SMASH PRODUCTS AND -SPACES
Manos Lydakis
February 23, 1998
1. INTRODUCTION
In this paper we construct a symmetric monoidal smash product of -spaces modeli*
*ng
the smash product of connective spectra. For the corresponding theory of ring-s*
*pectra,
we refer the reader to [Sch].
We give a brief review of -spaces. If n is a non-negative integer, the point*
*ed set n+
is the set {0; : :;:n} with 0 as the basepoint. The category opis the full subc*
*ategory of
the category of pointed sets, with objects all n+. The category GS of -spaces i*
*s the full
subcategory of the category of functors from opto pointed simplicial sets, with*
* objects
all F such that F (0+) ~=0+. A map of -spaces is a strict weak equivalence if i*
*t gives a
weak equivalence of simplicial sets for every n+ in op. Segal [Se] introduced -*
*spaces and
showed that they give rise to a homotopy category equivalent to the homotopy ca*
*tegory
of connective spectra. Bousfield and Friedlander [BF] later provided model cat*
*egory
structures for -spaces. We follow the terminology of [BF ] (only the class of *
*special
-spaces, and its subclass of very special -spaces, are considered in [Se], wher*
*e they are
called "-spaces" and "-spaces A such that A(1) has a homotopy inverse"). Segal *
*proved
that the homotopy category of connective spectra is equivalent to the category *
*with
objects the very special -spaces and morphisms obtained by inverting the strict*
* weak
equivalences of -spaces. Bousfield and Friedlander proved that the category obt*
*ained
from all of GS by inverting the stable weak equivalences (definition 5.4) is ag*
*ain equivalent
to the homotopy category of connective spectra, in such a way that very special*
* -spaces
correspond to omega-spectra.
One advantage of the approach of [BF ] in relating -spaces and spectra, is t*
*hat GS
has the structure of a closed simplicial model category and this structure is r*
*elated by
a Quillen-pair of functors to a similar one that the category of spectra has. T*
*he main
results of this paper can be summarized as follows: The smash product of -space*
*s is
compatible with the model category structures of [BF ], and corresponds to the *
*smash
product of spectra under the equivalence of homotopy categories of [BF ]. We r*
*emark
however that the proofs of the main results of this paper use neither [BF ] nor*
* model
categories (the main prerequisites for them are some basic facts about bisimpli*
*cial sets,
although sections 2, 3, 4, and 6, do not use much more simplicial theory than t*
*he definition
of simplicial objects). The parts of this paper labeled "remark" or "example" m*
*ay have
more prerequisites, and arguments there contain, in general, less details than *
*the rest of
the paper. On the other hand, the rest of the paper (which contains all the mai*
*n results)
is independent from these parts. The symbol " __|_|" denotes either the end of *
*the proof
or that the proof is easy and omitted.
1
This paper is organized as follows. In section 2 we define the smash product*
* of -
spaces, and prove that it gives GS the structure of a symmetric monoidal catego*
*ry. In
section 3 we study the filtration of a -space by its skeleta (definition 3.1). *
*The skeleta
of a -space are also considered in [BF ]. We prove that they are the stages of *
*a filtration
(i.e., that every skeleton injects in the next one), and describe in theorem 3.*
*10 how one
stage of the filtration is built from the next by attaching representable -spac*
*es.
The results of section 3 are useful in reducing the proof of whether a certa*
*in property
is shared by all -spaces, to proving that the representable -spaces have that p*
*roperty.
Questions about representable -spaces are usually very easy to answer, because *
*the
representable -spaces are very well-known functors. There is one of them for ea*
*ch non-
negative integer n, namely the functor n that takes the pointed set X to the n-*
*fold
cartesian product of X with itself. Reduction to representable -spaces can be u*
*sed to
give easy proofs of certain interesting facts, some of which seem to be more we*
*ll-known
than others. An example of the first kind (see proposition 5.20) is that every *
*-space
is a homotopy functor when extended degreewise (definition 2.10). An example o*
*f the
second kind (see proposition 5.21) is that, for any -space F and any connected *
*space X,
the assembly map from S1^ F (X) to F (S1^ X) (definition 2.12) is as connected *
*as the
suspension of S1^ X ^X (in fact, one may replace S1 by any connected space).
Reduction to representable -spaces is used in sections 4 and 5, to prove tha*
*t smashing
by a cofibrant -space (definition 3.1) preserves cofibrations (see theorem 4.6,*
* which
is actually a little stronger), strict weak equivalences (theorem 5.1), and sta*
*ble weak
equivalences (theorem 5.12), and that the smash product of -spaces corresponds *
*to the
smash product of spectra if one of the factors is cofibrant (theorem 5.11). In *
*the last
section we prove that our skeleta and our cofibrations agree with those in [BF *
*] (this is
not needed in the rest of the paper).
There is an interesting subclass of the class of all cofibrant -spaces, cons*
*isting of the
Q-cofibrant -spaces (see [Sch], especially Lemma A.3 and the paragraph immediat*
*ely
preceding it). The fact that the cofibrant -spaces are the cofibrant objects in*
* strict and
stable model category structures for GS [BF ], has its counterpart in the fact *
*that the
Q-cofibrant -spaces are also the cofibrant objects in strict and stable model c*
*ategory
structures for GS. The strict structure is a special case of a general constru*
*ction of
Quillen [Q ], and the stable structure is constructed by Schwede [Sch]. Being Q*
*-cofibrant
is a relatively strong condition on a -space. For example, if the -space is als*
*o discrete,
then it must be a sum of representable -spaces. On the other hand, this implies*
* that
the strict and stable notions of "Q-fibrant" are weaker than the corresponding *
*notions
in [BF ], and this makes the "Q-model category structures" ideal for the applic*
*ations of
this smash product in [Sch]. We work with the weaker notions of "cofibrant" in*
* this
paper, partly because we obtain more general theorems this way, and partly beca*
*use
there are interesting cofibrant -spaces which are not Q-cofibrant (for example,*
* certain
-spaces that Segal associates to categories with finite sums; see example 3.5).*
* There are
many more examples of cofibrant -spaces which are not Q-cofibrant (see example *
*3.3).
The definition and the basic properties of this smash product have been disc*
*overed
independently by Jeff Smith (unpublished). Most of the results of section 2 hav*
*e been
known to category-theorists for a long time and in greater generality, cf. [D ].
2. SMASH PRODUCTS AND FUNCTION OBJECTS
The following conventions will save us a lot of writing. A space is a pointed s*
*implicial set.
A space is discrete if its underlying simplicial set is constant. A map always *
*preserves
all the available structure. For example, if X and Y are spaces, then a map X !*
* Y is
the same thing as a pointed simplicial map X ! Y .
2
We choose a smash product functor from opxop to op, extend it to one for all
pointed sets, and write X ^Y for the smash product of the pointed sets X and Y*
* .
