and finitely presentable in SptC. Let ~ be an ordinal*
* and
denote by ~finits set of finite subsets. Now for every ~indexed collection of *
*cubical
C *spectra we need to show that the canonical map
` * `
colimfffinSH*(E, ff2fffinFff)//_SH(E, ff<~Fff) (43)
is an isomorphism. Injectivity of the map (43) holds because the inclusion of e*
*very
finite subcoproduct has a retraction. To prove surjectivity we use transfinite *
*induction.
The subcategory of finite subsets of ~ + 1 containing ~ is final in the categor*
*y of finite
subsets of ~ + 1. Thus the successor ordinal case holds and we may assume ~ is *
*a limit
ordinal such that (43) is surjective for every fi < ~. Without loss of generali*
*ty we may
`
assume Fffis bifibrant and hence that ff<~Fffis cofibrant. Since a filtered *
*colimit
of stably fibrant cubical C*spectra is stably fibrant applying a fibrant repla*
*cement
functor R yields a weak equivalence
` ` `
ff<~Fff= colimfi<~ ff>>
:oeq(k  p) + pifp + 1 k p + q.
101
To complete the definition of (50) one extends this construction in the natural*
* way to
all pointed cubical C*spaces. The monoidal structure is rigged so that (C, 0, *
*0, . .).is
the unit, and Sym (X) is freely generated by (0, X, 0, 0, . .).. Note that E F*
* represents
the functor which to G associates all p x qequivariant maps OEp,q:Ep Fq ! *
*Gp+q
in *C * Spc0. To show that is symmetric it suffices, by the Yoneda lemma, *
*to
define natural bijections *C * Spc0(E F , G) ! *C * Spc0(F E, G). In ef*
*fect, if
(OEp,q) 2 *C *Spc 0(E F , G) define (OE0p,q) 2 *C *Spc 0(F E, G) by the co*
*mmutative
diagrams:
OE0p,q
Fp Eq____//_Gp+q
OO
o cp,q
fflfflOEq,p
Eq Fp ____//_Gq+p
Here o is the symmetry isomorphism in *C *Spc 0and oep,qis the permutation def*
*ined
by 8
>>>
:k  p ifp + 1 k p + q.
Then OE0p,qis px qequivariant, and hence there exists a commutativity isomor*
*phism
oe: E F ! F E. According to (50) the associativity isomorphism for fol*
*low
using the associativity isomorphism for in *C * Spc0.
Now if En, Fn 2 *C * Spc0n the internal hom Hom__ n(En, Fn) exists for form*
*al
reasons as the equalizer of the two maps Hom__(En, Fn) ! Hom__( n x En, Fn) ind*
*uced
by the nactions on En and Fn. Internal hom objects in *C * Spc0 are defined *
*by
Y
Hom__ (E, F )k Hom__ n(Xn, Fn+k).
n
Note that Hom__ n(En, Fn+k) is the ninvariants of the internal hom Hom__(En, *
*Fn+k) in
*C * Spc0 for the action given by associating to oen 2 n the map
i j
Hom_(oen,Fn+k) Hom_En,OEn,k(oen,1)
Hom__(En, Fn+k)___//_Hom_(En, Fn+k)_//_Hom_(En, Fn+k).
With these definitions there are natural bijections
i j
*C * Spc0(E F , G) = *C * Spc0 E, Hom__(F , G) .
102
Small limits and colimits in the functor category *C * Spc0 exist and are f*
*ormed
pointwise. By reference to [17] or by inspection of the above constructions we *
*get the
next result.
Lemma 4.53: The triple (*C * Spc0, , Hom__) forms a bicomplete and closed sy*
*mmetric
monoidal category.
Example 4.54: For a pointed cubical C*space X the C*cubical symmetric sequence
Sym (X) is a commutative monoid in the closed symmetric structure on *C * Spc0.
Recall the monoid structure is induced by the canonical maps
p+qx px q(X p X q) ____//_X (p+q).
Lemma 4.53 and Example 4.54 imply the category of modules over Sym (X) in
*C * Spc0 is bicomplete and closed symmetric monoidal. The monoidal product
 Sym(X): Sym (X)  Mod x Sym (X)  Mod ____//_Sym(X)  Mod
is defined by coequalizers in *C * Spc0 of the form
Sym (X) F G_____////_F __G_//_F Sym(X)G
induced by Sym (X) F ! F and Sym (X) F G ! F Sym (X) G ! F G.
Moreover, the internal hom
Hom__Sym(X)(, ): (Sym (X)  Mod )opx Sym (X)  Mod__//_Sym(X)  Mod
is defined by equalizers in *C * Spc0 of the form
Hom__Sym(X)(F , G)__//_Hom_(F , G)___////_Hom_(Sym (X) F., G)
The first map in the equalizer is induced by the Sym (X)action on F and the se*
*cond
map is the composition of Sym (X)  and the Sym (X)action on G. Note that Sym *
*(X)
is the unit for the monoidal product.
Next we specialize these constructions to the projective cofibrant pointed c*
*ubical
C *space C = S1 C0(R).
103
Definition 4.55: The category of C*cubical symmetric spectra SptC is the categ*
*ory of
modules in *C *Spc 0over the commutative monoid Sym (C). In detail, a module o*
*ver
Sym (C) consists of a sequence of pointed cubical C*spaces E (En)n 0in *C **
* Spc0
together with nequivariant structure maps oen: C En ! En+1such that the compo*
*site
C p En ____//_C p1 En+1___//._._.//_En+p
is nx pequivariant for all n, p 0. A map of C*cubical symmetric spectra f *
*:E ! F
is a collection of nequivariant maps fn: En ! Fncompatible with the structure*
* maps
of E and F in the sense that there are commutative diagrams:
oen
C En ____//_En+1
C fn fn+1
fflffloen fflffl
C Fn ____//_Fn+1
There is a forgetful functor SptC ! *C * Spc0 and for all E 2 *C * Spc0 and
F 2 SptC a natural bijection SptC(E C, F ) ! *C * Spc0(E, F ). We denote by *
*^ the
monoidal product on SptC. Next we give a series of examples.
Example 4.56: If R: *C *Spc 0! *C *Spc 0is a cubical C*functor, define the i*
*nduced
functor R: SptC ! SptC by R(E)n R(En). Here nacts by applying R to the nac*
*tion
on En. The structure maps are given by the compositions C R(E)n ! R(C En) !
R(En+1). For a map E ! F between C*cubical symmetric spectra R(E ! F ) is the
sequence of maps R(En ! Fn) for n 0. In particular, using the tensor (3) and
cotensor (4) structures on pointed cubical C*spaces we obtain an adjoint funct*
*or pair:
 ^ K :SptC ____//_SptC:()Koo_
The C*cubical symmetric spectrum EK is defined in level n by hom *C*Spc0(K, E*
*n) and
the structure map Cp hom *C*Spc0(K, En) ! hom *C*Spc0(K, Ep+n) is the unique*
* map of
pointed cubical C*spaces making the diagram
Cp hom *C*Spc0(K, En) _K__//_hom*C*Spc0(K, Ep+n) K
 
 
fflffl fflffl
Cp En ______________________//_Ep+n
104
commute. This construction is natural in K and E, and for all pointed cubical s*
*ets K
and L there is a natural isomorphism
EK *Set*L= (EL)K.
Example 4.57: The cubical function complex hom SptC(E, F ) of C*cubical symmet*
*ric
spectra E and F is defined by
hom SptC(E, F )n SptC(E *n+, F ).
By definition, a 0cell of hom SptC(E, F ) is a map E ! F . A 1cell is a cubic*
*al homotopy
H :E *1+! F from H O (E ^ i0) to H O (E ^ i1) where i0 and i1 are the two inc*
*lusions
*0 ! *1. The 1cells generate an equivalence relation on SptC(E, F ) and the qu*
*otient
is ss0hom SptC(E, F ). Note there is an adjoint functor pair:
E ^ : *Set* ____//_SptC:homSptC(E,o)o_
Moreover, there exist natural isomorphisms
hom SptC(E ^ K, F ) = hom SptC(E, F K) = hom SptC(E, F )K.
Example 4.58: The internal hom of C*cubical symmetric spectra E and F are defi*
*ned
by Hom__Spt(E, F ) Hom__ (E, F ). There are natural adjunction isomorphisms
C Sym(C)
i j
SptC(E ^ F , G) = SptC E, Hom__Spt(F , G) .
C
In addition, there are natural cubical and internal isomorphisms
i j
hom SptC(E ^ F , G) = hom SptCE, Hom__Spt(F , G) ,
C
i j
Hom__Spt(E ^ F , G) = Hom__ E, Hom__ (F , G) .
C SptC SptC
If X is a pointed cubical C*space and E is a C*cubical symmetric spectrum, de*
*note
by Hom__Spt(X, E) the C*cubical symmetric spectrum with nth term the internal *
*hom
C
Hom__(X, En) with naction induced by the action on En. Define the nth structu*
*re map
oen: C Hom__(X, En) ! Hom__(X, En+1) as the adjoint of the composite of the e*
*valuation
C Ev: C Hom__(X, En) X ! C En with the structure map C En ! En+1. With
these definitions it follows that Hom__Spt(X, ) is right adjoint to ^X as end*
*ofunctors
C
of SptC.
105
Example 4.59: The loop C*cubical symmetric spectrum of E is CE Hom__Spt(C, *
*E).
C
Note that the functor C is finitely presentable.
Example 4.60: Let k > 0. The C*cubical symmetric spectrum E[1] has nth term E1*
*+n
with naction given by 1 oen 2 k+n. That is, 1 oen(1) = 1 and 1 oen(i) *
*= 1 + oen(i  1)
for i > 1. The structure map Cp E[1]n ! E[1]p+nis defined to be the composite*
* of
Cp E1+n ! Ep+1+nwith oep,1 1, where oep,1is the cyclic permutation of order *
*p + 1.
Inductively, one defines E[k] E[k  1][1].
Definition 4.61: The nth evaluation functor Evn: SptC ! *C * Spc0 sends E to E*
*n.
Its left adjoint, the shift desuspension functor Frn:*C * Spc0 ! SptC is defin*
*ed by
setting FrnE eFrnE Sym (K), where eFrnE (0, . .,.0, [n]+ E, 0, 0, . .*
*)..
Example 4.62: For a C*cubical symmetric spectrum E, the pointed cubical set of*
* maps
hom SptC(FrnC, E) is naturally nequivariant isomorphic to hom *C*Spc0(C, Evn*
*E) = En.
In effect, FrnC is the Sym (C)moduleiSym (C) [n]+jand Sym (C) []+ def*
*ines a
functor op ! SptC so that hom SptCSym (C) []+, E is the underlying C*cu*
*bical
i j
symmetric sequence of E. In particular, hom SptCE ^ (Sym (C) []+), Y is*
* the
underlying C*cubical symmetric sequence of the internal hom Hom__Spt(E, Y).
C
The point is now to derive an alternate description of the structure maps of*
* E. Let
~: Fr1C ! Fr0C be the adjoint of the identity map C ! Ev 1Fr0C and consider the
induced map Hom__Spt(~, E): Hom__ (Fr0C, E) ! Hom__ (Fr1C, E). By evaluatin*
*g in
C SptC SptC
level n we get a map En ! Hom__*C*Spc0(C, En+1) which is adjoint to the struct*
*ure map
oen: C En ! En+1. In particular, Hom__Spt(FrkC, E) is the kshift of E; its u*
*nderlying
C *
symmetric sequence is the sequence of pointed cubical C spaces Ek, E1+k, . .,.*
*En+k. . .
with n acting on En+kby_restricting_the action of n+kto the copy of n that p*
*ermutes
the first n elements of n + k. The structure maps of the kshifted spectrum of *
*E are the
structure maps oen+k:C En+k! En+k+1.
Example 4.63: The adjoint of the nth structure map oen: C En ! En+1of E yield*
*s a
map "oen:En ! CEn+1= CE[1]nand there is an induced map of C*cubical symmetric
spectra E ! CE[1].
Denote by ( 1 ) E the colimit of the diagram:
E ____//_ CE[1]___//_ 2CE[2]__//. . .
106
For every K in *Set*there is a canonical map
i j
( 1 ) E ^ K____//_( 1 ) (K E)
so that ( 1 ) is a cubical C*functor. Hence there are induced maps of cubical*
* function
complexes
i j
hom SptC(E, F_)__//_homSpt( 1 ) E, ( 1 ) F .
C
Remark 4.64: The functor Evn has a right adjoint Rn: *C * Spc0 ! SptC. Indeed,
RnE Hom__ (Sym (C), eRnL), where eRnL is the C*cubical symmetric sequence w*
*hose
nth term is the cofree nobject E nand with the terminal object in all other d*
*egrees.
We are ready to define the level model structures on SptC.
Definition 4.65: A map f :E ! F between C*cubical symmetric spectra is a level
equivalence if Ev nf :En ! Fn is a C *weak equivalence in *C * Spc0 for every
n 0. And f is a level fibration (respectively level cofibration, level acycli*
*c fibration,
level acyclic cofibration) if Evnf is a projective C*fibration (respectively p*
*rojective
cofibration, C*acyclic projective fibration, C*acyclic projective cofibration*
*) in *C *
Spc 0for every n 0. A map is a projective cofibration if it has the left lift*
*ing property
with respect to every level acyclic fibration, and an injective fibration if it*
* has the right
lifting property with respect to every level acyclic cofibration.
Let I and J denote the generating cofibrations and generating acyclic cofibr*
*ations
S
in the homotopy invariant model structure on *C * Spc0. Set IC nFrnI and
S
JC nFrnJ. We get the next result following the usual script, cf. [36, Theor*
*em 8.2].
Theorem 4.66: The projective cofibrations, the level fibrations and the level e*
*quivalences
define a left proper combinatorial model structure on SptC. The cofibrations ar*
*e generated by
IC and the acyclic cofibrations are generated by JC.
Note that Evn takes level (acyclic) fibrations to (acyclic) fibrations, so E*
*vn is a right
Quillen functor and Frnis a left Quillen functor. From [36, Theorem 8.3] we hav*
*e:
Theorem 4.67: The category SptC equipped with its level projective model struct*
*ure is a
*C * Spc0model category.
With some additional work one arrives at the following model structure.
107
Theorem 4.68: The level cofibrations, the injective fibrations and the level eq*
*uivalences define
a left proper combinatorial model structure on SptC.
The C*stable model structures are now only one Bousfield localization away;*
* the
next definition emphasizes the role of Csymmetric spectra as the stably fibra*
*nt
objects. We leave the formulation of the injective version to the reader.
Definition 4.69: A level fibrant C*cubical symmetric spectrum G is C*stably f*
*ibrant if
the adjoints Gn ! Hom__(C, Gn+1) of the structure maps of G are C*weak equival*
*ences.
A map f :E ! F is a C*stable weak equivalence if for every C*stably fibrant G*
* there
is an induced weak equivalence of pointed cubical sets
hom SptC(Qf, G): hom SptC(QF , G)__//_homSptC(QE, G).
Theorem 4.70: The projective cofibrations and C *stable weak equivalences defi*
*ne a left
proper combinatorial monoidal model structure on SptC. In this model structure *
*C ^  is a
Quillen equivalence.
Proof.The existence of the projective C*stable model structure with these prop*
*erties
follows from Theorem 4.66 and results in [36, x8]. In effect, [36, Theorem 8.8]*
* shows
the C*stably fibrant objects a.k.a. Csymmetric spectra coincides with the fi*
*brant
objects in the stable model structure on SptC obtained from [36, Theorem 8.10].*
* The
latter model structure is defined by localizing the projective level model stru*
*cture at
the adjoints Frn+1(C X) ! Frn+1(X) of the maps C X ! Evn+1FrnC where X is e*
*ither
a domain or a codomain of the set of generating projective cofibrations I*C*Sp*
*c0. *
Remark 4.71: If C0is a projective cofibrant pointed C*space weakly equivalent *
*to C,
then [36, Theorem 9.4] implies there is a Quillen equivalence between the proje*
*ctive
C *stable model structures on SptC0and SptC. By replacing pointed cubical C*s*
*paces
with pointed simplicial C*spaces, but otherwise forming the same constructions*
*, we
may define the Quillen equivalent to SptCcategory of C*simplicial symmetric sp*
*ectra.
Our results for SptC, in particular Theorem 4.70, hold verbatim in the simplici*
*al
context. We leave the formulation of Lemma 4.30 for symmetric spectra to the re*
*ader.
Relying on [34] we may first refine Theorem 4.70 to categories of modules in*
* SptC.
A monoid E has a multiplication E ^ E ! E and a unit map 1 ! E from the sphere
C *spectrum subject to the usual associativity and unit conditions. It is comm*
*utative
108
if the multiplication map is unchanged when composed with the twist isomorphism
of E ^ E. The category Mod Eof modules over a commutative monoid E is closed
symmetric with unit E.
Theorem 4.72: Suppose E is a cofibrant commutative monoid in SptC.
o The category Mod E of Emodules is a left proper combinatorial (and cellul*
*ar) symmetric
monoidal model category with the classes of weak equivalences and fibratio*
*ns defined
on the underlying category of C*cubical symmetric spectra (cofibrations d*
*efined by the
left lifting property).
o If E ! F is a C*stable weak equivalence between cofibrant commutative mon*
*oid in
SptC , then the corresponding induction functor yields a Quillen equivalen*
*ce between
the module categories Mod E and Mod F .
o The monoidal Quillen equivalence between C *cubical and C *simplicial sy*
*mmetric
spectra yields a Quillen equivalence between Mod E and modules over the im*
*age of E in
C *simplicial symmetric spectra.
The result for algebras in SptC provided by [34] is less streamlined; refini*
*ng the
following result to the level of model structures is an open problem. An Ealge*
*bra is
a monoid in Mod E.
Theorem 4.73: Suppose E is a cofibrant commutative monoid in SptC.
o The category Alg Eof Ealgebras comprised of monoids in Mod Eequipped wit*
*h the
classes of weak equivalences and fibrations defined on the underlying cate*
*gory of C*
cubical symmetric spectra (cofibrations defined by the left lifting proper*
*ty) is a semimodel
category in the following sense: (1) CM 1CM 3 holds, (2) Acyclic cofibrat*
*ions whose
domain is cofibrant in Mod Ehave the left lifting property with respect t*
*o fibrations,
and (3) Every map whose domain is cofibrant in Mod E factors functorially*
* into a
cofibration followed by an acyclic fibration and as an acyclic cofibration*
* followed by a
fibration. Moreover, cofibrations whose domain is cofibrant in Mod E are c*
*ofibrations in
Mod E, and (acyclic) fibrations are closed under pullback.
o The homotopy category Ho (Alg E) obtained from AlgE by inverting the weak *
*equiva
lences is equivalent to the full subcategory of cofibrant and fibrant Eal*
*gebras modulo
homotopy equivalence.
109
o If E ! F is a C *stable weak equivalence between cofibrant commutative m*
*onoid
in Spt C, then the corresponding induction functor yields an equivalence b*
*etween
Ho (Alg E) and Ho (Alg F).
o The monoidal Quillen equivalence between C *cubical and C *simplicial sy*
*mmetric
spectra yields an equivalence between the homotopy categories of AlgE and *
*of algebras
over the image of E in C*simplicial symmetric spectra.
o Ho (Alg E) acquires an action by the homotopy category of simplicial sets.
Remark 4.74: For semimodel structures at large see the work of Spitzweck [76].
It turns out there is a perfectly good homotopical comparison between SptC a*
*nd
C *cubical symmetric spectra. The following result is special to our situation.
Theorem 4.75: The forgetful functor induces a Quillen equivalence between the p*
*rojective
C *stable model structures on SptC and SptC.
Proof.By [36, Theorem 10.1] there exists a zigzag of Quillen equivalences betw*
*een
SptC and SptC since the cyclic permutation condition holds for C by Lemma 4.23.
The improved result stating that the forgetful and symmetrization adjoint funct*
*or
pair between SptCand SptCdefines a Quillen equivalence is rather long and invol*
*ves
bispectra and layer filtrations. We have established the required ingredients n*
*eeded
for the arguments in [40, x4.4] to go through in our setting. *
* *
The next result concerning monoidalness of the stable C*homotopy category i*
*s a
consequence of Theorem 4.75.
Corollary 4.76: The total left derived functor of the smash product ^ on SptC y*
*ields a
symmetric monoidal product ^L on the stable homotopy category SH *. In additio*
*n, the
suspension functor Fr0induces a symmetric monoidal functor
i j i j
H*, L, C_____//SH*, ^L, 1 .
With the results for SptC in hand we are ready to move deeper into our treat*
*ment
of the triangulated structure of SH *. The notion we are interested in is that*
* of a
closed symmetric monoidal category with a compatible triangulation, as introduc*
*ed
in [55]. The importance of this notion is evident from the next theorem which *
*is a
consequence of our results for SptC and specialization of the main result in [5*
*5].
110
Theorem 4.77: The Euler characteristic is additive for distinguished triangles *
*of dualizable
objects in SH *.
Next we define Euler characteristics and discuss the content of Theorem 4.77.
The dual of E in SH *is DE SH_*_(E, 1) where SH_*_(E, F ) is the derived v*
*ersion of
the internal hom of E and F in SptC, i.e. take a cofibrant replacement Ecof E a*
* fibrant
replacement F fof F and form Hom__Spt(Ec, F f). Let fflE: DE ^L E ! 1 denote t*
*he
C
evident evaluation map. There is a canonical map
*
DE ^L E _____//SH_(E, E). (51)
Recall that E is called dualizable if (51) is an isomorphism. For E dualizable *
*there is
a coevaluation map jE: 1 ! E ^L DE. The Euler characteristic O(E) of E is defin*
*ed as
the composite map
jE o fflE
1 ____//_E ^L DE___//_DE ^L E___//_1. (52)
Here, o denotes the twist map. The categorical definition of Euler characteris*
*tics
given above and the generalization reviewed below, putting trace maps in algebra
and topology into a convenient framework, was introduced by Dold and Puppe [20].
Theorem 4.77 states that for every distinguished triangle
E ____//_F___//_G___//_ S1E
of dualizable objects in SH *, the formula
O(F ) = O(E) + O(G) (53)
holds for the Euler characteristics (52) in the endomorphism ring of the sphere*
* C*
spectrum. Note that if E and F are dualizable, then so is G. As emphasized in [*
*55],
the proof of the additivity theorem for Euler characteristics makes heavily use*
* of
the stable model categorical situation, so that a generalization of the formula*
* (53) to
arbitrary triangulated categories seems a bit unlikely. In order to explain thi*
*s point
in some details we shall briefly review the important notion of a closed symmet*
*ric
monoidal category with a compatible triangulation in the sense specified by May*
* [55].
Remark 4.78: In our treatment of zeta functions of C*algebras in Section 5.6 w*
*e shall
make use of Euler characteristics in "rationalized" stable homotopy categories.
111
If E is dualizable as above and there is a "coaction" map E: E ! E ^LCE for*
* some
object CE of SH *, typically arising form a comonoid structure on CE, define th*
*e trace
tr(f ) of a self map f of E by the diagram:
jE o
1 _____//E ^L DE_______//_DE ^L E
tr(f) D(f)^LE
fflffl fflE^Lid fflffl
CE oo___1 ^L CEoo___ DE ^L E ^L CE
For completeness we include some of the basic properties of trace maps proven in
[55]. Define a map
(f, ff): (E, E) ! (F , F )
to consist of a pair of maps f :E ! F and ff: CE ! CF such that the following
diagram commutes:
E L
E _____//E ^ CE
f  f^Lff
fflffl F fflffl
F ____//_F ^L CF
Lemma 4.79: The trace satisfies the following properties, where E and F are dua*
*lizable and
E and F are given.
o If f is a self map of the sphere C*spectrum, then O(f ) = f .
o If (f, ff) is a self map of (E, E), then ff O tr(f ) = tr(f ).
i r
o If E ! F ! E is a retract, f a self map of E, and (i, ff) a map (E, E) *
*! (F , F ), then
ff O tr(f ) = tr(i O f O r).
o If f and g are self maps of E and F respectively, then tr(f ^L g) = tr(f )*
* ^L tr(g), where
E^LF = (id^L o ^L id) O ( E ^L F ) with o the transposition.
o If h: E _ F ! E _ F induces f :E ! E and g: F ! F by inclusion and retract*
*ion,
then tr(h) = tr(f ) + tr(g), where CE = CF = CE_F and E_F = E _ F .
o For every self map f , tr( S1f ) = tr(f ), where S1E = S1 E.
The following additivity theorem was shown by May in [55, Theorem 1.9].
112
Theorem 4.80: Let E and F be dualizable in SH *, E and F be given, where C = *
*CE = CF .
Let (f, id) be a map (E, E) ! (F , F ) and extend f to a distinguished triang*
*le
f g h
E _____//F____//_G___// S1E.
Assume given maps OE and _ that make the left square commute in the first of th*
*e following
two diagrams:
f g h
E ____//_F____//G___//_ S1E
OE _ ! S1OE
fflfflfflfflfflfflffflfflgh
E ____//_F____//G___//_ S1E
f g h
E _________//F_________//G_________// S1E
E  F G S1 E
fflfflf^Lifflffldg^Lifflffldh^Lidfflffl
E ^L C____//_F ^L C___//_G ^L C___//_ S1(E ^L C)
Then there are maps ! and G as indicated above rendering the diagrams commutat*
*ive and
tr(_) = tr(OE) + tr(!).
Additivity of Euler characteristics follows from this theorem by starting ou*
*t with
the data of a distinguished triangle.
The proof of Theorem 4.80 uses the fact that SH *is the homotopy category of*
* a
closed symmetric monoidal stable model structure such that the smash product ^L*
* is
compatible with the triangulated structure in the sense made precise by the axi*
*oms
(TC1)(TC5) stated in [55, x4].
The axiom (TC1) asserts there exists a natural isomorphism ff: E ^L S1 ! S1E
such that the composite map
_ff1//_ __o_//_ _ff_//_ 1
S1S1 S1 ^L S1 S1 ^L S1 S1S
is multiplication by 1, while (TC2) basically asserts that smashing or taking *
*internal
hom objects with every object of SH *preserves distinguished triangles. These a*
*xioms
are analogs of the elementary axioms (T1), (T2) for a triangulated category, an*
*d are
easily verified.
113
Next we formulate the braid axiom (TC3): Suppose there exist distinguished
triangles
f g h
E _____//F____//_G___// S1E,
and
f0 g0 0h0 0
E0 ____//_F_0__//_G___//_ S1E .
Then there exist distinguished triangles
p1 q1 L 0f^Lh0 L 0
F ^L E0 ____//_H___//_E ^ G____// S1(F ^ E ),
p2 q2 g^Lg0
1S1(G ^L G0)__//_H____//F ^L F_0__//_G ^L G0,
p3 q3 L h0^Lf0 L 0
E ^L F 0____//_H0___//_G ^ E___//_ S1(E ^ F ),
such that the following diagrams commute:
1S1(F ^L G0) E ^L E0 1(G1^L F 0) (54)
____________________________PPf^Lidnnnid^Lf0PPPPSn ______________
____________________________PPPPnnnPPP nnnn ________________*
*____________
___________________PPPnnnn PPPP nnn ________________*
*_________________
1S1(id^Lh0)_______________________________________________P1(g^Lid)PPPnn*
*nnPPPPn1(id^Lg0)nnn_11(h^Lid)________________________________
____________________________PPPPnnnnnnn PPP __S1____________*
*_______S1_______ S
_______________nnnnPPP nnn PPPP ________________*
*_____________
,,_______________vvnn((1 L vvn0 ((P _______________
F ^L_E0____________ S1(G ^ G ) E ^L F 0
______________________________CCCQQQQCC mmm _________________*
*____________
______________________________CCp1QQQCCp3mmm _________________*
*____________
______________________________CCQQQQCCmmmm ________________*
*______________
______________________________CCQQQp2CCmmm ________________*
*______________
______________________________id^Lf0CCQQQQCCCmmmmf^Lid_______*
*_______________________
______________________________CCCQQQCCmmm ________________*
*______________
__________________CCQQ((QfflfflvvCCmmmm ________________*
*_________________
g^Lid_______________________________________________CCCCid^Lg0______*
*___________________________HQ
______________________________CCmmCCCmmQQQQ ________________*
*_____________
______________________________CCC11(id^Lh0)C11(h^Lid)C______*
*________________________mmmQQQ
______________________________CCSmmmmCCQQQSQ ________________*
*______________
______________________________CCmmmmq2CCQQQQ ________________*
*______________
______________________________CCCq3mmCCCmmm_________________*
*____________________________q1QQQQQ
~~______________________________!!CC""""CC!!________________*
*______________vvmmmmfflffl((QQQ
G ^L_E0_____ F ^L F 0 E ^L G0
______________________________QQQg^Lidmmmid^Lg0QQQQmm______________*
*________
______________________________QQQQmmmQQQ mmmm ________________*
*_____________
_________________QQQmmmm QQQQ mmm ________________*
*________________
id^Lf0_____________________________________________QQQQmmmmQQQQmmmmf^Lid*
*_______________________________
__________________________h^LidQQQQQmmmmQQQQQid^Lh0mmmmm__________*
*______________________________
_,,________________________((QQvvmmmmvvmmm QQ(( ________________*
*_________
G ^L F 0 S1(E ^L E0) F ^L G0
The axiom (TC3) is more complicated than (TC1) and (TC2) in that it involves a
simultaneous use of smash products and internal hom objects, a.k.a. desuspensio*
*ns.