Such a functor exists, because op is equivalent to the category of finite point*
*ed sets.
We write X+ for the space obtained from the simplicial set X by attaching a dis*
*joint
basepoint. If F is a -space and Y is a space, the -space F ^Y takes n+ to F (n+*
*) ^Y .
We write S* for the category of spaces. We write C(A; B) for the set of morphis*
*ms from A
to B in a category C.
2.1. Definition: We introduce function objects in GS . The first one is t*
*he
pointed set GS(F; F 0). The second is the space hom (F; F 0) that has GS(F ^(q *
* )+; F 0)
as its pointed set of q-simplices. The third function object is the -space Hom *
*(F; F 0)
that takes m+ to hom (F; F 0(m+ ^ )), where we define F 0(m+ ^ ) evaluated at n*
*+ to
be F 0(m+ ^n+).
2.2. Theorem: There exists a functor from GS xGS to GS, whose value at (F;*
* F 0)
we call the smash product F ^F 0of F and F 0, as well as an isomorphism
GS (F ^F 0; F 00) ~=GS(F; Hom (F 0; F)00)
natural in the -spaces F , F 0, and F 00.
Proof. We need the category GGS of x-spaces. It is the category of functo*
*rs
from opx op to S* taking (0+; 0+) to a point. Given F 00in GS , let RF 00in GGS
take (m+; n+) to F 00(m+ ^n+). The external smash product F e^F 0of the -space*
*s F
and F 0takes (m+; n+) to F (m+) ^F 0(n+). Note that
GS(F; Hom (F 0; F)00)~=GGS(F e^F 0; RF 00):
This isomorphism is similar to the one in
S*(X; hom*(Y; Z)) ~=S*(X ^Y; Z);
where the space hom*(X; Y )has S*(X ^(q )+; Y ) as its pointed set of q-simpli*
*ces for X
and Y spaces. Thus, if R has a left adjoint L: GGS ! GS, then we may take F ^F *
*0to
be L(F e^F 0). But L does exist: Given F 000in GGS let LF 000(n+) be the colimi*
*t over all
i+ ^j+ ! n+ of F 000(i+; j+ ) (to see that LF 000(0+) is a point, note that the*
* identity map
of 0+ is terminal among the maps of the form i+ ^j+ ! 0+). *
*__|_|
2.3. Corollary: There exist isomorphisms
hom (F ^F 0; F 00) ~=hom (F; Hom (F 0; F 00))
Hom (F ^F 0; F 00) ~=Hom (F; Hom (F 0; F 00))
natural in the -spaces F , F 0, and F 00.
Proof. The first isomorphism is similar to the one in theorem 2.2. For the s*
*econd,
use the first and the isomorphism Hom (F 0; F 00)(n+ ^ ) ~=Hom (F 0; F 00(n+ ^ *
*)): __|_|
2.4. Remark: The smash product of the -spaces F and F 0is the universal -
space F 00with a map of x-spaces F e^F 0! RF 00. We indicate below a similarity
between this definition and the definition of the tensor product of abelian gro*
*ups. We
first recall from [Se] that abelian groups embed as a full subcategory of -spac*
*es.
Fix an abelian group A. It determines a -space HA, where HA(n+) = A eZ[n+]
and eZ[n+] is the reduced free abelian group on the pointed set n+. Fix another*
* abelian
3
group A0. A map of sets f : A ! A0is a group homomorphism if and only if there *
*is a
map of -spaces
ef: HA!HA0
such that ef1+= f. Further, this efis unique, if it exists.
The above observation expresses linearity in terms of -spaces. The following*
* observa-
tion does this for bilinearity. Fix a third abelian group A00. A map of sets g:*
* A ^A0!A00
is bilinear if and only if there is a map of x-spaces
eg: HAe^HA0 !RHA00
such that eg(1+;1+)= g. Further, this egis unique, if it exists.
2.5. Definition: Given n+ in op, define the -space n by n(m+) = op(n+; m+).
2.6. Lemma: If n is a non-negative integer and F is a -space, then hom (n; *
*F ) is
isomorphic to F (n+).
Proof. This follows from the Yoneda lemma, the isomorphism
opxop n q
GS(n ^(q )+; F ) ~=SET ( x ; F )
(where we wrote F also for the associated functor from opxop to sets), and the *
*fact
that (n+; [q] ) represents nxq . __*
*|_|
2.7. Definition: The -space S is the inclusion of opin the category of spac*
*es (we
identify a pointed set with its associated discrete space).
2.8. Proposition: The -spaces S and 1 are isomorphic. __|*
*_|
2.9. Lemma: The above smash product is associative and commutative, up to n*
*atural
isomorphism, and S acts as a unit, up to natural isomorphism.
Proof. We claim that F ^S ~=F . This follows from theorem 2.2, proposition*
* 2.8,
and the fact that Hom (1; F 00) is isomorphic to F 00(this is essentially a spe*
*cial case of
lemma 2.6).
To check that F ^F 0~=F 0^F , we check that GS(F ^F 0; F 00) ~=GS(F 0^F; F 0*
*0) for all
F 002 obGS . This follows from the isomorphisms
GGS(F e^F 0; RF 00) ~=GGS(F 0e^F; RF 00O T )
and RF 00~=RF 00O T , where T is the obvious involution of opxop.
Finally, we compare both (F ^F 0) ^F 00and F ^(F 0^F 00) to a more symmetric*
* -
space F ^F 0^F 00, where the space (F ^F 0^F 00)(n+) is defined to be the colim*
*it of
F (i+) ^F 0(j+ ) ^F 00(k+ ) over i+ ^j+ ^k+ ! n+ (to simplify the exposition, w*
*e write as
if the smash product of pointed sets was associative instead of associative up *
*to unique
natural isomorphism). The isomorphism
GS((F ^F 0) ^F 00; F 000) ~=GS(F ^F 0^F 00; F 000)
is obtained by observing that
GS ((F ^F 0) ^F 00; F~000)=GS(F ^F 0; Hom (F 00; F 000))
~= GGS(F e^F 0; R Hom (F 00; F 000))
~= GGGS (F e^F 0e^F 00; R0F 000)
4
where GGGS is the category of functors from opxopxop to S* taking (0+; 0+; 0+)
to 0+, and R0F 000takes (i+; j+ ; k+ ) to F 000(i+ ^j+ ^k+ ). The second isomo*
*rphism is
similar. __|_|
2.10. Definition: Given a -space F , we obtain an extended functor from spa*
*ces
to spaces, which we again denote by F , as follows. If X is any pointed set, de*
*fine F (X)
as the colimit of F (n+) over n+ ! X. If X is any space, define F (X) as the di*
*agonal
of the pointed bisimplicial set which, evaluated at [p], yields F (Xp). We say *
*that F is
extended degreewise.
2.11. Convention: Consider all functors from spaces to spaces that satisfy *
*the
following three conditions. First, they are determined by their behavior on dis*
*crete spaces
by using degreewise evaluation and diagonalization, just as in the preceding de*
*finition.