114
If E and E0are subobjects of F and F 0then H is typically the pushout of E ^L F*
* 0and
F ^L E0along E ^L E0, q2:H ! F ^L F 0the evident inclusion, while q1:H ! E ^L G0
and q2: H ! G ^L E0 are obtained by quotienting out by F ^L E0 and E ^L F 0
respectively. For an interpretation of axiom (TC3) in terms of Verdier's axiom *
*(T3) for
a triangulated category we refer to [55]. There is an equivalent way of formula*
*ting
axiom (TC3) which asserts the existence of distinguished triangles
r1 s1 L h0^Lg0 L 0
E ^L G0 ____//_H0___//_G ^ G___//_ S1(E ^ G ),
r2 s2 L 0S1(f^Lf0) L 0
G ^L G0)____//_H0___//_ S1(E ^ E_)_//_ S1(G ^ G ),
r3 s3 L 0g^Lh0 L 0
G ^L E0_____//H0____//F ^ G____//_ S1(G ^ E ),
such that the diagram (54) corresponding to the distinguished triangles ( 1S1*
*h, f, g)
and ( 1S1h0, f 0, g0) commutes [55, Lemma 4.7]. This axiom is called (TC3')
The additivity axiom (TC4) concerns compatibility of the maps qiand riin the
sense that there is a weak pushout and weak pullback diagram:
q2
H _____________//F ^L F 0
(q1,q3) r2
fflffl (r1,r3)fflffl
(E ^L G0) _ (G ^L E0)_____//_H0
In particular, r2Oq2 = r1Oq1+r3Oq3. Recall that weak limits and weak colimits s*
*atisfy the
existence but not necessarily the uniqueness part in the defining universal pro*
*perty
of limits and colimits respectively. We refer to [55] for the precise definitio*
*n of the
subtle braid duality axiom (TC5) involving DE, DF and DG, and the duals of the
diagrams appearing in the axioms (TC3) and (TC3'). Assuming axioms (TC1)(TC5),
additivity of Euler characteristics is shown in [55, x4].
We shall leave the straightforward formulations of the corresponding base ch*
*ange
and also the equivariant generalizations of the results in this section to the *
*interested
reader, and refer to [49] for further developments on the subject of May's axio*
*ms.
115
4.6 C *functors
The purpose of this section is to construct a convenient and highly structured *
*enriched
functor model for the stable C*homotopy category. Although there are several r*
*ecent
works on this subject, none of the existing setups of enriched functor categori*
*es as
models for homotopy types apply directly to the stable C*homotopy category. Mo*
*re
precisely, this subject was initiated with [53] and vastly generalized in [22] *
*with the
purpose of including examples arising in algebraic geometry. A further developm*
*ent
of the setup is given in [6] and [7]. Dealing effectively with enriched functor*
*s in stable
C *homotopy theory in its present state of the art requires some additional in*
*put,
which in turn is likely to provide a broader range of applications in homotopy *
*theory
at large. The algebrogeometric example of motivic functors in [23] has been pi*
*votal
in the construction of a homotopy theoretic model for motives [67], [68].
For background in enriched category theory we refer to [50]. We shall be wor*
*king
with the closed symmetric monoidal category *C * Spc of cubical C*spaces rela*
*tive
to some (essentially small) symmetric monoidal *C * Spcsubcategory f*C * Spc.
Denote by [f*C * Spc, *C * Spc] the *C * Spccategory of *C * Spcfunctors *
*from
f*C * Spc to *C * Spc equipped with the projective homotopy invariant model
structure. It acquires the structure of a closed symmetric monoidal category [*
*17].
Every object X of f*C * Spc represents a *C * Spcfunctor which we, by abuse *
*of
notation, denote by *C * Spc(X, ).
Theorem 4.81: There exists a pointwise model structure on [f*C * Spc, *C * Sp*
*c] defined
by declaring S ! T is a pointwise fibration or weak equivalence if S(X) ! T (X)*
* is so in
*C *Spc for every member X of f*C *Spc . The pointwise model structure is com*
*binatorial
and left proper. The cofibrations are generated by the set consisting of the ma*
*ps
f *C * Spc(X, )
where f runs through the generating cofibrations of *C * Spc and X through the*
* objects of
f*C * Spc. Likewise, the acyclic cofibrations are generated by the set consist*
*ing of maps of
the form
g *C * Spc(X, )
where g runs through the generating acyclic cofibrations of *C * Spc.
116
Proof.Most parts of the proof is a standard application of Kan's recognition le*
*mma
[35, Theorem 2.1.19]. The required smallness assumption in that result holds be*
*cause
[f*C * Spc, *C * Spc] is locally presentable. It is clear that the pointwise*
* weak
equivalences satisfy the twooutofthree axiom and are closed under retracts. *
*Let Wpt
denote the class of pointwise weak equivalences. An evident adjunction argument
shows
{f *C * Spc(X, )}  inj= {g *C * Spc(X, )}  inj\ Wpt.
It remains to show there is an inclusion
{g *C * Spc(X, )}  cell {f *C * Spc(X, )}  cof\ Wpt.
We note that it suffices to show maps of {f Hom__(X, Y)}cell are weak equival*
*ences in
*C *Spc , where f is a generating acyclic cofibration and X, Y are objects of *
*f*C *Spc :
Using the inclusion {g *C *Spc (X, )}cof {f *C *Spc (X, )}cof it suffi*
*ces to
show that maps of {g *C * Spc(X, )}  cellare pointwise weak equivalences. *
*Since
colimits in [f*C * Spc, *C * Spc] are formed pointwise this follows immediate*
*ly
from the statement about maps of {f Hom__(X, Y)}cell. To prove the remaining*
* claim
we shall employ the injective homotopy invariant model structure. Recall the we*
*ak
equivalences in the injective model structure coincides with the weak equivalen*
*ces
in the projective model structure, but an advantage of the former is that Hom__*
*(X, Y)
is cofibrant. Thus every map f Hom__(X, Y) as above is an acyclic cofibration*
* in the
injective homotopy invariant model structure, and hence the same holds on the l*
*evel
of cells; in particular, these maps are C*weak equivalences.
Left properness follows provided cofibrations in [f*C * Spc, *C * Spc] are*
* point
wise cofibrations in the (left proper) injective homotopy invariant model struc*
*ture. To
prove this we note that the generating cofibrations f *C * Spc(X, ) are poi*
*ntwise
cofibrations, so that every cofibration is a pointwise cofibration. *
* *
For every object X of f*C * Spc the functor  *C * Spc(X, ) is a left Q*
*uillen
functor because evaluating at X clearly preserves fibrations and acyclic fibrat*
*ions.
There is an evident pairing
*C * Spc x [f*C * Spc, *C * Spc]___//_[f*C * Spc, *C * Spc]. (55)
Lemma 4.82: The pairing (55) is a Quillen bifunctor with respect to the pointwi*
*se model
structure on [f*C * Spc, *C * Spc] and the projective homotopy invariant mode*
*l structure
on *C * Spc.
117
Proof.For the pushout product of S ! T and h *C * Spc(X, ) there is a natur*
*al
isomorphism
i j i j
(S ! T )* h *C * Spc(X, ) = (S ! T )*h *C * Spc(X, ).
i j
Since *C * Spc is monoidal by Proposition 3.89 it follows that (S ! T )*h is*
* a
cofibration and an acyclic cofibration if either S ! T or h *C * Spc(X, ), *
*and
hence h, is so. This finishes the proof because  *C * Spc(X, ) is a left Q*
*uillen
functor. *
As we have noted repeatedly the next type of result is imperative for a high*
*ly
structured model structure.
Lemma 4.83: The pointwise model structure on [f*C * Spc, *C * Spc] is monoida*
*l.
Proof.The inclusion of f*C * Spc into *C * Spc is a *C * Spcfunctor and the*
* unit
of [f*C * Spc, *C * Spc]. It is cofibrant because the unit of *C * Spc is co*
*fibrant.
The natural isomorphism
i j i j
f *C * Spc(X, ) * g *C * Spc(Y, ) = (f *g) *C * Spc(X Y, )
combined with the facts that *C * Spc is monoidal and  *C * Spc(X Y, ) *
*is a
left Quillen functor finishes the proof. *
* *
The following model structure on C*functors takes into account that f*C * *
*Spc
has homotopical content in the form of weak equivalences (as a full subcategory*
* of
*C * Spc). A homotopy C *functor is an object of [f*C * Spc, *C * Spc] whi*
*ch
preserves weak equivalences. What we shall do next is localize the pointwise mo*
*del
structure in such a way that the fibrant objects in the localized model structu*
*re are
precisely the pointwise fibrant homotopy C *functors. It will be convenient t*
*o let
X ! X0denote a generic weak equivalence in f*C * Spc.
Theorem 4.84: There is a homotopy functor model structure on [f*C *Spc , *C **
*Spc ] with
fibrant objects the pointwise fibrant homotopy C*functors and cofibrations the*
* cofibrations in
the pointwise model structure. The homotopy functor model structure is combinat*
*orial and
left proper.
118
Proof.The existence and the properties of the homotopy functor model structure
follow by observing that a pointwise fibrant C*functor S is a homotopy C*func*
*tor if
and only if the naturally induced map of simplicial sets
hom [f*C*Spc,*C*Spc](Y *C * Spc(X0, ), S)
____//_hom[f*C*Spc,*C*Spc](Y *C * Spc(X, ), S)
is a weak equivalence for every domain and codomain Y of the generating cofibra*
*tions
of *C * Spc. That is, the homotopy functor model structure is the localization*
* of the
pointwise model structure with respect to the set of map
Y *C * Spc(X0, )____//_Y *C * Spc(X, ).
*
Remark 4.85: In the above there is no need to apply a cofibrant replacement fun*
*ctor
Q in the pointwise projective model structure on *C * Spc to Y since all the d*
*omains
and codomains of the generating cofibrations of *C * Spc are cofibrant accordi*
*ng
to Lemma 3.6. However, using the same script for more general model categories
requires taking a cofibrant replacement.
We shall refer to the weak equivalences in the homotopy functor model struct*
*ure
as homotopy functor weak equivalences.
Corollary 4.86: If Y is projective cofibrant in *C * Spc then the naturally in*
*duced map
Y *C * Spc(X0, )_____//Y *C * Spc(X, )
is a homotopy functor weak equivalence.
We shall leave implicit the proofs of the following three results which the *
*interested
reader can verify following in outline the proofs of the corresponding results *
*for the
pointwise model structure.
Lemma 4.87: The pairing (55) is a Quillen bifunctor with respect to the homotop*
*y functor
model structure on [f*C * Spc, *C * Spc] and the projective homotopy invarian*
*t model
structure on *C * Spc.
119
In what follows, assume that every object of f*C * Spc is cofibrant.
Lemma 4.88: The functor  *C * Spc(X, ) is a left Quillen functor with resp*
*ect to the
homotopy functor model structure.
Proposition 4.89: The homotopy functor model structure is monoidal.
Although the work in [6] which makes a heavy use of [11] and [22] does not a*
*pply
directly to our setting, it offers an approach which we believe is worthwhile t*
*o pursue
when the model categories in question are not necessarily right proper. We shal*
*l give
such a generalization by using as input the recent paper [77].
Stanculescu [77] has shown the following result.
Theorem 4.90: Let M be a combinatorial model category with localization functor*
* fl : M !
Ho (M). Suppose there is an accessible functor F : M ! M and a natural transfor*
*mation
ff : id! F satisfying the following properties:
A 1: The functor F preserves weak equivalences.
A 2: For every X 2 M, the map F(ffX) is a weak equivalence and fl(ffF(X)) is a *
*monomorphism.
Then M acquires a left Bousfield localization with Fequivalences as weak equiv*
*alences.
The assumption that F be an accessible functor allows one to verify the hypo*
*thesis
in Smith's main theorem on combinatorial model categories:
Theorem 4.91: Suppose M is a locally presentable category, W a full accessible *
*subcategory
of the morphism category of M, and I a set of morphisms of M such that the foll*
*owing
conditions hold:
C 1: W has the threeoutoftwo property.
C 2: I  inj W.
C 3: The class I  cof\ W is closed under transfinite compositions and pushouts.
Then M acquires a cofibrantly generated model structure with classes of weak eq*
*uivalences
W, cofibrations I  cof, and fibrations (I  cof\ W)  inj.
120
Remark 4.92: The "only if" implication follows since every accessible functor s*
*atisfies
the solutionset condition, see [1, Corollary 2.45], and every class of weak eq*
*uivalences
in some combinatorial model category is an accessible subcategory of its morphi*
*sm
category.
In order to prove Theorem 4.90, note that conditions (C1) and (C2) hold, so *
*it
remains to verify (C3). This follows from the characterization of acyclic Fcof*
*ibrations
by the left lifting property described in [77, Lemma 2.4].
Suppose ()ptis an accessible fibrant replacement functor in the pointwise m*
*odel
structure on C*functors. To construct the homotopy functor model structure usi*
*ng
Theorem 4.90, we set
Fht(S) S O ()pt.
The verification of the axioms A 1 and A 2 for Fhtfollows as in [7, Proposition*
* 3.3].
Lemma 4.93: The Fhtmodel structure coincides with the homotopy functor model s*
*tructure.
Proof.The model structures have the same cofibrations and fibrant objects. *
* *
By using the same type of localization method we construct next the stable m*
*odel
structure on [f*C * Spc, *C * Spc]. We fix an accessible fibrant replacement *
*functor
()htin the homotopy functor model structure. Let C0denote the right adjoint fu*
*nctor
of  C  again denoted by C in what follows  given by cotensoring with C. No*
*te
that C0commutes with filtered colimits and homotopy colimits because C is small.
Define the endofunctor Fstof [f*C * Spc, *C * Spc] by setting
i j
Fst(S) hocolimn C0 nO (S)htO C n .
This is an accessible functor and it satisfies the axioms A 1 and A 2 by [6, Le*
*mma 8.9].
We are ready to formulate the main result in this section. Most parts of thi*
*s result
should be clear by now, and more details will appear in a revised version of the
general setup in [6] dodging the right properness assumption.
Theorem 4.94: The following holds for the stable model structure on the enriche*
*d category
of C*functors
[f*C * Spc, *C * Spc].
121
o It is a combinatorial and left proper model category.
o It is a symmetric monoidal model category.
o When f*C * Spc is the category of Cspheres then there exists a Quillen e*
*quivalence
between the stable model structures on C*functors and on cubical C*spect*
*ra.
Remark 4.95: Recall that the category of Cspheres is the full subcategory of **
*C *Spc
comprising objects X for which there exists an acyclic cofibration C n ! X in t*
*he
projective homotopy invariant model structure on *C * Spc. This is the "minima*
*l"
choice of f*C * Spc. It is not clear whether the full subcategory of finitely *
*presentable
cubical C*spaces fp*C * Spc gives a Quillen equivalent model structure [22, x*
*7.2].
This point is also emphasized in [6, x10].
122
5 Invariants
In what follows we employ C*homotopy theory to define invariants for C*algebr*
*as.
Section 5.1 introduces briefly bigraded homology and cohomology theories at lar*
*ge.
The main examples are certain canonical extensions of KKtheory, see Section 5.*
*2, and
local cyclic homology theory, see Section 5.3, to the framework of pointed simp*
*licial
C *spaces. We also observe that there is an enhanced ChernConnesKaroubi char*
*acter
between KKtheory and local cyclic theory on the level of simplicial C*spectra*
*. Section
5.5 deals with a form of Ktheory of C*algebras which is constructed using the*
* model
structures introduced earlier in this paper. This form of Ktheory is wildly di*
*fferent
from the traditional 2periodic Ktheory of C*algebras [13, II] and relates to*
* topics
in geometric topology. Finally, in the last section we discuss zeta functions *
*of C*
algebras.
5.1 Cohomology and homology theories
We record the notions of (co)homology and bigraded (co)homology theories.
Definition 5.1: oA homology theory on SH *is a homological functor SH *! Ab
which preserves sums. Dually, a cohomology theory on SH *is a homological
functor SH *op! Ab which takes sums to products.
o A bigraded cohomology theory on SH *is a homological functor from SH *op
to Adams graded graded abelian groups which takes sums to products together
with natural isomorphisms
(E)p,q ( S1E)p+1,q
and
(E)p,q ( C0(R)E)p,q+1
such that the diagram
(E)p,q________//_ ( S1E)p+1,q
 
 
fflffl fflffl
( C0(R)E)p,q+1__//_ ( CE)p+1,q+1
123
commutes for all integers p, q 2 Z.
Bigraded homology theories are defined likewise.
The category of graded abelian groups refers to integergraded objects subje*
*ct to
the Koszul sign rule a b = (1)abb a. In the case of bigraded cohomolog*
*y theories
there is a supplementary graded structure. The category of Adams graded graded
abelian groups refers to integergraded objects in graded abelian groups, but n*
*o sign
rule for the tensor product is introduced as a consequence of the Adams grading*
*. It
is helpful to think of the Adams grading as being even.
As alluded to in the introduction the bigraded cohomology and homology theor*
*ies
associated with a C*spectrum E are defined by the formulas
i j
Ep,q(F ) SH *F , Spq C0(Rq) E , (56)
and i j
Ep,q(F ) SH * 1CSpq C0(Rq), F ^L E . (57)
When E is the sphere C*spectrum 1 then (56) defines the stable cohomotopy grou*
*ps
ssp,q(F ) 1p,q(F ) and (57) the stable homotopy groups ssp,q(F ) 1p,q(F ) o*
*f F . Invoking
the symmetric monoidal product ^L on SH *there is a pairing
ssp,q(F ) ssp0,q0(F/0)/_ssp+p0,q+q0(F ^L F 0). (58)
More generally, there exists formally defined products
^: Ep,q(F ) E0p0,q0(F_0)//_(E ^L E0)p+p0,q+q0(F ^L F 0),
0,q00 // L 0 p+p0,q+q0 L 0
[: Ep,q(F ) E0p (F_)___(E ^ E ) (F ^ F ),
=: Ep,q(F ^L F 0) E0p0,q0(F_0)//_(E ^L E0)pp0,qq0(F ),
\: Ep,q(F ) E0p0,q0(F ^L_F_0)//_(E ^L E0)p0p,q0q(F 0).
When E = E0is a monoid in SH *composing the external products with E ^L E ! E
yields internal products. The internalization of the slant product \ is a type *
*of cap
product. We refer the interested reader to [55] and the references therein for*
* more
details concerning the formal deduction of the above products using function sp*
*ectra
or derived internal hom objects depending only on the structure of SH *as a sym*
*metric
monoidal category with a compatible triangulation, and the corresponding constr*
*ucts
in classical stable homotopy theory.
124
5.2 KKtheory and the EilenbergMacLane spectrum
The construction we perform in this section is a special case of "twisting" a c*
*lassical
spectrum with KKtheory: By combining the integral EilenbergMacLane spectrum
representing classical singular cohomology and homology with KKtheory we deduce
a C*symmetric spectrum which is designed to represent Khomology and Ktheory
of C*algebras. Certain parts involved in this example depend heavily on a theo*
*ry of
noncommutative motives developed in [62]. Throughout this section we work with
simplicial objects rather than cubical objects, basically because we want to em*
*phasize
the (simplicial) DoldKan equivalence.
In the companion paper [62] we construct an adjoint functor pair:
()KK : C * Spc0____//_oCo*_SpcKK0:U (59)
Here C * SpcKK0is the category of pointed simplicial C*spaces with KKtransf*
*ers,
i.e. additive functors from Kasparov's category of KKcorrespondences to simpli*
*cial
abelian groups. This is a closed symmetric monoidal category enriched in abeli*
*an
groups and the symmetric monoidal functor ()KK is uniquely determined by
(A n+)KK KK (A, ) eZ[ n+]. (60)
The right adjoint of the functor adding KKtransfers to pointed simplicial C*s*
*paces
is the lax symmetric monoidal forgetful functor U.
With these definitions there are isomorphisms
i j i j
C * SpcKK0(A n+)KK , Y= C * SpcKK0KK (A, ), Hom__(Z[ n+], Y)
i j
= C * Spc0 A, UHom__(Z[ n+], Y)
i j
= C * Spc0 A, Hom__( n+, UY)
= C * Spc0(A n+, UY).
The above definition clearly extends KKtheory to a functor on pointed simpl*
*icial
C *spaces. Moreover, for pointed simplicial C*spaces X and Y there exist cano*
*nically
induced maps
X YKK _____//XKK YKK____//_(X Y)KK .
125
In particular, when X equals the preferred suspension coordinate C = S1 C0*
*(R)
in C*homotopy theory and Y its nth fold tensor product C n, there is a map
C (C n)KK____//_(C n+1)KK .
The above defines the structure maps in the C*algebra analog of the stably fib*
*rant
EilenbergMacLane spectrum
HZ = {n O___//_eZ[Sn]}
studied in stable homotopy theory. This description clarifies the earlier remar*
*k about
twisting the integral EilenbergMacLane spectrum with KKtheory. A straightforw*
*ard
analysis based on Bott periodicity in KKtheory reveals there exist isomorphism*
*s of
simplicial C*spaces
8
>>>
:K1() eZ[Sn] n 1 odd.
The main result in this section shows the spectrum we are dealing with is st*
*ably
fibrant.
Theorem 5.2: The simplicial C*spectrum
KK {nO___//_(C n)KK }
is stably fibrant.
Proof.We shall note the constituent spaces (C n)KK of KK are fibrant in the pr*
*ojective
homotopy invariant model structure on C * Spc0. First, EvA(C n)KK is a simpli*
*cial
abelian group and hence fibrant in the model structure on Set*, so that (C n)K*
*K is
projective fibrant. For KKtheory of C*algebras, homotopy invariance holds tri*
*vially,
while matrix invariance and split exactness hold by [32, Propositions 2.11,2.12*
*]. The
same properties hold for the KKtheory of the pointed simplicial C*spaces C n *
*using
(60). These observations imply that KK is level fibrant.
It remains to show that for every A 2 C* Alg and m 0, there is an isomorp*
*hism
i j i j
H* A Sm, (C n)KK____//_H* C A Sm, (C n+1)KK . (61)
126
This reduction step follows directly from Theorem 3.97; the latter shows that t*
*he
set of isomorphism classes of cubical C*algebras of the form A Sm generates *
*the
homotopy category H*. In the next step of the proof we shall invoke the categor*
*y of
noncommutative motives M(KK ) [62]. Its underlying category Ch (KK ) consist*
*s of
chain complexes of pointed C*spaces with KK transfers. Every C*algebra A has*
* a
corresponding motive M(A) and likewise for C A. The category of motives is in*
* fact
constructed analogously to H*. Via the adjunction (59), the shift operator [] *
*on chain
complexes identifies the map (61) with
i j i j
M(KK ) M(A), Z(n)[2n  m]_____//M(KK ) M(C A)[2], Z(n + 1)[2n  m] ,
where i j
Z(1) M C0(R) [1]
is the socalled C*algebraic Bott object [62]. Thus, using theisymmetric mono*
*idalj
product in Ch(KK ), the map identifies with  Z(1) from M(KK ) M(A), Z(n)[2n*
*m]
i j
to M(KK ) M(A) Z(1), Z(n)[2n  m] Z(1) . With these results in hand, it re*
*mains
to note that  Z(1) is an isomorphism by Bott periodicity. *
* *
Lemma 5.3: There is an isomorphism
i j i j
KK p,q(A) SH * 1CA, KK Spq C0(Rq) = M(KK ) M(A), Z(q)[p] .
i *
* j
Proof.Fix some integer m p  q, q. Then SH * 1CA C m Sqpm C0(Rmq), KK
i j
is isomorphic to H* A Sm+qp C0(Rmq), (C m)KK and hence, by Theorem 5.2, to
i i j j
M(KK ) M(A) M C0(Rmq) [q  m][m  q], Z(m)[p + (m  q)] ,
or equivalently i j
M(KK ) M(A) Z(m  q), Z(m)[p] .
By Bott periodicity, tensoring with Z(m  q) implies the identification. *
* *
The simplicial C*spectrum KK is intrinsically a simplicial C*symmetric spe*
*ctrum
via the natural action of the symmetric groups on the tensor products (C n)KK .*
* It is
straightforward to show that KK is a ring spectrum in a highly structured sens*
*e.
Lemma 5.4: KK is a commutative monoid in the category of C*symmetric spectra.
127
The construction of motives and KK alluded to above works more generally for
G
G C *algebras. In particular, there exists an equivariant KKtheory ()KK fo*
*r pointed
simplicial G  C*spaces. For completeness we state the corresponding equivaria*
*nt
result:
Theorem 5.5: The simplicial G  C*spectrum
O___//_ n KKG
KK G {n (C ) }
is stably fibrant and a commutative monoid in the category of G  C*symmetric *
*spectra.
Remark 5.6: It is possible to give an explicit model of KK as a C*functor.
128
5.3 HLtheory and the EilenbergMacLane spectrum
For the background material required in this section and the next we refer to [*
*57],
[64] and [80]. In particular, when C*algebras are viewed as bornological alge*
*bras
we always work with the precompact bornology in order to ensure that local cycl*
*ic
homology satisfy split exactness, matrix invariance and homotopy invariance.
Setting
(A n+)HL HL (A, ) eZ[ n+] (62)
extends local cyclic homology HL of C*algebras to pointed simplicial C*spaces*
*. With
respect to the usual composition product in local cyclic homology, this gives r*
*ise to a
symmetric monoidal functor taking values in pointed simplicial C*spaces equipp*
*ed
with HL transfers, and an adjoint functor pair where U denotes the lax symmetr*
*ic
monoidal forgetful functor:
()HL : C * Spc0____//_oCo*_SpcHL0:U (63)
The existence of (63) is shown using (62) by following the exact same steps as *
*for the
KK theory adjunction displayed in (59). In analogy with KK we may now define *
*the
structure maps
C (C n)HL____//_(C n+1)HL
in a simplicial C*spectrum we shall denote by HL . Moreover, the natural nac*
*tion
on (C n)HL equips HL with the structure of a commutative monoid in SptC.
Using the multiplication by (2ssi)1 Bott periodicity isomorphism in local c*
*yclic
homology and an argument which runs in parallel with the proof of Theorem 5.2, *
*we
deduce that also the local cyclic homology twisted EilenbergMacLane spectrum is
stably fibrant:
Theorem 5.7: The simplicial C*spectrum
HL {nO___//_(C n)HL}
is stably fibrant and a commutative monoid in SptC.
We shall leave the formulation of the equivariant version of Theorem 5.7, fo*
*r G
totally disconnected, to the reader.
129
5.4 The ChernConnesKaroubi character
Local cyclic homology of C*algebras defines an exact, matrix invariant and hom*
*otopy
invariant functor into abelian groups. Thus the universal property of KK theo*
*ry
implies that there exists a unique natural transformation between KK theory and
local cyclic homology. Moreover, it turns out this is a symmetric monoidal natu*
*ral
transformation. By the definition of KK theory and local cyclic homology for p*
*ointed
simplicial C*spaces in terms of left Kan extensions, if follows that there exi*
*sts a unique
symmetric monoidal natural transformation
()KK ____//_()HL.
We have established the existence of the ChernConnesKaroubi character.
Theorem 5.8: There exists a ring map of simplicial C*symmetric spectra
KK ____//_HL. (64)
Since the local cyclic homology of a C*algebra A is a complex vector space,*
* the
ChernConnesKaroubi character for A induces a Clinear map
KK *(C, A) Z C ! HL *(C, A). (65)
By naturality there exists an induced map of simplicial C*symmetric spectra
KK Z C _____//HL. (66)
The constituent spaces in the spectrum on left hand side in (66) are n 7! (C n)*
*KK Z C.