Second, they commute with filtered colimits. Third, they take one-point spaces *
*to one-
point spaces. These functors are precisely the functors from spaces to spaces *
*which
are isomorphic to (the degreewise extension of) a -space. In fact, given any tw*
*o such
functors F and F 0, restriction to op gives a bijection between the degreewise *
*extended
maps from F to F 0and the maps of -spaces from the restriction of F to the rest*
*riction
of F 0. This allows us to identify any such functor with the degreewise extensi*
*on of its
associated -space, and we do this in what follows without any other comment. F*
*or
example, we identify S(X) with X, for any space X.
2.12. Definition: We define a natural map F ^F 0! F O F 0, which we call the
assembly map (here F denotes the extended functor from spaces to spaces defined*
* in 2.10,
so that the composition F OF 0makes sense). This map is an isomorphism when F 0*
*equals
some n (proposition 2.16).
We first define a map F (X) ^Y! F (X ^Y,)natural in the spaces X and Y . Thi*
*s is
the map that most resembles other assembly maps in the literature, and it is th*
*e special
case F 0= S ^Y . Using degreewise extension, it suffices to define a map of th*
*e form
F (n+) ^m+ ! F (n+ ^m+). We do this as follows. Given i in m+ and x in F (n+), *
*the
element OE(x ^i) is defined as *(x), where denotes the map from n+ to n+ ^m+ t*
*hat
takes j to j ^i. There is a similar map X ^F (Y )! F (X ^Y ), natural in the s*
*paces
X and Y . We use the name special assembly map to refer to any of these two map*
*s.
To handle the general case, by definition of the smash product, it is enough*
* to specify
a natural map F (n+) ^F 0(m+) ! F (F 0(n+ ^m+)). This is defined as the composi*
*tion
F (n+) ^F 0(m+) ! F (n+ ^F 0(m+)) ! F (F 0(n+ ^m+));
where the first map is a special assembly map, and the second is given by apply*
*ing F to
a special assembly map.
2.13. Remark: We are now able to identify the -spaces that represent "algeb*
*ras
over the sphere spectrum". Surprisingly enough, these turn out to be well-known.
Let us say that a -space F is a Gamma-ring, if there are maps : F ^F ! F and
j: S ! F satisfying the usual associativity and unit conditions. Then correspo*
*nds to
e: F e^F !RF , i.e., to a map
e: F (X) ^F (Y )!F (X ^Y )
natural in X and Y in op, which extends degreewise to give a similar map, denot*
*ed
again by e, natural in the spaces X and Y . In fact, the -space F is a Gamma-r*
*ing
if and only if it is an FSP, as defined by B"okstedt in [B"o], under e, j, and *
*the special
assembly map of definition 2.12. Further, this defines a full embedding of Gamm*
*a-rings
5
in FSPs, and one can show that all connective FSPs are in the image of this emb*
*edding,
up to stable weak equivalence of FSPs.
2.14. Convention: If I is a small category, F is a functor from I to pointe*
*d sets,
i is an object of I, and x 2 F (i), we denote the image of x in colimF by [i; x*
*].
2.15. Proposition: For any non-negative integers m and n, the -spaces mn
and m ^n are isomorphic.
Proof. An isomorphism from mn to m ^n is defined by mapping f: m+ ^n+ !k+
to [f; 1 ^1], where the notation is as in 2.14. Its inverse takes [g; OE ^ ] to*
* g O (OE ^ ). __|_|
2.16. Proposition: The assembly map F ^F 0! F O F 0is an isomorphism, when-
ever F 0equals some n.
Proof. The case F = m follows from the previous proposition. Note that, f*
*or
fixed F 0, the functor F O F 0preserves all limits and colimits, and the functo*
*r F ^F 0
preserves all colimits, since it is a left adjoint. The conclusion will follow*
* if we show
that any F is an iterated colimit of diagrams involving only -spaces of the for*
*m m for
some m. There is a standard trick to write F this way, which we recall below.
We claim that F is isomorphic to the coequalizer of the two obvious maps
_ _
m ^F (l+ ) -!-! m ^F (m+):
l+!m+ m+
To see this, we may assumeWthat F is discrete. Write for the canonical map to*
* the
above coequalizer from m+ m ^F (m+). A map in one direction takes (f ^x) to f**
*(x).
Its inverse takes x 2 F (m+) to (1 ^x), where 1 denotes the identity map of m+.*
* __|_|
2.17. Remark: It follows from the proof of proposition 2.16 that there is a*
* second
description of the value of the -space F on the space X, namely F (X) is natura*
*lly
isomorphic to the coequalizer of
_ _
m (X) ^F (l+ ) -!-! m (X) ^F (m+);
l+!m+ m+
where the two maps are induced from the corresponding maps in the proof of prop*
*o-
sition 2.16. The above statement is true if we take m (X) to be as defined in *
*defini-
tion 2.10, or if we let m (X) equal the m-fold cartesian product of X with itse*
*lf (see
convention 2.11).
Recall from [Bor, definition 6.1.2] the definition of a symmetric monoidal c*
*ategory.
2.18. Theorem: The category of -spaces is symmetric monoidal with respect to
the above smash product.
Proof. We have already verified in lemma 2.9 that the smash product of -spac*
*es is
associative, commutative, and unital. It remains to show that, given a positive*
* integer N
and -spaces F1; : :;:FN , certain natural automorphisms of F1^ . .^.FN equal th*
*e iden-
tity. In fact, every natural automorphism of F1^ . .^.FN equals the identity. T*
*he case
N = 2 should suffice to explain the proof, so we assume N = 2 below, and we wri*
*te
F and F 0instead of F1 and F2.
We may assume that F and F 0are discrete. Let OE be a natural automorphism
of F ^F 0. Given a non-negative integer m and x 2 F (m+), write ^xfor the map f*
*rom m
to F that takes f: m+ ! k+ to f*x. By definition of the smash product, every z
in (F ^F 0)(k+ ) is of the form (^x^^y)[f; 1m+ ^1n+] for some f: m+ ^n+ ! k+ in*
* op,
6
where the notation is as in 2.14. Thus it suffices to show that, for all non-n*
*egative
integers m and n, if F = m and F 0= n then OE = 1.
Proposition 2.16 implies that GS (m ^n; m ^n) ~=op(m+ ^n+; m+ ^n+). Let
OE = f* for some f: m+ ^n+ ! m+ ^n+ in op. We show that all i ^j in m+ ^n+ are
fixed under f. Note that this is trivially true if m = n = 1, since OE is an au*
*tomorphism.