Recall that the map (65) is an isomorphism provided A is a member of the socal*
*led
bootstrap category comprising the C*algebras with a KK equivalence to a membe*
*r of
the smallest class of nuclear C*algebras that contains C and is closed under c*
*ountable
colimits, extensions and KK equivalences. Equivalently, A is in the bootstrap *
*category
if and only if it is KK equivalent to a commutative C*algebra. This implies (*
*66) is a
pointwise weak equivalence when restricted to the bootstrap category.
For second countable totally disconnected locally compact groups the work of
Voigt [80] allows us to construct as in (64) an equivariant ChernConnesKaroubi
character.
130
5.5 Ktheory of C *algebras
The prerequisite to the Ktheory of C*algebras proposed here is Waldhausen's w*
*ork
on Ktheory of categories with cofibrations and weak equivalences [81]. In what
follows we tend to confine the general setup to cofibrantly generated pointed m*
*odel
categories M in order to streamline our presentation. Throughout we consider the
homotopy invariant projective model structure on *C * Spc0 and the stable model
structures on S1spectra of pointed cubical C*algebras.
A Waldhausen subcategory of M is a full subcategory N M of cofibrant objec*
*ts
including a zeroobject * with the property that if X ! Y is a map in N and X !*
* Z
`
is a map in M, then the pushout Y XW belongs to N. With notions of cofibratio*
*ns
and weak equivalences induced from the model structure on M it follows that N is
a category with cofibrations and weak equivalences. For those not familiar with*
* the
axioms for cofibrations and weak equivalences in Ktheory we state these in det*
*ail.
Definition 5.9: A category with cofibrations and weak equivalences consists of a
pointed category C equipped with two subcategories of cofibrations cofC and weak
equivalences weq C such that the following axioms hold.
Cof 1: Every isomorphism is a cofibration.
Cof 2: Every object is cofibrant. That is, * ! X is in cofC for every object X *
*of C.
Cof 3: If X ! Y is a cofibration, then the pushout of every diagram of the form
Z oo___X ____//_Y
in C exists, and the cobase change map Z ! Z [X Y is in cofC.
Weq 1: Every isomorphism is a weak equivalence.
Weq 2: The gluing lemma holds. That is, for every commutative diagram
Z oo___X _____//Y
  
  
fflffl fflfflfflffl
Z0 oo___X0 ____//_Y0
in C where the vertical maps are weak equivalences and the right hand
horizontal maps are cofibrations, Z [X Y ! Z0[X0Y0is in weq C.
131
Applying the Soconstruction we obtain a simplicial category wSo(N). Taking *
*the
nerve produces a simplicial space [n] 7! NwSn(N), and K(N) is defined by the lo*
*ops
 NwSo(N)  on the realization of this simplicial space. The algebraic Kgrou*
*ps of
N are defined as Kn(N) ssnK(N). With these definitions one finds that the abe*
*lian
group K0(N) is generated by symbols [X] where X is an object of N, subject to t*
*he
relations [X] = [Y] if there is a weak equivalence X ! Y and [Z] = [X] + [Y] if*
* there
is a cofibration sequence X ! Z ! Y.
The algebraic Ktheory spectrum of N is defined by iterating the Soconstruc*
*tion
forming  NwS(n)o(N)  for n 1. It is not difficult to verify that there is a*
* symmetric
spectrum structure on the Ktheory spectrum of N. Since we will not make use of
this important extra structure here we refer the reader to [74] for details.
The identity map on the Ktheory of the full subcategory of cofibrant object*
*s Mcof
is nullhomotopic by a version of the Eilenberg swindle. For this reason, Kth*
*eory
deals with subcategories of Mcofdefined by finiteness conditions which are typi*
*cally
not preserved under infinite coproducts. The cube lemma for cofibrant objects a*
*s in
[35, Lemma 5.2.6] implies the full subcategory fpMcofof finitely presentable ob*
*jects is
also a Waldhausen subcategory of M. With these choices of subcategories of poin*
*ted
model categories we get, by combining [21, Corollary 3.9] and [72, Theorem 3.3]*
*, the
next result.
Corollary 5.10: Every Quillen equivalence M ! N between pointed stable model ca*
*tegories
induces a weak equivalence
K(fpMcof) __~_//_K(fpNcof).
A functor between categories with cofibrations and weak equivalences is call*
*ed
exact if it preserves the zeroobject, cofibrations, weak equivalences and coba*
*se change
maps along cofibrations. An exact functor F is a Ktheory equivalence if the in*
*duced
map  NwSo(F)  is a homotopy equivalence. Every left Quillen functor induces *
*an
exact functor between the corresponding full subcategories of finitely presenta*
*ble and
cofibrant objects.
We give the following widely applicable characterization of cofibrant and fi*
*nitely
presentable objects in terms of cell complexes.
132
Lemma 5.11: Suppose the domains and codomains of the generating cofibrations IM*
* are
finitely presentable. Then an object X of M is cofibrant and finitely presentab*
*le if and only if
it is a retract of a finite IM cell complex Z of the form
* = Z0_____//Z1____//._._.//_Zn = Z,
where Zi! Zi+1is the pushout of a generating projective cofibration.
Proof.If X is a retract of a finite IM cell complex, then X is cofibrant and f*
*initely
presentable since a finite colimit of finitely presentable objects is finitely *
*presentable
and a retract of a finitely presentable object is finitely presentable.
Conversely, every cofibrant object X is a retract of the colimit X1 of an IM*
* cell
complex
* = X0 ____//_X1__//_X2___//_._._.//_Xn__//_. . .
for pushout diagrams where the top map is a coproduct of generating cofibration*
*s:
` `
~2~is~___//_~2~it~ (67)
 
 
fflffl fflffl
Xi ________//_Xi+1
If ~0i ~idefine X(~0i) Xi+1by taking the pushout along the attaching maps s~*
* ! Xi
for ~ 2 ~0ias in (67). Note that, since X is finitely presentable, there exists*
* a factoring
X ! X(~0i) ! X1 for ~0i ~ia finite subset. Likewise, since the coproduct is fi*
*nitely
`
presentable, ~2~0is~ ! Xifactors through X(~00i1) for some finite subset ~00*
*i1of ~i1.
Clearly X(~0i) is the filtered colimit of X(~0i1)(~0i) for finite ~0i1 ~i1c*
*ontaining ~00i1.
Hence the map X ! X(~0i) factors through some X(~0i1)(~0i).
Iterating this argument we find a factoring of the form
X ____//_X(~00)(~01) . .(.~0i1)(~0i)//_X1 ,
as desired. *
Remark 5.12: The last part of the proof does not require that the codomains of *
*IM are
finitely presentable.
133
Lemma 5.13: Suppose M is weakly finitely generated and X and Y are finitely pre*
*sentable
cofibrant objects in M. Let idM ! R denote the fibrant replacement functor on M*
* obtained
by applying the small object argument to the set J0M. Then for every map X ! RY*
* there exists
a finitely presentable and cofibrant object Y0and a commutative diagram with ho*
*rizontal weak
equivalences:
X _______X
 
 
fflffl fflffl
Y //___//Y0//_//_RY
Proof.Denote by Riall diagrams of the form
s0~___//_Yi1
 
 
fflffl fflffl
t0~_____//*
where s0~! t0~is a map in J0M. Then RY is the colimit of the diagram
Y = Y0 //~_//_Y1//~//_Y2//~//_././.~//_Yn//~//_. . .
for pushout diagrams
` `
s0~___//_t0~
 
 
fflffl fflffl
Yi1_____//_Yi
indexed by the set Ri. Since X is finitely presentable the map X ! RY factors t*
*hrough
some Yi. Now the trick is to observe that Yiis a filtered colimit indexed by th*
*e finite
~ ~
subsets of Riordered by inclusion. Hence there is a factoring X ! Y~i Yi RY
for some finite subset ~i Ri. Iterating this argument we find finite subsets ~*
*k Rk
~ ~ ~ ~
for 1 k i, and some factoring X ! Y0 Y~1 . . . Y~i Yi RY. By
~ 0 0
construction, there is an acyclic cofibration Y Y and Y is both finitely pre*
*sentable
and cofibrant. *
Next we recall a much weaker homotopical finiteness condition first introduc*
*ed
in special cases in [81, x2.1]. An object of M is called homotopy finitely pres*
*entable
if it is isomorphic in the homotopy category of M to a finitely presentable cof*
*ibrant
object.
134
Let hfpMcofdenote the full subcategory of M of homotopy finitely presentable
cofibrant objects. Note that X is homotopy finitely presentable if and only if*
* there
exist finitely presentable cofibrant objects Y and Z where Z is fibrant and weak
~ ~
equivalences X ! Z Y. Equivalently, there exists a finitely presentable cof*
*ibrant
object Y and a weak equivalence from X to a fibrant replacement RY of Y.
With no additional assumptions on the model structure on M one cannot expect
that hfp Mcofis a category with cofibrations and weak equivalences in the same
way as fpMcof. The only trouble is that a pushout of homotopy finitely presenta*
*ble
objects need not be homotopy finitely presentable. However, the next result whi*
*ch is
reminiscent of [72, Proposition 3.2] covers all the cases we shall consider in *
*this paper.
Lemma 5.14: Suppose M is cubical, weakly finitely generated and  *1+preserve*
*s finitely
presentable objects. Then hfpMcofis a Waldhausen subcategory of M.
Proof.Suppose X, Y and Z are homotopy finitely presentable cofibrant objects and
there are maps Z X Y. We show the pushout is homotopy finitely presentable
by constructing a commutative diagram with vertical weak equivalences:
Z oo_____X //____//Y
  
  
fflffl fflffl fflffl
ReZOoo___RX0O//__//_WOOOO
  
  
  
Z0 oo____X0 //___//_Y0
Applying the gluing lemma for cofibrant objects [28, II Lemma 8.8] or the cu*
*be
`
lemma [35, Lemma 5.2.6] we deduce that the induced map of pushouts Y XZ !
` `
Y0 X0Z0 is a weak equivalence. This shows that Y XZ is homotopy finitely
presentable provided X0, Y0 and Z0are finitely presentable cofibrant objects. N*
*ext,
existence of the middle column in the diagram where X0is finitely presentable a*
*nd
cofibrant follows because X is homotopy finitely presentable. For the same rea*
*son
there exists a weak equivalence Z ! ReZ for some finitely presentable and cofib*
*rant
object eZ. Since X maps to the fibrant object ReZ there exists a map RX0! ReZ b*
*y the
lifting axiom in M.
135
Now Lemma 5.13 shows the composite map
X0 ____//_RX0___//_ReZ
factors through some finitely presentable and cofibrant object Z0which maps by *
*an
acyclic cofibration to ReZ. It remains to construct W and Y0.
Since Y is homotopy finitely presentable there exists a finitely presentable*
* cofibrant
object eYand a weak equivalence from Y to ReY. The lifting axiom CM 4 in M yiel*
*ds a
map RX0! ReY and by Lemma 5.13 the composite X0! RX0! ReY factors through
some finitely presentable cofibrant object eY0. Using the cubical mapping cyli*
*nder
we may factor the latter map as a cofibration X0 Y0 composed with a cubical
homotopy. Note that Y0is finitely presentable and cofibrant by the assumption t*
*hat
 *1+preserves finitely presentable objects. By the factorization axiom CM 5 *
*in M
we deduce there is a cofibration RX0 W, so that W is cofibrant, and an acyclic
~ e
fibration W RY .
Finally, the weak equivalences between Y, W and Y0 follow since there exist
liftings in the following diagrams (the lower horizontal maps are weak equivale*
*nces
and so are the right vertical fibrations):
*//___//W *//___//_W
 
   
 fflffl fflffl
fflffl fflfflfflffl fflffl
Y ____//_ReY Y0 //__//_ReY
*
Note that hfpMcofcontains more fibrant objects than fpMcofsince nonconstant
fibrant objects need not be finitely presentable. The next result follows easil*
*y from a
version of Waldhausen's approximation theorem [72, Theorem 2.8] and Lemma 5.14.
For the convenience of the reader we recall the setup. A category with cofib*
*rations
and weak equivalences C is equipped with special objects if there is a full sub*
*category
C0 C and a functor Q: C ! C0 together with a natural transformation idC ! Q
such that X ! QX is a cofibration and a weak equivalence for every object X of *
*C.
Cofibrant replacement functors in model categories furnish the prime examples of
special objects.
136
Axioms App 1 and App 2 formulated below are used in the original formulation*
* of
the approximation theorem [81, Theorem 1.6.7]. In the recent slightly modified *
*version
[72, Theorem 2.8] which we shall refer to as the special approximation theorem,*
* axiom
SApp 2 replaces App 2.
Definition 5.15: Let F: C ! D be an exact functor.
App 1: F reflects weak equivalences.
App 2: Every map F(X) ! Z in D factors as F(X ! Y) for a cofibration X ! Y in C
composed with a weak equivalence F(Y) ! Z in D.
SApp 2: Suppose C is equipped with special objects. Then App 2 holds if Z is a
special object.
Proposition 5.16: Suppose M is cubical, weakly finitely generated and  *1+pr*
*eserves
finitely presentable objects. Then (fp hfp)Mcofinduces an equivalence in Kth*
*eory.
Next we turn to the examples arising in C*homotopy theory. Let (X, *C * Sp*
*c, X)
denote the retract category of a cubical C *space X. The homotopy theory of t*
*his
category was worked out in Lemma 3.111. We are interested in the Ktheory of it*
*s full
subcategory of finitely presentable cofibrant objects.
Definition 5.17: The Ktheory of a cubical C*space X is
i j
K(X) K fp(X, *C * Spccof, X) .
Lemma 5.14 implies hfp(*C * Spc)cofis a Waldhausen subcategory of *C * Spc.
By applying Proposition 5.16 we get the next result.
Lemma 5.18: The special approximation theorem applies to the inclusion
(fp hfp)(X, *C * Spccof, X).
Thus for every cubical C*space X there is an induced equivalence
i j
K(X) ____//_K hfp(X, *C * Spccof, X) .
137
In the remaining of this section we consider the Ktheory of the trivial C**
*algebra.
Adopting the work of R"ondigs [66] to our setting we show a fundamental result:
Theorem 5.19: Denote by SptS1the category of S1spectra of pointed cubical C*s*
*paces. Then
the Ktheory of the trivial C*algebra is equivalent to the Ktheory of hfp(Spt*
* S1)cof.