Let I: 1+ ! m+ and J: 1+ ! n+ correspond to i and j. It suffices to show th*
*at
f O (I ^J) = I ^J. But
f O (I ^J)= (f O (I ^J))*1m+ ^n+
= (I ^J)*f*1m+ ^n+
= (I ^J)*OE1m+ ^n+
= (I* ^J*)OE1m+ ^n+
= OE(I* ^J*)1m+ ^n+
= OE(I ^J)
= I ^J;
where the fourth equality follows from identifying mn with m ^n using proposi-
tion 2.16, the fifth equality follows from naturality, and the last equality fo*
*llows from the
last line of the previous paragraph. _*
*_|_|
2.19. Remark: The composition product (F; F 0) 7! F O F 0is associative and*
* unital
(with unit S), up to natural isomorphism, and the assembly map is compatible wi*
*th as-
sociativity and unit isomorphisms. In the language of monoidal categories, the *
*assembly
map makes the identity functor of GS a lax monoidal functor from the monoidal c*
*ategory
(GS; O; S) to the monoidal category (GS; ^; S).
3. THE SKELETON FILTRATION
In this section we define a filtration
[1
* = F (0) F (1) . . . F (m)= F
m=0
of a -space F , and we prove in theorem 3.10 that there is a pushout square giv*
*ing F (m)
in terms of F (m-1)and m .
3.1. Definition: Let m be a non-negative integer and F be a -space. The m-
skeleton of F is the -space F (m)defined as follows. In case F is discrete, def*
*ine
F (m)(n+) = {f*x| x 2 F (k+ ); f: k+ !n+; k m};
and extend this definition degreewise in the general case.
Let n be the group of automorphisms of n+ in op. If n acts freely (off the
basepoint) on F=F (n-1)(n+) for all positive integers n, we say that the -space*
* F is
cofibrant. A map of -spaces is a cofibration provided that it is injective and*
* that its
cofiber is cofibrant.
3.2. Proposition: The -space n is cofibrant for all n.
Proof. The elements of (n)(m-1)(m+) are the non-surjective maps from n+ to m*
*+.
We want to show that given x: n+ ! m+ and oe 2 m , if oe 6= 1 and x is onto then
oe O x 6= x. But if oei 6= i and xj = i then oexj = oei 6= i = xj. *
* __|_|
3.3. Example: If f : F ! F 0is an injective map of -spaces and F 0is cofibr*
*ant, then
both F and the cofiber of f are cofibrant, and f is a cofibration. This is true*
* because, if
7
G is any group, all subobjects and cofibers of free pointed G-sets are free (in*
* the pointed
sense). Thus the inclusions (n)(n-1) n and n_m ! n+m are cofibrations. These
are probably the easiest examples of cofibrations that are not Q-cofibrations (*
*since their
cofibers are not sums of representables).
3.4. Example: Recall the -space HZ of remark 2.4, and let e : n+ ! eZ[n+] b*
*e the
canonical map. Thus eZ[n+] = Ze(1) + . .+.Ze(n). Then HZ is not cofibrant. In f*
*act,
the element e(1) + e(2) of HZ(2+) is fixed by 2, and does not belong to HZ(1)(2*
*+) =
Ze(1) [ Ze(2). A similar argument shows that HA is not cofibrant, for any non-t*
*rivial
abelian group A.
3.5. Example: We recall certain well known -spaces associated to finite sum*
*s in
a category C . We show that, under a mild assumption on C, they are cofibrant (*
*despite
a formal similarity to the -spaces of the previous example; see below). We also*
* show
that, in all interesting cases, these are not Q-cofibrant.
Given an object n+ of op, let Pn be the category of pointed subsets of n+ and
inclusions. Let C be a small category with a chosen initial object *. Then C de*
*termines
a -space F whose value at n+ is the nerve of the following category. Its objec*
*ts are
all functors from Pn to C that preserve sums (in the sense that they take a dia*
*gram
S0 S S00in Pn expressing S as the sum of S0and S00to a similar diagram in C) a*
*nd
take 0+ to *. Its morphisms are the isomorphisms (of functors from Pn to C) bet*
*ween
its objects. The -spaces F of this type were among the important examples consi*
*dered
in [Se]. For example, if C is the category of finite based sets, then the homot*
*opy groups
of F (S1) are the stable homotopy groups of spheres, and if C is the category o*
*f finitely
generated projective modules over some ring R, then the homotopy groups of F (S*
*1) are
the algebraic K-theory groups of R. If we assume that finite sums exist in C, t*
*hen F (n+)
is homotopy equivalent to the product F (1+)n,and we see a similarity with the *
*-spaces
of example 3.4. In fact, the sum in C makes F (1+) an abelian H-space, in parti*
*cular an
abelian monoid in the homotopy category ho(S*). The construction HA of remark 2*
*.4
is possible for any abelian monoid A in a category D having finite products and*
* a zero
object, and it produces a functor op! D. Finally, F lifts H(F (1+)), in the sen*
*se that
F and H(F (1+)) are isomorphic as functors op!ho(S*).
We now show that F is cofibrant if and only if the initial object of C is un*
*ique
(thus we may always replace C by an equivalent category whose associated -space*
* is
cofibrant). Let oe be a non-trivial element of n and let C in F0(n+) be fixed *
*by oe.
Choose a non-trivial cycle for oe, i.e., choose an injection i : Z=m ! n+ such *
*that
m 2 and oei(a) = i(a + 1) for all a in Z=m. Let S be the image of i, so that *
*we
have a representation of C(S) as the sum of all Ca = C({i(a)}), with associated*
* maps
fa : Ca ! C(S). Since oeC = C, all maps fa are equal. For any object C0 of C an*
*d any
maps g and g0 from C0 to C0, let h : C(S) ! C0 be the unique map such that hf0 *
*= g
and hfa = g0 for all non-zero elements a of Z=m. Then g = hf0 = hf1 = g0, i.e.,*
* C0 is
an initial object. The conclusion follows since the vertices of F (n-1)(n+) con*
*sist of those
functors C such that for some element i of n+ we have C({i}) = *, and since act*
*ions on
nerves of categories are free if and only if they are free on objects.
To conclude this example, we show that if C has finite sums and more than on*
*e object,
then F is not Q-cofibrant. By lemma A.3 of [Sch], the zero-simplices of a Q-cof*
*ibrant
-space split as a sum of various n. Thus, if P is the discrete -space given by*
* the
zero-simplices of F , it suffices to show that P has no non-trivial maps to n, *
*for any n.
An element x of n(m+) is trivial, if so are its images under all maps n ! 1. Th*
*us
we may assume that n = 1. Fix a map f : P ! 1 and an element C in P (n+). We
8
show that f(C) = 0. Let m = 2n. Let p and q be the maps m+ ! n+ which take all
i n, resp. all i > n, to 0 and such that, for 1 i n, p(n + i) = i, and q(i) *
*= i.
Because C has finite sums, there exists D in P (m+), such that p*D = q*D = C (i*
*.e.,
given a pointed subset S of m+, the object D(S) is some choice of the sum over *
*i 2 S
of D(i), with D(i) = C(i) if i 2 n*, else D(i) = C(i - n); further, this choice*
* is fixed
if S n+ (then D(S) = C(S)) or if S \ n+ = 0+ (then D(S) = Cp(S))). In case
f(D) 2 n+ we have f(C) = fp*(D) = p*f(D) = 0. In case f(D) 62 n+ we have
f(C) = fq*(D) = q*f(D) = 0.
3.6. Proposition: Suppose that D is a pullback square of pointed sets and t*
*hat F is
a functor from pointed sets to pointed sets. Then F (D) is also a pullback squa*
*re, provided
that all four maps of D are injective.
Proof. Suppose that D is the square below
X -f! Y
i # #j
Z -g! W
where we may assume that all maps are inclusions. Choose a retraction u to i an*
*d extend
it to a retraction v to j. In other words, the equalities fu = vg and vj = 1 ho*
*ld. This can
be done because X = Y \Z. Because every injective map of pointed sets is split,*
* all maps
in F (D) are again injective. It remains to show that given y in F (Y ) and z i*
*n F (Z), if
they map to the same element in F (W ) then they lift to F (X). Because g* is i*
*njective,
it suffices to lift y. But u*(z) is such a lift, since f*u*(z) = v*g*(z) = v*j**
*(y) = y. __|_|
n
3.7. Definition: For any non-negative integers n and m, let m denote the *
*set
of those injective maps from m+ to n+ which are increasing with respect to the *
*usual
order.
3.8. Proposition: For any -space F and any positive integers m and n, there*
* is
a pushout square
n + (m-1) + n + +
m ^ F (m ) m ^F (m )
# #
F (m-1)(n+) F (m)(n+)
n +
where the map m ^ F (m+) ! F (m)(n+) takes f ^x to f*x.
Proof. We may assume that F is discrete. Since the square above is commutati*
*ve, we
obtain a map from the pushout of the truncated square to F (m)(n+), which is su*
*rjective.
It remains to show that this map is injective.
Since the map from nm+ ^F (m-1)(m+) to F (m-1)(n+) is onto, it suffices to *
*show
that, given f*x = g*y in F (m)(n+) with x and y in F (m+) and f and g in nm, i*
*f f ^x 6=
g ^y then x and y are in F (m-1)(m+). If f = g, then x = y since f* is injectiv*
*e, so there
is nothing to prove in this case. Assume now that f 6= g and let p+ be their pu*
*llback.
Since f and g are distinct increasing injections, they have distinct images, an*
*d therefore
p < m. The conclusion now follows from proposition 3.6. *
*__|_|
3.9. Definition: We view m as a -m -space, that is, a functor from op to
m -spaces taking 0+ to a point, by using the mapping space action (that is, the*
* image
of f 2 mkunder the action of oe 2 m is f O oe-1).
3.10. Theorem: For any -space F and any positive integer m, there is a push*
*out
9
square
@(m ^F (m+))=m (m ^F (m+))=m
# #
F (m-1) F (m)
where @(m ^F (m+)) is defined as
m ^F (m-1)(m+) [(m)(m-1)^F(m-1)(m+)(m )(m-1)^F (m+)
and the map (m ^F (m+))=m ! F (m)takes the orbit of f ^x to f*x.
Proof. The cofiber of the inclusion @(m ^F (m+)) m ^F (m+) is isomorphic
to m =(m )(m-1)^(F=F (m-1))(m+). Further, the inclusion of nm+ in m (n+) induc*
*es
an isomorphism between nm+ and the m -orbits of m =(m )(m-1)(n+). Finally, the
action of m on m (n+)=(m )(m-1)(n+) is free. The conclusion now follows from pr*
*opo-
sition 3.8, since a commutative square of pointed sets with horizontal maps inj*
*ective is
a pushout if and only if the induced map on horizontal cofibers is an isomorphi*
*sm. __|_|
3.11. Proposition: If F is a discrete cofibrant -space then there exists a *
*pointed
set S such that F (n)is obtained from F (n-1)by attaching n ^S along (n)(n-1)^S.
Proof. Let S = F=F (n-1)(n+)=n. For every non-basepoint element s in S, choo*
*se
a representative x(s) in F (n+), so that the orbit nx(s) equals s, where we den*
*oted the
image of x(s) in F=F (n-1)(n+) again by x(s). Consider the diagram below, where*
* the
map S ! F (n+) is defined by s 7! x(s).
(n)(n-1)^S n ^S
# #
@(n ^F (n+))=n (n ^F (n+))=n
# #
F (n-1) F (n)
The top square is a pullback square with all four maps injective, and every ele*
*ment
of (n ^F (n+))=n can be lifted either to @(n ^F (n+))=n or to n ^S. It follows *
*that
the top square is a pushout, therefore so is the composed square, since the bot*
*tom square
is a pushout by theorem 3.10. __
|_|
4. SMASH PRODUCTS AND COFIBRATIONS
In this section we show that the smash product of -spaces behaves well with res*
*pect to
injective maps and cofibrations.
4.1. Lemma: Let f: F ! F 0be a map of -spaces and n be a positive integer s*
*uch
that fn+ is injective. Then the equality f(F (n+)) \ (F 0)(m)(n+) = f(F (m)(n+)*
*) holds,
for all non-negative integers m n.
Proof. Fix x 2 F (n+), k m, y 2 Fk0, s: k+ ! n+, and suppose s*y = fn+x.
Write s = s0s00with s0 injective and s00surjective. Replacing y by (s00)*y and *
*s by s0,
we see that we may assume that s is injective. Choose r: n+ ! k+ with rs = 1. T*
*hen
fn+s*r*x = s*r*fn+x = s*r*s*y = s*y = fn+x, and therefore s*r*x = x since fn+ is
injective, i.e. fn+x is in f(F (m)(n+)). *
* __|_|
4.2. Lemma: Let f: F ! F 0be a map of -spaces and n be a positive integer s*
*uch
that for all non-negative integers m n the map fm+ is injective. Then the map *
*f(m)l+
is injective, for all non-negative integers m n and all non-negative integers *
*l.
10
Proof. The proof is by induction on m, the case m = 0 being trivial. Since f*
*(m-1)l+is
injective by induction, it suffices to show that f(m)l+=f(m-1)l+is injective. T*
*his follows from
proposition 3.8, since fm+ is injective by hypothesis, f(m-1)m+is injective by*
* induction,
and f(F (m+)) \ (F 0)(m-1)(m+) = f(F (m-1)(m+)) by lemma 4.1. _*
*_|_|
4.3. Proposition: For any -space F , smashing with F preserves injective ma*
*ps.
Proof. Fix an injective map of -spaces f : F 0! F 00. Since the smash produc*
*t F ^F 0
of the -spaces F and F 0can be evaluated degreewise in F , we may assume that F*
* is
discrete. Similarly, we may assume that F 0and F 00are discrete. Since inject*
*ions of
pointed sets are preserved by filtered colimits, it suffices to show that for a*
*ll non-negative
integers m the map F ^f(m) is injective. This will be shown by induction on m, *
*the case
m = 0 being trivial.