To begin with, consider a sequential diagram of categories with cofibrations*
* and
weak equivalence
M0 ____//_M1___//_._._.//_Mn___//_Mn+1___//_... . (68)
It is straightforward to check that the colimit M1 of (68) taken in the cate*
*gory
of small categories is a category with cofibrations and weak equivalences: A ma*
*p in
M1 is a cofibration if some representative of it is a cofibration, and likewise*
* for weak
equivalences. Moreover, with this definition the canonical functor Mn ! M1 is e*
*xact
and there is a naturally induced isomorphism
colimnSoMn ____//_SoM1
of simplicial categories with cofibrations and weak equivalences. Now specializ*
*e to
the constant sequential diagram with value fp(*C * Spc0)cofand transition map *
*the
suspension functor S1 . Let S1fp(*C * Spc0)cofdenote the corresponding coli*
*mit
with cofibrations and weak equivalences as described above.
Lemma 5.20: The canonical functor
fp(*C * Spc0)cof___//_S1fp(*C * Spc0)cof
is a Ktheory equivalence.
Proof.Since the category fp(*C *Spc 0)cofhas a good cylinder functor, the susp*
*ension
functor S1  induces a Ktheory equivalence [81, Proposition 1.6.2]. *
* *
Next we relate the target of the Ktheory equivalence in Lemma 5.20 to S1sp*
*ectra
of pointed cubical C*spaces. An object E of SptS1is called strictly finitely p*
*resentable
if En is finitely presentable in *C * Spc0 for every n 0, and there exists a*
*n integer
n(E) such that the structure maps of E are identity maps for n n(E). Every fi*
*nitely
presentable S1spectrum is isomorphic to a strictly finitely presentable one. I*
*t implies
that the inclusion functor sfp(Spt S1)cof,! fp(Spt S1)cofis an equivalence of c*
*ategories,
and therefore:
138
Lemma 5.21: The inclusion functor
sfp(Spt S1)cof__//_fp(Spt S1)cof
is a Ktheory equivalence.
Define the functor
: sfp(Spt S1)cof__//S1fp(*C * Spc0)cof
by sending E to (En, n) and f :E ! F to (fn, n) for n n(E), n(F ). Note that *
* does
not extend to a functor from fp(Spt S1)cofto S1fp(*C * Spc0)cof.
Proposition 5.22: The functor is exact and has the approximation property.
Proof.It is clear that preserves the point and also (projective) cofibrations*
* because
if E ! F is a cofibration in SptS1, then En ! Fn is a cofibration of pointed cu*
*bical
C *spaces for every n 0. The interesting part of the proof consists of showi*
*ng that
preserves stable weak equivalences. More precisely, if E ! F is a stable weak
equivalence of finitely presentable cofibrant S1spectra of pointed cubical C**
*spaces,
then En ! Fn is a C*weak equivalence for all n >> 0. This uses that the stable*
* model
structure on SptS1is weakly finitely generated.
Next we show that has the approximation property: It clearly detects weak
equivalences. For a map (E) ! (X, m) in S1fp(*C * Spc0)cofwe may choose a
representative En ! Y, and there exists an integer k such that Sk+m Y = Sk+n *
* X.
The map En ! Y factors through cyl(En ! Y) for the good cylinder functor on
fp(Spt S1)cof. Define the strictly finitely presentable cofibrant S1spectrum c*
*yl(E ! Y)
of pointed cubical C*spaces by
8
>>>E m < n
>>< m
cyl(E ! Y)m >>cyl(En ! Y) m = n
>>>
:S1 cyl(E ! Y)m1 m > n.
The structure maps of cyl(E ! Y) are given by the structure maps of E if m < n *
* 1,
by S1 En1! En ! cyl(En ! Y) if m = n  1, and by the appropriate identity map
if m n. Clearly, E ! cyl(E ! Y) is a (projective) cofibration which provides *
*the
required factoring by applying . *
139
In order to conclude that is a Ktheory equivalence, note that sfp(Spt S1)*
*cofinherits
a good cylinder functor from fp(Spt S1)cofso that Lemma 5.21 allows us to apply*
* the
approximation theorem.
Lemma 5.23: The functor
: sfp(Spt S1)cof__//S1fp(*C * Spc0)cof
is a Ktheory equivalence.
Proposition 5.16 takes care of the remaining Ktheory equivalence needed to *
*finish
the proof of Theorem 5.19.
Lemma 5.24: The inclusion functor
fp(Spt S1)cof__//hfp(Spt S1)cof
is a Ktheory equivalence.
Localization techniques imply our last result in this section.
Theorem 5.25: The Ktheory of fp(*Set*)cofis a retract of the Ktheory of fp(*C*
* * Spc0)cof
up to homotopy.
Remark 5.26: Theorem 5.25 connects C *homotopy theory to geometric topology
since the Ktheory of fp(*Set)cofis Waldhausen's A(*) or the Ktheory of the sp*
*here
spectrum [81, x2]. The spectrum A(*) is of finite type [24] and rationally equi*
*valent to
the algebraic Ktheory of the integers. We refer to [70] for a recent survey an*
*d further
references. Theorem 5.25 shows the Ktheory of the trivial C*algebra as define*
*d by
C *homotopy theory carries highly nontrivial invariants.
140
5.6 Zeta functions
Our definition of zeta functions of C*algebras is deeply rooted in algebraic g*
*eometry.
If X is a quasiprojective variety over a finite field Fqthen its HasseWeil ze*
*ta function
is traditionally defined in terms of the number of Fqnpoints of X by the formu*
*la
i tnj
iX(t) exp 1n=1#X(Fqn)__ .
n
The symmetric group n on n letters acts on the nfold product X x . .x.X and
the symmetric power Sym n(X) of X is a quotient quasiprojective variety over F*
*q.
For example, the higher dimensional affine spaces AnFq= Sym n(A1Fq) and project*
*ive
spaces PnFq= Sym n(P1Fq) arise in this way. Using symmetric powers the HasseWe*
*il
zeta function can be rewritten as the formal power series
iX(t) = 1n=0#Sym n(X)(Fq)tn.
Kapranov [45] has generalized the whole setup by incorporating multiplicative E*
*uler
characteristics with compact support in the definition of zeta functions. That *
*is, if ~
is an invariant of quasiprojective variety over Fq with values in a ring R for*
* which
~(X) = ~(X r Y) + ~(Y) for every Y X closed and ~(X x Y) = ~(X)~(Y), then the*
* zeta
function of X with respect to ~ is the formal power series
iX,~(t) 1n=0~Sym n(X)tn 2 R[[t]].
A typical choice of the ground ring is Grothendieck's K0group of varieties ove*
*r Fq
with ring structure induced by products of varieties.
In our first setup the ring R will be a K0group of the thick symmetric mono*
*idal
triangulated subcategory SH *cQof compact objects in the rationalized stable ho*
*motopy
category of C*algebras. The Eilenberg swindle explains why we restrict to comp*
*act
`
objects in SH *Q: If [E] is a class in K0(SH *) and E an infinite coproduct o*
*f copies of
` `
E, the identification E E = E implies the class of E is trivial. A crux s*
*tep toward
the definition of zeta functions of C*algebras is to note there exist symmetri*
*c powers
of compact objects giving rise to a ~structure on the ring K0(SH *cQ) with mul*
*tiplication
induced by the monoidal product on SH *. It is the ~structure that ultimately *
*allows
us to push through the definition of zeta functions by a formula reminiscent of*
* the
classical one in algebraic geometry.
141
Throughout what follows we shall tactically replace the category SH *Qwith i*
*ts
idempotent completion, turning it into a pseudoabelian symmetric monoidal Qlin*
*ear
category. Working with the idempotent completion, so that every projector acqui*
*res
an image, allows us to construct symmetric powers and wedge powers by using
Young's work on the classical representation theory of symmetric groups dealing
with idempotents and partitions. We review this next, cf. [38] and [82] for det*
*ails.
Recall the set of irreducible representations of n over Q is in bijection w*
*ith the set
of partitions of n. Moreover, there exists a set of orthogonal idempotents e *
* in the
group ring Q[ n] called the Young symmetrizer so that e = 1Q[ n]and e induc*
*es
the corresponding representation of n (up to isomorphism). For every element E*
* of
SH *Qthere is an algebra map from Q[ n] to the endomorphisms of the nfold prod*
*uct
E(n) E^L. .^.LE given by sending oe 2 n to the endomorphism Eoethat permutes *
*the
factors accordingly. That is, writing oe 2 n as a product of elementary transp*
*ositions
(i, i + 1) for 1 i < n and letting the latter act on E(n)by applying the comm*
*utativity
constrain between the ith and i + 1st factor yields a welldefined action. The *
*identity
Ee = idE(n)follows immediately, where Ee is the endomorphism of E(n)obtained
from e . Since e2 = e and we are dealing with a pseudoabelian category, the n*
*fold
product of E splits into a direct sum of the images of the idempotents Ee in the
endomorphism ring SH *Q(E(n), E(n)).
Definition 5.27: The Schur functor S of a partition of n is the endofunctor *
*of SH *Q
defined by S (E) Ee(E(n)). We say that E is Schur finite if there exists an i*
*nteger n
and a partition of n such that S (E) = 0.
The notion of Schur finiteness was introduced by Deligne [19]. See [26, A 2*
*.5]
for more background. We thank Mazza for discussions about Schur finiteness in t*
*he
algebrogeometric setting of motives [56].
Next we define, corresponding to the partition (n) of n, the nth symmetric p*
*ower
of E by
1 (n)
Sym n(E) S(n)(E) = Ee(n)(E(n)) = __ Eoe(E ).
n!oe2 n
Similarly, corresponding to the partition (1, . .,.1) of n, we define the nth w*
*edge power
of E by
1 (n)
Altn(E) S(1,...,1)(E) = Ee(1,...,1)(E(n)) = __ sgn(oe)Eoe(E *
*).
n!oe2 n
142
Remark 5.28: The corresponding notions for rationalized Chow groups introduced
in [51] is the subject of much current research on motives in algebraic geometr*
*y; in
particular, the following notions receive much attention: E is negative or posi*
*tive
finite dimensional provided Sym n(E) = 0 respectively Altn(E) = 0 for some n, a*
*nd
finite dimensional if there exists a direct sum decomposition E = E+ E where*
* E+
is positive and E is negative finite dimensional. The monoidal product of two *
*finite
dimensional objects in SH Q is finite dimensional, and the same holds for Schur*
* finite
objects. The main result in [31] shows that negative and positive finite dimens*
*ional
objects satisfy the twooutofthree property for distinguished triangles. A th*
*orough
study of Schur finite and finite dimensional objects in some rigid setting of C*
**algebras
remains to be conducted.
Recall that the Grothendieck group K0(SH *cQ) is the quotient of the free ab*
*elian
group generated by the isomorphism classes [E] of objects of SH *cQby the subgr*
*oup
generated by the elements [F ][E][G] for every distinguished triangle E ! F !*
* G.
Due to the monoidal product ^L on SH *Qthere is an induced multiplication on K0*
*(SH *cQ)
which turns the latter into a commutative unital ring.
The main result in Guletski~i's paper [30] shows that the wedge and symmetric
power constructions define opposite ~structures on K0(SH *cQ). Next we recall *
*these
notions. The interested reader can consult the papers by Atiyah and Tall [4] an*
*d by
Grothendieck [29] for further details on this subject (which is important in K*
*theory).
Let R be a commutative unital ring. Then a ~ring structure on R consists of*
* maps
~n: R ! R for every integer n 0 such that the following conditions hold:
o ~0(r) = 1 for all r 2 R
o ~1 = idR
o ~n(r + r0) = ~i(r)~j(r0)
i+j=n
A ~ring structure on R induces a group homomorphism
~t:R ____//_1 + tR[[t]];Or//_1 + n 1~n(r)tn, (69)
from the underlying additive group of R to the multiplicative group of formal p*
*ower
series in an indeterminate t over R with constant term 1, i.e. ~t(r + r0) = ~t(*
*r)~t(r0).
143
The ring 1 + tR[[t]] acquires a ~ring structure when the addition is define*
*d by
multiplication of formal power series, multiplication and ~operations are give*
*n by
universal polynomials, which in turn are uniquely determined by the identities
iYk jiYl j Yk Yl
(1 + ait) (1 + bjt) = (1 + aibjt)
i=1 j=1 i=1 j=1
and
iYk j Y i Y j
~n (1 + ait) = 1 + t aj .
i=1 S {1,...,k},#S=nj2S
Ring homomorphisms between ~rings commuting with all the ~operations are
called ~ring homomorphisms. And a ~ring R is called special if the map (69) i*
*s a
~ring homomorphism. In particular, the ~ring 1 + tR[[t]] is special, as was n*
*oted by
Grothendieck [4].
Every ring homomorphism OE: R ! 1+tR[[t]] for which OE(r) = 1+rt+higher degr*
*ee
terms defines a ~ring structure on R. The opposite ~ring structure of a ~rin*
*g R is the
~ring structure associated with the ring homomorphism OE(r) ~t(r) = ~t(r)*
*1.
In the example of K0(SH *cQ) we set
~n([E]) [Sym n(E)]. (70)
The main result in [30] shows that (70) defines a ~ring structure on K0(SH *cQ*
*). Its
opposite ~structure arises by replacing the class of Sym n(E) by the class of *
*Altn(E) in
the definition (70). The class [Sym 0(E)] is the unit in K0(SH *cQ).
Definition 5.29: Let
K0(SH *cQ)___//1 + tK0(SH *cQ)[[t]];O[E]//_1 + n 1[Sym n(E)]tn
be the ~ring homomorphism determined by the ~ring structure on K0(SH *cQ) in *
*(70).
The zeta function of E in SH *cQis the formal power series
iE(t) n 0[Sym n(E)]tn. (71)
Remark 5.30: We trust the first part of this section makes it plain that our de*
*finition
of zeta functions of C*algebras is deeply rooted in algebraic geometry.
144
The definition of zeta functions makes it clear that the following result ho*
*lds.
Lemma 5.31: If E ! F ! G is a distinguished triangle in SH *cQthen
iF = iEiG.
Definition 5.32: A power series f (t) 2 K0(SH *cQ)[[t]] is (globally) rational *
*if there exists
polynomials g(t), h(t) 2 K0(SH *cQ)[t] such that f (t) is the unique solution o*
*f the equation
g(t)x = h(t).
Corollary 5.33: If E ! F ! G is a distinguished triangle in SH *cQand two of th*
*e three zeta
functions iF, iE and iG are rational, then so is the third.
For classes [E] and [F ] in K0(SH *cQ), if the zeta functions i[E], i[F ]are*
* rational then
so is i[E] [F.] Moreover, since the ~structure on K0(SH *cQ) is special, it fo*
*llows that
i[E^LF ]= iE * iF is also rational, where the product * on the right hand side *
*of the
equation is given by the multiplication in the ~ring 1+tK0(SH *cQ)[[t]]. Thus *
*rationality
of zeta functions are closed under addition and multiplication in K0(SH *cQ). M*
*oreover,
the shift functor in the triangulated structure on SH *Qpreserves rationality. *
*The next
result follows easily from the equality
i ji j
n 0[Altn(E)](t)n n 0[Sym n(E)]tn = 1
in K0(SH *cQ)[[t]].
Lemma 5.34: o If E is negative finite dimensional, then iE(t) is a polyno*
*mial.
o If E+ is positive finite dimensional, then iE+(t)1is a polynomial.
o If E is finite dimensional, then iE(t) is rational.
For the purpose of showing a functional equation for zeta functions of C*al*
*gebras
we shall shift focus to the rationalized category M(KK )Q of KKmotives. The l*
*atter is
the homotopy category of a stable monoidal model structure on nonconnective or*
* Z
graded chain complexes Ch (KK ) of pointed C*spaces with KK transfers constr*
*ucted
similarly to the homotopy invariant model structure on C*spaces. We leave open*
* the
question of comparing zeta functions defined in terms of SH *cQand M(KK )cQ, b*
*ut note
that the properties shown so far in this section hold for K0(M(KK )cQ) and hen*
*ce iE(t),
where now E is a compact object in M(KK )Q.
145
Next we outline the construction of the category of KKmotives. More details*
* will
appear in [62].
There is a category KK of C*algebras with maps A ! B the elements of KK (A*
*, B)
and composition provided by the intersection product in KKtheory. We note that
KK is symmetric monoidal and enriched in abelian groups, but it is not abelian*
*. Let
Ch (KK ) denote the abelian category of additive functors from KK to nonconn*
*ective
chain complexes Ch of abelian groups equipped with the standard projective model
structure [35, Theorem 2.3.11]. By [22, Theorem 4.4] there exists a pointwise m*
*odel
structure on Ch (KK ). Moreover, the pointwise model structure is stable becau*
*se the
projective model structure on Ch is stable.
Next we may localize the pointwise model structure in order to construct the
exact, matrix invariant and homotopy invariant symmetric monoidal stable model
structures on Ch (KK ). With due diligence these steps can be carried out as f*
*or C*
spaces. The discussion of Euler characteristics in stable C*homotopy theory ca*
*rries
over to furnish M(KK ) with an additive invariant related to triangulated stru*
*cture.
Moreover, we may replace abelian groups with modules over some commutative
ring with unit. In particular, there is a rationalized category of motives M(K*
*K )Q
corresponding to nonconnective chain complexes of rational vector spaces. In t*
*his
category we may form symmetric powers Sym n(E) and wedge powers Altn(E) for
every object E and n 0. Finally, we note that the endomorphism ring of the un*
*it
M(KK )Q(1, 1) is a copy of the rational numbers.
Recall that E is negative or positive finite dimensional provided Sym n(E) =*
* 0
respectively Altn(E) = 0 for some n, and finite dimensional if there exists a d*
*irect sum
decomposition E = E+ E where E+ is positive and E is negative finite dimensio*
*nal.
We denote by M(KK )fdQthe thick subcategory of finite dimensional objects in M*
*(KK )Q.
The following result shows in particular that the Euler characteristics of n*
*egative
and positive finite dimensional rational motives are integers.
Proposition 5.35: oIf E is finite dimensional, then a direct sum decompositi*
*on
E = E+ E
is unique up to isomorphism.
146
o If E is negative finite dimensional, then O(E) is a nonpositive integer an*
*d the smallest
n such that Sym n(E) = 0 equals 1  O(M).
o If E is positive finite dimensional, then O(E) is a nonnegative integer an*
*d the smallest
n such that Altn(E) = 0 equals 1 + O(M).
Proof.The first part follows as in the proof of [51, Proposition 6.3], cf. [3, *
*Proposition
9.1.10], and the last part as in [3, Theorems 7.2.4, 9.1.7], cf. [18, x7.2]. *
* *
Definition 5.36: If E is finite dimensional, define
O+(E) = O(E+) and O(E) = O(E).
The first part of the next definition is standard while the second part can *
*be found
in [52, Definition 8.2.4].
Definition 5.37: In M(KK )Q we make the following definitions.
o An object E is invertible if there exists an object F and an isomorphism
E ^L F = 1.
o An object E is 1dimensional if it is either (1) negative finite dimension*
*al and
O(E) = 1, or (2) positive finite dimensional and O(E) = 1.
Note that if E is invertible, then the dual DE of E is an inverse F which is*
* unique
up to unique isomorphism. The unit object 1 is clearly 1dimensional; for a pro*
*of we
refer to [52, Example 8.2.5]. In fact, Alt2(1) = 0 since the twist map on 1 ^L *
*1 is the
identity map.
Lemma 5.38: The following hold in M(KK )Q.
o An object E is invertible if and only if it is 1dimensional.
o If E is negative finite dimensional, then Sym O(E)(E) is invertible.
o If E is positive finite dimensional, then AltO(E)(E) is invertible.
147
Proof.The first part is clear from [52, 8.2.6, 8.2.9]. It remains to show that *
*Sym O(E)(E)
and AltO(E)(E) are 1dimensional objects. This follows from the easily verified*
* formulas
i j _O(E) + n  1!
O Sym n(E) =
n
and _ !
i j O(E)
O Altn(E) =
n
found in [2, x3] and [18, x7.2]. *
* *
Next we define the determinant of a finite dimensional rational motive in an*
*alogy
with determinants of algebrogeometric motives appearing in [43, Definition 2] *
*and
[52, Definition 8.4.3].
Definition 5.39: If E is a finite dimensional rational motive, define the deter*
*minant
of E by
det(E) AltO+(E)(E+) ^L DSym O(E)(E).
The determinant of E is welldefined up to isomorphism.
According to the combinatorics of the LittlewoodRichardson numbers [54, I9]*
* the
following identities hold for the determinant, cf. [2, x3], [18, x1].
Proposition 5.40: Suppose E and F are finite dimensional objects of M(KK )Q. T*
*hen
o det(E F ) = det(E)det(F ).
o det(E ^L F ) = det(E)O(F )^L det(F )O(E).
o det(DE) = Ddet(E).
i j
o det Altn(E) = det(E)r, where r = n O(E)n=O(E).
i j
o det Sym n(E) = det(E)s, where s = n O(E)+n1n=O(E).
Lemma 5.41: o If E is negative finite dimensional, there is an isomorphism
Sym n(DE) ' Sym O(E)n(E) ^L Ddet(E)
for all n 2 [0, O(E)].
148
o If E is positive finite dimensional, there is an isomorphism
Altn(DE) = AltO(E)n(E) ^L Ddet(E)
for all n 2 [0, O(E)].
Proof.Note that Altn(DE) is isomorphic to DAltn(E). Using the evident map
Altn(E) ^L AltO(E)n(E)__//det(E) (72)
we get
i j
Altn(E) ^L AltO(E)n(E) ^L Ddet(E)____//_1. (73)
Likewise, replacing E by its dual in (72) yields
i O(E)n j n
1 ____//_Alt (E) ^L Ddet(E) ^L Alt (E). (74)
The maps in (73) and (74) satisfy the DoldPuppe duality axioms in [20], cf. [4*
*3]. The
proof for the symmetric powers is entirely similar. *
* *
We are ready to formulate and prove a functional equation for zeta functions*
* of
finite dimensional rational motives. The result follows using the same steps as*
* in the
proof of the main result in [43].
Theorem 5.42: Suppose E is finite dimensional. Then the zeta functions of E and*
* its dual
are related by the functional equation
iDE (t1) = (1)O+(E)det(E)tO(E)iE(t).
Proof.We may assume E is negative or positive finite dimensional since the zeta
function is an additive invariant of the triangulated structure on M(KK )Q. In*
* what
follows we use that symmetric powers and wedge powers define opposite special
~structures on K0(M(KK )cQ).
o If E is negative finite dimensional, then
O(E)X
iDE (t1) = Sym n(DE)tn.
n=0
149
By Lemma 5.41, the sum equals
O(E)X O(E)X
[Ddet(E)] [Sym O(E)n(E)]tn= [Ddet(E)] [Sym n(E)]tn+O(E)
n=0 n=0X
= [Ddet(E)]tO(E) [Sym n(E)]tn
n 0
= [Ddet(E)]tO(E)iE(t).
o If E is positive finite dimensional, then
O(E)X
iDE (t1)1 = [Altn(DE)](t)n.
n=0
By Lemma 5.41, the sum equals
O(E)X O(E)X
[Ddet(E)] [AltO(E)n(E)](t)n= [Ddet(E)] [Altn(E)](t)nO(E)
n=0 n=0
O(E)X
= [Ddet(E)](t)O(E) [Altn(E)](t)n
n=0
= [Ddet(E)](t)O(E)iE(t)1.
It remains to note that E and its dual DE have the same sign. *
* *
150
6 The slice filtration
In this section we construct a sequence of full triangulated subcategories
. . . 1CSH*,eff SH *,eff 1CSH*,eff . . . (75)
of the stable C*homotopy category SH *. Here, placed in degree zero is the sma*
*llest
triangulated subcategory SH *,effof SH *that is closed under direct sums and co*
*ntains
every suspension spectrum 1CX, but none of the corresponding desuspension spec*
*tra
nC 1CX for any n 1. We shall refer to SH *,effas the effective stable C*ho*
*motopy
category. If q is an integer, we define the category qCSH*,effas the smallest *
*triangulated
full subcategory of that is closed under direct sums and contains for all t  m*
* q the
C *spectra of the form i j
Frm Ss C0(Rt) X . (76)
With these definitions we deduce the slice filtration (75) which can be viewed *
*as a
Postnikov tower. The analogous construction in motivic homotopy theory is due to
Voevodsky [79]. Much of the current research in the motivic theory evolves arou*
*nd
his tantalizing set of conjectures concerning the slice filtration.
In order to make precise the meaning of "filtration" in the above we note th*
*at the
smallest triangulated subcategory of SH *that contains qCSH*,efffor every inte*
*ger q
coincides with SH *since the latter is a compactly generated triangulated categ*
*ory.
Likewise, at each level of the slice filtration we have the following result.
Lemma 6.1: The category qCSH*,effis a compactly generated triangulated categor*
*y with
the set of compact generators given by (76). Thus a map f :E ! F in qCSH*,eff*
*is an
isomorphism if and only if there is a naturally induced isomorphism
i i j j q i i j j
qCSH*,effFrmSs C0(Rt) X , E____//_ CSH *,effFrmSs C0(Rt) X , F
i j
for every compact generator Frm Ss C0(Rt) X .
The "effective" sstable C*homotopy category SH *,effsis defined similarly *
*to SH *,eff
by replacing Csuspension spectra with S1suspension spectra.
We are ready to discuss certain functors relating qCSH*,effand SH *.
151
Proposition 6.2: For every integer q the full inclusion functor
iq: qCSH*,eff___//_SH*
acquires an exact right adjoint
rq:SH *_____// qCSH*,eff
such that the following hold:
o The unit of the adjunction id! rqO iq is an isomorphism.
o By defining fq iq O rq there exists a natural transformation fq+1 ! fq a*
*nd fq+1 =
fq+1O fq.
Proof.Existence of the right adjoint rqfollows by combining Lemma 6.1 with a ge*
*neral
result due to Neeman [60, Theorem 4.1] since the inclusion functor iq is clearl*
*y exact
and preserves coproducts. The unit of the adjunction is an isomorphism because *
*iqis
a full embedding, while the last claim follows by contemplating the diagram:
rq+1 q+1 *,effiq+1*
SH *____//_ C SH //__//_SH
 fflffl 
  
  
 rq q fflffl*,effiq
SH *_____// CSH //___//SH*
*
Next we discuss some properties of fq and the counit of the adjunction.
Lemma 6.3: For every integer q and map f :E ! F in SH *the induced map fq:fqE !*
* fqF
is an isomorphism in SH *if and only if there is a naturally induced isomorphism
i i j j q i i j j
qCSH*,effFrmSs C0(Rt) X , E____//_ CSH *,effFrmSs C0(Rt) X , F
i j
for every compact generator Frm Ss C0(Rt) X .
Proof.This follows from Lemma 6.1. *
152
Lemma 6.4: For every integer q the counit of the adjunction fq ! idevaluated at*
* E yields
an isomorphism
i i j j i i j j
SH *Frm Ss C0(Rt) X , fqE____//_SH*FrmSs C0(Rt) X , E
i j q
for every compact generator Frm Ss C0(Rt) X of CSH *,eff.
Proof.This follows by using the canonical isomorphism
i i j j i i j j
SH *Frm Ss C0(Rt) X , F = SH *iqFrm Ss C0(Rt) X , F
and the adjunction between iq and rq. *
* *
Theorem 6.5: For every integer q there exists an exact functor
sn: SH *____//SH*.
There exist natural transformations fq ! sq and sq ! S1fq+1such that the follo*
*wing hold:
o For every E there exists a distinguished triangle in SH *
fq+1E____//_fqE__//_sqE__//_ S1fq+1E.
o The functor sq takes values in the full subcategory qCSH*,effof SH *.
o Every map in SH *from an object of q+1CSH*,effto sqE is trivial.
o The above properties characterizes the exact functor sq up to canonical is*
*omorphism.
Proof.Compact generatedness of the triangulated categories q+1CSH*,effand qCS*
*H*,eff
imply the above according to [61, Propositions 9.1.8, 9.1.19] and standard argu*
*ments.
*
Definition 6.6: The nth slice of E is snE.
Remark 6.7: We note that snE is unique up to unique isomorphism. If E 2 nCSH*,*
*eff
and q n, then fqE = E and the qth slice sqE of E is trivial for all q < n.
153
The functor sqis compatible with the smash product in the sense that for E a*
*nd F
there is a natural map
sq(E) ^L sq0(F_)__//sq+q0(E ^L F ).
In particular, there is a map
s0(1) ^L sq(E)___//sq(E).
This shows the zeroslice of the sphere spectrum has an important property in t*
*his
setup.
Lemma 6.8: For every integer q and map f :E ! F in SH *the induced map sq:sqE !*
* sqF
is an isomorphism in SH *if and only if there is a naturally induced isomorphism
i i j j i i j j
SH * Frm Ss C0(Rt) X , E____//_SH*FrmSs C0(Rt) X , F
i j
for every compact generator FrtqSs C0(Rt) X .