Note that a map of cofibration sequences of pointed sets is injective if it *
*is injective on
subobjects and quotient objects, and that taking m -orbits preserves injective *
*maps. Ap-
plying theorem 3.10 we see that it suffices to show that F ^m =(m )(m-1)^fm+ =f*
*(m-1)m+
is injective. This amounts to showing that fm+ =f(m-1)m+is injective. Note that*
* fm+ is in-
jective by hypothesis and that f(m-1)m+is injective, by lemma 4.2. The conclusi*
*on follows
because f(F 0(m+)) \ (F 00)(m-1)(m+) = f((F 0)(m-1)(m+)), by lemma 4.1. *
* __|_|
4.4. Proposition: If F ! F 0and eF! eFa0re two injections of -spaces, then *
*the
canonical map F ^eF[0F ^eFF 0^eF! F 0^eFi0s injective.
Proof. Consider the diagram below.
F ^eF 0 = F ^eF 0
# #
F ^eF[0F ^eFF 0^eF! F 0^eF 0
# #
(F 0=F ) ^eF ! (F 0=F ) ^eF 0
It follows from proposition 4.3 that the top right vertical map is injective, a*
*s is the map
F ^eF! F 0^eF, and its cobase change, the top left vertical map. Thus the colum*
*ns give
cofibration sequences of spaces for each n+ in op, and the conclusion follows s*
*ince the
bottom horizontal map is injective, by proposition 4.3 again. *
* __|_|
4.5. Lemma: If F and F 0are cofibrant -spaces, then so is F ^F 0.
Proof. We may assume that F and F 0are discrete. Assume for the moment that
F = m .
If F 0= n, the conclusion follows from proposition 2.15 and proposition 3.2.*
* It follows
from proposition 4.3 that the map F ^(n)(n-1)! F ^n is a cofibration, because i*
*t is
an injection of -spaces with cofibrant target. Note that cofibrations are close*
*d under
cobase change and sequential colimits. The case F 0is any discrete cofibrant -s*
*pace now
follows using proposition 3.11.
This completes the proof in case F = m . The proof of the general case proce*
*eds as
in the previous paragraph. __|_|
4.6. Theorem: If F ! F 0and eF! eF 0are two cofibrations of -spaces, then t*
*he
canonical map : F ^eF[0F ^eFF 0^eF! F 0^eFi0s a cofibration.
Proof. The cofiber of is isomorphic to (F 0=F ) ^(Fe0=Fe). The conclusion n*
*ow follows
from lemma 4.5 and proposition 4.4. __|*
*_|
11
5. SMASH PRODUCTS AND WEAK EQUIVALENCES
There are three main results in this section. The following theorem is the firs*
*t one.
5.1. Theorem: Smashing with a cofibrant -space preserves strict weak equiva*
*lences.
5.2. Proposition: For any strict weak equivalence of -spaces F ! F 0and any
non-negative integer m, the map F (m)! (F 0)(m) is a strict weak equivalence.
Proof. This follows immediately from proposition 3.8. *
* __|_|
Proof of theorem 5.1. Since the smash product of the -spaces F and F 0can be
evaluated degreewise in F , we may assume that F is discrete. By proposition 4*
*.3 it
suffices to show that, for all non-negative integers n, smashing with F (n)pres*
*erves strict
weak equivalences. We prove this by induction on n. The case n = 0 is trivial*
*. It
follows from proposition 3.11 that F (n)is obtained from F (n-1)by attaching n *
*^S
along (n)(n-1)^S, for some pointed set S. The conclusion follows because smash*
*ing
with F (n-1), (n)(n-1), and n preserves strict weak equivalences, since the ind*
*uctive
hypothesis applies to the first two, and, by proposition 2.16, smashing with n *
*is the
same as composing with n, and composing with any -space preserves strict weak
equivalences. __|_|
Before we state the remaining main results of this section, we define spectr*
*a, stable
weak equivalences of -spaces, and the (naive) smash product of spectra.
5.3. Definition: A spectrum E consists of a sequence of spaces En and a seq*
*uence
of maps Enn+1: S1^ En ! En+1, for n = 0; 1; : :,:where S1 is the space 1 =@1 .*
* A map
f: E !E0 of spectra is a sequence of maps fn: En!E0nsuch that
fn+1 O Enn+1= (E0)nn+1O (S1^ fn):
A spectrum determines direct systems
. .!.ssm (En) ! ssm+1 (En+1) ! . .;.
and we define the homotopy groups of E by ssn(E) = colimkssn+k(Ek). A map of sp*
*ectra
is called a weak equivalence provided that it induces isomorphisms on homotopy *
*groups.
5.4. Definition: Define the spectrum F (S) associated to the -space F by se*
*t-
ting F (S)n = F (Sn), where Sn+1 is defined recursively as S1^ Sn, and where the
maps F (S)nn+1are obtained from the special assembly map. A map f of -spaces is
called a stable weak equivalence provided that f(S) is a weak equivalence of sp*
*ectra.
5.5. Example: Recall from lemma 2.6 and remark 2.8 that the maps of -spaces
from S to F are given by the vertices of F (1+). Now let F be as in example 3.5*
* with
C = op. Thus F (1+) may be identified with the nerve of the isomorphisms of op.*
* The
map S ! F determined by the object 1+ of op is a stable weak equivalence. This *
*is
essentially the version of the Barratt-Priddy-Quillen-Segal theorem proven in [*
*Se].
5.6. Remark: The analog of proposition 5.2 for stable weak equivalences is *
*false.
A counterexample for m = 1 and F 0= * is provided by setting F (n+) equal to n+*
* ^n+.
5.7. Remark: We are now able to describe 1 and 1 functors. The functor 1
associates to a space X the -space 1 X, that takes n+ to n+ ^X. The functor 1
associates to a -space F the space F (1+). Then 1 is left adjoint to 1 , and t*
*akes
weak equivalences of spaces to stable (in fact, strict) weak equivalences of -s*
*paces, when
restricted to cofibrant objects. This is no restriction at all, because (recall*
* that "space"
12
means "pointed simplicial set" in this paper) all spaces are cofibrant. We add *
*it for the
sake of symmetry, because in order for the functor 1 to take stable weak equiva*
*lences
to weak equivalences of spaces we have to restrict it to stably fibrant objects*
* (see [BF ]),
or at least to stably Q-fibrant objects (see [Sch]). These classes of -spaces a*
*re contained
in the class of very special -spaces, and for very special -spaces the notions *
*of stable
and strict weak equivalence coincide (see sections 4 and 5 of [BF ]).
5.8. Definition: Let t be the natural isomorphism that interchanges the sec*
*ond
and third factors in the smash product of spaces X ^Y ^Z ^W .
5.9. Definition: The smash product E ^E0 of the spectra E and E0 is the spe*
*c-
trum given by (E ^E0)2n = En ^E0n, (E ^E0)2n+1 = S1^ (E ^E0)2n, (E ^E0)2n2n+1= *
*1,
and (E ^E0)2n+12n+2= (Enn+1^(E0)nn+1) O t.
5.10. Proposition: Smashing with a spectrum preserves weak equivalences.
Proof. This follows essentially because ssn commutes with sequential colimit*
*s of spaces
(recall that "space" means "pointed simplicial set", and ssn(X) is the set of p*
*ointed
homotopy classes of maps from Sn to the singular complex of the realization of *
*X). __|_|
The following two theorems are the remaining main results of this section.
5.11. Theorem: There is a map of spectra F (S) ^F 0(S) ! (F ^F 0)(S) , natu*
*ral in
the -spaces F and F 0, which is a weak equivalence if one of the factors is cof*
*ibrant.
5.12. Theorem: Smashing with a cofibrant -space preserves stable weak equiv*
*a-
lences.
Proof. This follows immediately from theorem 5.11 and proposition 5.10. *
* __|_|
5.13. Definition: An S2-spectrum E consists of a sequence of pointed simpli*
*cial
sets E2n and a sequence of pointed maps E2n2n+2: S2^ E2n!E2n+2, for n = 0; 1; :*
* :.:We
define maps, homotopy groups, and weak equivalences of such objects, so that fo*
*rgetting
the odd terms of a spectrum gives a functor E 7! E* that preserves weak equival*
*ences. A
-space F determines an S2-spectrum F (St), where F (St)2n = F (S2n). The struct*
*ural
maps are given by the composition
S1^ S1^ F (Sn ^Sn) ! F (S1^ S1^ Sn ^Sn) t*!F (S1^ Sn ^S1^ Sn);
where the first map is the special assembly map.
5.14. Proposition: The S2-spectra F (S)* and F (St) are naturally isomorphi*
*c.
Proof. Define the spectrum S by S = S(S), and the S2-spectrum St by St = S(S*
*t).
Note that the conclusion of the proposition is true in the special case F = S, *
*i.e., S* and St
are isomorphic. The general case follows because F (S)*, respectively F (St), i*
*s obtained
from a functorial construction which, to a -space F and an S2-spectrum E, assoc*
*iates
the S2-spectrum F (E), by letting E = S*, respectively E = St. *
* __|_|
The only reason we consider F (St), and, in fact, the only reason we conside*
*r S2-
spectra, is to be able to write the map in the statement of Theorem 5.11 as a c*
*omposition
of simpler maps. One of these simpler maps is given by the proposition above, a*
*nd another
one is given by the proposition below.
5.15. Proposition: There is a non-trivial map (F (S) ^F 0(S))* ! (F ^F 0)(S*
*t),
natural in the -spaces F and F 0.
Proof. Recall that there is a natural map F (m+) ^F 0(n+) ! (F ^F 0)(m+ ^n+)
(in fact, F ^F 0is essentially defined by saying that it is universal with this*
* property).
13
This map extends degreewise to a natural map F (X) ^F 0(Y ) ! (F ^F 0)(X ^Y ), *
*where
X and Y are spaces. The map we want is obtained from this map by evaluating on
spheres. __|_|
5.16. Lemma: If F and F 0are -spaces and F is cofibrant, then the map
(F (S) ^F 0(S))* ! (F ^F 0)(St)
is a weak equivalence.
Proof of theorem 5.11. Note that the forgetful functor E 7! E* from spectra *
*to S2-
spectra has a left adjoint L, such that (LE)2n = E2n, (LE)2n+1= S1^ E2n, (LE)2n*
*2n+1=
1, and (LE)2n+12n+2= E2n2n+2. The conclusion now follows from proposition 5.14,*
* lemma 5.16,
and the fact that, for any spectra E and E0, the equality E ^E0= L(E ^E0)* hold*
*s. __|_|
5.17. Definition: We say that a -space F is o(n)-connected provided that for
any simply-connected space X the space F (X) is as connected as X ^n. We say th*
*at a
x-space F is o(n)-connected provided that so is its restriction to the diagonal*
*. We
say that a map of -spaces, resp. x-spaces, is o(n)-connected provided that so i*
*s its
(pointwise) homotopy cofiber.
5.18. Lemma: The map of x-spaces given by
F (X) ^F 0(Y ) ! (F ^F 0)(X ^Y )
is o(3)-connected, for any -spaces F and F 0with F cofibrant.
Proof of lemma 5.16. By lemma 5.18, there is a constant c such that the map
F (Sn) ^F 0(Sn) ! (F ^F 0)(S2n) is (3n + c)-connected for n > 1. *
* __|_|
5.19. Proposition: For any -space F there is a natural strict weak equivale*
*nce
F c! F with F ccofibrant.
Proof. We may assume that the -space F is discrete. Let C(n+) be the followi*
*ng
category. Its objects are the pairs (f; x) such that f : m+ ! n+ and x 2 F (m+)*
*. There
is one morphism from (f; x) to (g; y) for each h in opsuch that gh = f and h*(x*
*) = y.
Define F 0(n+) as the nerve of C(n+). There is a map F 0(n+) ! F (n+) taking (*
*f; x)
to f*(x), and is the disjoint union of projections of nerves of categories to t*
*heir terminal
objects, in particular a weak equivalence. Note that
a
Fq0~= kqx op(k+q-1; k+q) x . .x.op(k+0; k+1) x F (k+0):
(k0;:::;kq)
The required -space F cis given by a pointed version of this. Let
_
Fqc= kq^op(k+q-1; k+q) ^. .^.op(k+0; k+1) ^F (k+0)
(k0;:::;kq)
so that we have a canonical map F 0! F c. Then it is still true that there is *
*a map
F c! F , and that, for fixed n+, there is a section F (n+) ! F c(n+), as well a*
*s a map
F c(n+) ^(1)+ ! F c(n+) which is a homotopy between the identity and the compos*
*ition
F c(n+) ! F (n+) ! F c(n+). All these maps are compatible with the canonical m*
*ap
F 0! F c, and exist essentially because op is a pointed category and F is a poi*
*nted
functor. __|_|
Proof of lemma 5.18. Suppose first F = n.
14
Suppose also F 0= m . The map in the statement of the lemma corresponds, un-
der the isomorphism of proposition 2.15 between n ^m and nm , to the map that
takes (f; g) to f ^g. Define a filtration
* = F0n F1n . . .Fnn = n
of n by letting Fkn(m+) consists of those f: n+ ! m+ such that the cardinality
of f-1(0) is at least (n+1-k). Then Fkn=Fk-1n(X) is isomorphic to nk+^X ^k. In
particular, the map Fkn n is o(k + 1)-connected, and F1n is isomorphic to n ^1.