The distinguished triangles in SH *
fq+1E_____//fqE___//_sqE__//_ S1fq+1E
induce in a standard way an exact couple and a spectral sequence with input the
groups ssp,n(sqE) where the rth differential go from tridegree (p, n, q) to (p *
* 1, n, q + r).
It would be interesting to work out concrete examples of such spectral sequence*
*s.
154
References
[1] J. Ad'amek and J. Rosick'y. Locally presentable and accessible categories,*
* volume 189
of London Mathematical Society Lecture Note Series. Cambridge University P*
*ress,
Cambridge, 1994.
[2] Y. Andr'e. Motifs de dimension finie (d'apr`es S.I. Kimura, P. O'Sulliva*
*n. .)..
Ast'erisque, (299):Exp. No. 929, viii, 115145, 2005. S'eminaire Bourbaki*
*. Vol.
2003/2004.
[3] Y. Andr'e, B. Kahn, and P. O'Sullivan. Nilpotence, radicaux et struct*
*ures
mono"idales. Rend. Sem. Mat. Univ. Padova, 108:107291, 2002.
[4] M. F. Atiyah and D. O. Tall. Group representations, ~rings and the J
homomorphism. Topology, 8:253297, 1969.
[5] C. Barwick. On (enriched) left Bousfield localizations of model categorie*
*s.
Preprint, arXiv 0708.2067.
[6] G. Biedermann. Lstable functors. Preprint, arXiv 0704.2576.
[7] G. Biedermann, B. Chorny, and O. R"ondigs. Calculus of functors and model
categories. Adv. Math., 214(1):92115, 2007.
[8] B. Blackadar. Ktheory for operator algebras, volume 5 of Mathematical Sc*
*iences
Research Institute Publications. Cambridge University Press, Cambridge, se*
*cond
edition, 1998.
[9] B. Blackadar. Operator algebras, volume 122 of Encyclopaedia of Mathematic*
*al Sci
ences. SpringerVerlag, Berlin, 2006. Theory of C*algebras and von Neumann
algebras, Operator Algebras and Noncommutative Geometry, III.
[10] F. Borceux. Handbook of categorical algebra. 2, volume 51 of Encyclopedia *
*of Mathe
matics and its Applications. Cambridge University Press, Cambridge, 1994. *
*Cate
gories and structures.
[11] A. K. Bousfield and E. M. Friedlander. Homotopy theory of spaces, spectr*
*a, and
bisimplicial sets. In Geometric applications of homotopy theory (Proc. Con*
*f., Evanston,
155
Ill., 1977), II, volume 658 of Lecture Notes in Math., pages 80130. Sprin*
*ger, Berlin,
1978.
[12] D. C. Cisinski. Les pr'efaisceaux comme mod`eles des types d'homotopie.
Ast'erisque, (308):xxiv+390, 2006.
[13] A. Connes. Noncommutative geometry. Academic Press Inc., San Diego, CA, 19*
*94.
[14] A. Connes, C. Consani, and M. Marcolli. Noncommutative geometry and mo
tives: the thermodynamics of endomotives. Adv. Math., 214(2):761831, 2007.
[15] A. Connes and N. Higson. D'eformations, morphismes asymptotiques et K
th'eorie bivariante. C. R. Acad. Sci. Paris S'er. I Math., 311(2):101106,*
* 1990.
[16] J. Cuntz. A general construction of bivariant Ktheories on the category o*
*f C*
algebras. In Operator algebras and operator theory (Shanghai, 1997), volum*
*e 228 of
Contemp. Math., pages 3143. Amer. Math. Soc., Providence, RI, 1998.
[17] B. Day. On closed categories of functors. In Reports of the Midwest Cate*
*gory
Seminar, IV, Lecture Notes in Mathematics, Vol. 137, pages 138. Springer,*
* Berlin,
1970.
[18] P. Deligne. Cat'egories tannakiennes. In The Grothendieck Festschrift, V*
*ol. II,
volume 87 of Progr. Math., pages 111195. Birkh"auser Boston, Boston, MA, *
*1990.
[19] P. Deligne. Cat'egories tensorielles. Mosc. Math. J., 2(2):227248, 2002. *
*Dedicated
to Yuri I. Manin on the occasion of his 65th birthday.
[20] A. Dold and D. Puppe. Duality, trace, and transfer. In Proceedings of the *
*Inter
national Conference on Geometric Topology (Warsaw, 1978), pages 81102, Wa*
*rsaw,
1980. PWN.
[21] D. Dugger and B. E. Shipley. Ktheory and derived equivalences. Duke Math.*
* J.,
124(3):587617, 2004.
[22] B. I. Dundas, O. R"ondigs, and P. A. Ostvaer. Enriched functors and stable*
* homo
topy theory. Doc. Math., 8:409488 (electronic), 2003.
156
[23] B. I. Dundas, O. R"ondigs, and P. A. Ostvaer. Motivic functors. Doc. Mat*
*h.,
8:489525 (electronic), 2003.
[24] W. G. Dwyer. Twisted homological stability for general linear groups. Ann.*
* of
Math. (2), 111(2):239251, 1980.
[25] W. G. Dwyer and J. Spalin'ski. Homotopy theories and model categories. In
Handbook of algebraic topology, pages 73126. NorthHolland, Amsterdam, 19*
*95.
[26] D. Eisenbud. Commutative algebra, with a view toward algebraic geometry, v*
*olume
150 of Graduate Texts in Mathematics. SpringerVerlag, New York, 1995.
[27] S. I. Gelfand and Y. I. Manin. Methods of homological algebra. Springer Mo*
*nographs
in Mathematics. SpringerVerlag, Berlin, second edition, 2003.
[28] P. G. Goerss and J. F. Jardine. Simplicial homotopy theory, volume 174 of *
*Progress
in Mathematics. Birkh"auser Verlag, Basel, 1999.
[29] A. Grothendieck. La th'eorie des classes de Chern. Bull. Soc. Math. France*
*, 86:137
154, 1958.
[30] V. Guletski~i. Zeta functions in triangulated categories. Preprint, arXiv *
*0605040.
[31] V. Guletski~i. Finitedimensional objects in distinguished triangles. J.*
* Number
Theory, 119(1):99127, 2006.
[32] N. Higson. A characterization of KKtheory. Pacific J. Math., 126(2):2532*
*76, 1987.
[33] P. S. Hirschhorn. Model categories and their localizations, volume 99 of M*
*athematical
Surveys and Monographs. American Mathematical Society, Providence, RI, 200*
*3.
[34] M. Hovey. Monoidal model categories. Preprint, arXiv 9803002.
[35] M. Hovey. Model categories, volume 63 of Mathematical Surveys and Monograp*
*hs.
American Mathematical Society, Providence, RI, 1999.
[36] M. Hovey. Spectra and symmetric spectra in general model categories. J. Pu*
*re
Appl. Algebra, 165(1):63127, 2001.
157
[37] M. Hovey, B. E. Shipley, and J. Smith. Symmetric spectra. J. Amer. Math. S*
*oc.,
13(1):149208, 2000.
[38] G. James and A. Kerber. The representation theory of the symmetric group, *
*volume 16
of Encyclopedia of Mathematics and its Applications. AddisonWesley Publis*
*hing
Co., Reading, Mass., 1981. With a foreword by P. M. Cohn, With an introduc*
*tion
by Gilbert de B. Robinson.
[39] J. F. Jardine. Representability for model categories. Preprint.
[40] J. F. Jardine. Motivic symmetric spectra. Doc. Math., 5:445553 (electroni*
*c), 2000.
[41] J. F. Jardine. Categorical homotopy theory. Homology, Homotopy Appl., 8(1)*
*:71144
(electronic), 2006.
[42] M. Joachim and M. W. Johnson. Realizing Kasparov's KKtheory groups as the
homotopy classes of maps of a Quillen model category. In An alpine antholo*
*gy of
homotopy theory, volume 399 of Contemp. Math., pages 163197. Amer. Math. *
*Soc.,
Providence, RI, 2006.
[43] B. Kahn. Motivic zeta functions of motives. Preprint, arXiv 0606424.
[44] T. Kandelaki. Algebraic Ktheory of Fredholm modules and KKtheory. J. Hom*
*o
topy Relat. Struct., 1(1):195218 (electronic), 2006.
[45] M. Kapranov. The elliptic curve in the sduality theory and Eisenstein ser*
*ies for
KacMoody groups. Preprint, arXiv 0001005.
[46] G. G. Kasparov. Hilbert C*modules: theorems of Stinespring and Voiculescu*
*. J.
Operator Theory, 4(1):133150, 1980.
[47] G. G. Kasparov. The operator Kfunctor and extensions of C*algebras. Izv.*
* Akad.
Nauk SSSR Ser. Mat., 44(3):571636, 719, 1980.
[48] G. G. Kasparov. Equivariant KKtheory and the Novikov conjecture. Invent.
Math., 91(1):147201, 1988.
[49] B. Keller and A. Neeman. The connection between May's axioms for a trian
gulated tensor product and Happel's description of the derived category of*
* the
quiver D4. Doc. Math., 7:535560 (electronic), 2002.
158
[50] G. M. Kelly. Basic concepts of enriched category theory. Repr. Theory Appl*
*. Categ.,
(10):vi+137 pp. (electronic), 2005. Reprint of the 1982 original [Cambridg*
*e Univ.
Press, Cambridge; MR0651714].
[51] S.I. Kimura. Chow groups are finite dimensional, in some sense. Math. Ann*
*.,
331(1):173201, 2005.
[52] S.I. Kimura. A note on finite dimensional motives. In Algebraic cycles *
*and
motives. Vol. 2, volume 344 of London Math. Soc. Lecture Note Ser., pages *
*203213.
Cambridge Univ. Press, Cambridge, 2007.
[53] M. Lydakis. Simplicial functors and stable homotopy theory. Preprint, Ho*
*pf
Topology Archive.
[54] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathemat*
*ical
Monographs. The Clarendon Press Oxford University Press, New York, second
edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publica*
*tions.
[55] J. P. May. The additivity of traces in triangulated categories. Adv. Math.*
*, 163(1):34
73, 2001.
[56] C. Mazza. Schur functors and motives. KTheory, 33(2):89106, 2004.
[57] R. Meyer. Local and analytic cyclic homology, volume 3 of EMS Tracts in Ma*
*thematics.
European Mathematical Society (EMS), Zu"rich, 2007.
[58] J. Milnor. The geometric realization of a semisimplicial complex. Ann. of*
* Math.
(2), 65:357362, 1957.
[59] F. Morel and V. Voevodsky. A1homotopy theory of schemes. Inst. Hautes 'Et*
*udes
Sci. Publ. Math., (90):45143 (2001), 1999.
[60] A. Neeman. The Grothendieck duality theorem via Bousfield's techniques and
Brown representability. J. Amer. Math. Soc., 9(1):205236, 1996.
[61] A. Neeman. Triangulated categories, volume 148 of Annals of Mathematics St*
*udies.
Princeton University Press, Princeton, NJ, 2001.
[62] P. A. Ostvaer. Noncommutative motives. In preparation.
159
[63] G. K. Pedersen. Pullback and pushout constructions in C*algebra theory. *
* J.
Funct. Anal., 167(2):243344, 1999.
[64] M. Puschnigg. Excision in cyclic homology theories. Invent. Math., 143(2):*
*249
323, 2001.
[65] D. G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43.
SpringerVerlag, Berlin, 1967.
[66] O. R"ondigs. Algebraic Ktheory of spaces in terms of spectra. Diplomarb*
*eit,
University of Bielefeld.
[67] O. R"ondigs and P. A. Ostvaer. Motives and modules over motivic cohomology.
C. R. Math. Acad. Sci. Paris, 342(10):751754, 2006.
[68] O. R"ondigs and P. A. Ostvaer. Modules over motivic cohomology. Adv. Math.,
219(2):689727, 2008.
[69] J. Rosenberg. The role of Ktheory in noncommutative algebraic topology. *
* In
Operator algebras and Ktheory (San Francisco, Calif., 1981), volume 10 of*
* Contemp.
Math., pages 155182. Amer. Math. Soc., Providence, R.I., 1982.
[70] J. Rosenberg. Ktheory and geometric topology. In Handbook of Ktheory. Vo*
*l. 1,
2, pages 577610. Springer, Berlin, 2005.
[71] J. Rosick'y. Generalized Brown representability in homotopy categories. Th*
*eory
Appl. Categ., 14:no. 19, 451479 (electronic), 2005.
[72] S. Sagave. On the algebraic Ktheory of model categories. J. Pure Appl. Al*
*gebra,
190(13):329340, 2004.
[73] C. Schochet. Topological methods for C*algebras. III. Axiomatic homology.
Pacific J. Math., 114(2):399445, 1984.
[74] S. Schwede. An untitled book project about symmetric spectra. In preparati*
*on.
[75] S. Schwede and B. E. Shipley. Algebras and modules in monoidal model cate
gories. Proc. London Math. Soc. (3), 80(2):491511, 2000.
160
[76] M. Spitzweck. Operads, algebras and modules in general model categories.
Preprint, arXiv 0101102.
[77] A. E. Stanculescu. Note on a theorem of Bousfield and Friedlander. Topol*
*ogy
Appl., 155(13):14341438, 2008.
[78] R. W. Thomason and T. Trobaugh. Higher algebraic Ktheory of schemes and
of derived categories. In The Grothendieck Festschrift, Vol. III, volume 8*
*8 of Progr.
Math., pages 247435. Birkh"auser Boston, Boston, MA, 1990.
[79] V. Voevodsky. Open problems in the motivic stable homotopy theory. I. In
Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), volum*
*e 3 of Int.
Press Lect. Ser., pages 334. Int. Press, Somerville, MA, 2002.
[80] C. Voigt. Equivariant local cyclic homology and the equivariant ChernConn*
*es
character. Doc. Math., 12:313359 (electronic), 2007.
[81] F. Waldhausen. Algebraic Ktheory of spaces. In Algebraic and geometric to*
*pology
(New Brunswick, N.J., 1983), volume 1126 of Lecture Notes in Math., pages *
*318419.
Springer, Berlin, 1985.
[82] H. Weyl. The classical groups. Princeton Landmarks in Mathematics. Princet*
*on
University Press, Princeton, NJ, 1997. Their invariants and representatio*
*ns,
Fifteenth printing, Princeton Paperbacks.
Department of Mathematics, University of Oslo, Norway.
email: paularne@math.uio.no
161