The restriction of to F1n(X) ^F1m (Y ) is an isomorphism onto F1nm (X ^Y ). T*
*he
conclusion follows from this, together with the fact that F1k and k are o(1)-co*
*nnected
(for any k).
Suppose now that F 0is discrete. We prove that the lemma is true in this cas*
*e, by
proving that it is true for all (F 0)(m). As usual, the proof will be by induct*
*ion on m, the
case m = 0 being trivial. The inductive step follows from the previous paragrap*
*h and
proposition 3.11.
In case F 0is any cofibrant -space, the lemma is true because diagonalizatio*
*n pre-
serves connectivity. The complete special case F = n now follows from theorem 5*
*.1 and
propositions 3.2 and 5.19. The rest of the proof is similar, i.e., the discrete*
* case is done
by induction on skeleta, and then the general case of an arbitrary cofibrant -s*
*pace F
follows because diagonalization preserves connectivity. *
* __|_|
We conclude this section by proving certain interesting statements about -sp*
*aces.
5.20. Proposition: Any -space F preserves connectivity, i.e., if f is a k-c*
*onnected
map of spaces, then so is F (f). In particular, F is a homotopy functor, i.e., *
*it preserves
weak equivalences of spaces.
Proof. This follows from propositions 3.11 and 5.19. *
* __|_|
5.21. Proposition: If X and Y are connected spaces and F is a -space, the m*
*ap
F (X) ^Y ! F (X ^Y ) is as connected as the suspension of X ^2^Y .
Proof. The proof is similar to, but easier than, the proof of lemma 5.18. *
* __|_|
5.22. Proposition: If X; Y are connected spaces, and F; F 0are -spaces with*
* F
cofibrant, then the map F (X) ^F 0(Y ) ! (F ^F 0)(X ^Y ) is as connected as the*
* suspen-
sion of (X ^2^Y ) _ (X ^Y ^2).
Proof. The proof is the same as the proof of lemma 5.18 *
* __|_|
5.23. Proposition: The assembly-map F ^F 0! F OF 0is a stable weak equivale*
*nce,
whenever F or F 0is cofibrant.
Proof. Given the previous proposition, it suffices to show that the associa*
*ted map
f : F (X) ^F 0(Y ) ! (F O F 0)(X ^Y ) is highly connected, if so are X and Y . *
*This follows
from the definition of f, together with propositions 5.20 and 5.21. *
* __|_|
6. COFIBRATIONS AND STRICT COFIBRATIONS
In this section we show that our definitions of the skeleta and the cofibration*
*s of -spaces
are equivalent to those found in [BF , p. 89 and p. 91]. We recall the latter b*
*elow.
6.1. Definition: Let m be a non-negative integer. The functor skm : GS ! GS*
* is
defined as follows. If F is a -space and n is a non-negative integer, define (s*
*km F )(n+)
to be the colimit over all k+ ! n+ with k m of F (k+ ).
15
There is a map skm F ! F , given for f : k+ ! n+ by f*: F (k+!)F (n+). A map*
* of
-spaces f : F ! F 0is called a strict cofibration provided that for all positiv*
*e integers n
the map
gn : F (n+) [(skn-1F)(n+)(skn-1F 0)(n+) ! F 0(n+)
is injective, and the action of n is free off the image of gn.
6.2. Proposition: For any -space F and any non-negative integer m, the -spa*
*ces
F (m)and skm F are isomorphic.
Proof. The map skm F ! F induces a surjective map skm F ! F (m), which we
show is also injective. Fix a non-negative integer n, as well as maps f: k+ ! *
*n+ and
g: l+ !n+ with k; l m. We have to show that if f*x = g*y in F (n+), then [f; x*
*] = [g; y]
in (skm F )(n+), where the notation is as in 2.14. By arguing as in the proof o*
*f lemma 4.1,
we may assume that f and g are injective. By proposition 3.6, if p+ is the pull*
*back of
f and g, then x and y can be lifted to z in F (p+). If h: p+ ! n+ is the assoc*
*iated
canonical map, both [f; x] and [g; y] equal [h; z]. *
* __|_|
6.3. Lemma: Strict cofibrations of -spaces are injective.
Proof. Let f: F !F 0be a strict cofibration of -spaces. We prove by inductio*
*n on n
that fm+ is injective for all m n. The case n = 0 is trivial.
By lemma 4.2 and the inductive hypothesis, F (n-1)(n+) injects into (F 0)(n-*
*1)(n+).
Therefore F (n+) injects into F (n+) [F(n-1)(n+)(F 0)(n-1)(n+), which in turn i*
*njects
into F 0(n+) by the definition of strict cofibration and proposition 6.2. *
* __|_|
6.4. Proposition: A map of -spaces f: F !F 0is a cofibration if and only if*
* it is
a strict cofibration.
Proof. For any injection of -spaces f: F !F 0and any positive integer n, the*
* map gn
of definition 6.1 is isomorphic over F 0(n+) to the inclusion
f(F (n+)) [ (F 0)(n-1)(n+) F 0(n+):
This follows from proposition 6.2, and the fact that, by lemma 4.1,
f(F (n+)) \ (F 0)(n-1)(n+) = f(F (n-1)(n+)):
The conclusion now follows from lemma 6.3. __*
*|_|
Acknowledgements_ The author is grateful to Tom Goodwillie and Stefan Schwede f*
*or many helpful
discussions. He is also grateful to the SFB 343 at the University of Bielefeld*
*, especially Friedhelm
Waldhausen, for their hospitality while this paper was written.
References
[B"o] B"okstedt, M.: Topological Hochschild homology, preprint, Bielefeld, 198*
*5.
[BF] Bousfield, A. K., Friedlander, E. M.: Homotopy theory of -spaces, spec-
tra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, S*
*pringer,
Berlin, 1978, pp. 80-130.
[Bor] Borceux, F.: Handbook of Categorical Algebra 2, Categories and Structure*
*s,
Encyclopedia of Mathematics and its Applications, Vol. 51, Cambridge Uni*
*v.
Press, Cambridge, 1994.
16
[D] Day, B.: On closed categories of functors, Reports of the Midwest Catego*
*ry
Seminar IV, Springer Lecture Notes in Math., Vol. 137, Springer, Berlin,*
* 1970,
pp. 1-38.
[Q] Quillen, D. G.: Homotopical Algebra, Springer Lecture Notes in Math., Vo*
*l.
43, Springer, Berlin, 1967.
[Se] Segal, G.: Categories and cohomology theories, Topology 13 (1974), 293-3*
*12.
[Sch] Schwede, S.: Stable homotopical algebra and -spaces, Math. Proc. Cam. Ph*
*il.
Soc., to appear.
Fakult"at f"ur Mathematik
Universit"at Bielefeld
Postfach 100131
33501 Bielefeld
Germany
manos@math206.mathematik.uni-bielefeld.de
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