YET ANOTHER DELOOPING MACHINE
BERNARD BADZIOCH, KUERAK CHUNG, AND ALEXANDER A. VORONOV
Abstract.We suggest a new delooping machine, which is based on recog-
nizing an n-fold loop space by a collection of operations acting on it, *
*like the
traditional delooping machines of Stasheff, May, Boardman-Vogt, Segal, a*
*nd
Bousfield. Unlike in the traditional delooping machines, which carefully*
* select
a nice space of such operations, we consider all naturalWoperationsWon n*
*-fold
loop spaces, resulting in the algebraic theory Map*( oSn, oSn). The ad-
vantage of this new approach is that the delooping machine is universal *
*in a
certain sense, the proof of the recognition principle is more conceptual*
*, works
the same way for all values of n, and does not need the test space to be
connected.
1.Introduction
The goal of this paper is to give a proof of the following characterization of
n-fold loop spaces. In the categoryWSpaces*of pointed spaces, consider the full
subcategoryWgenerated by thenwedges kSn of n-dimensional spheres for k 0
(whereW 0Sn = *). Let T S denote the opposite category,nsee Figure 1. Since
k Sn is a k-fold coproduct of Sn's in Spaces*, in T S it is a k-fold categori*
*cal
product of Sn's.
Theorem 1.1. A space Y 2 Spaces*is weakly equivalentnto an n-fold loop space, i*
*ff
there exists a product preserving functor eX:T S ! Spaces*such that eX(Sn) ' Y .
The category T Sn is in fact an algebraic theory (see 2.1). From this point *
*of
view, one can regard the above theorem as a recognition principle: a loop space*
* n
structure is detected by the structure of an algebra over the algebraic theory *
*T S .
We will actually prove a stronger versionn(see Theorem 4.8) of Theorem 1.1:
given a product preserving functor eX: T S ! Spaces*, one can construct a space
BnXe such that nBnXe ' eX(Sn), thereby delooping the space eX(Sn).
____________
Date: March 4, 2004.
The third author was supported in part by NSF grant DMS-0227974.
W W n
Figure 1. A morphism 3 Sn ! 4 Sn in category T S .
1
2 B. BADZIOCH, K. CHUNG, AND A. A. VORONOV
This description of iterated loop spacesnis in some sense an extreme delooping
machine. By Yoneda's lemma the theory T S encodes all natural maps ( nX)k !
( nX)l, and we use all this structure in order to detect loop spaces. This stan*
*ds in
contrast to the approach of Stasheff [Sta63], May [May72 ], Boardman-Vogt [BV73*
* ],
Segal [Seg74], or Bousfield [Bou92 ], where only carefully chosen sets of maps *
*of of
loop spaces are used for the same purpose. Our indiscriminate method however
brings some advantages. First of all, as in [Bou92 ], Theorem 1.1 is true for a*
*ll, not
necessarily connected, loop spaces. Also, since we avoid making particular choi*
*ces
of operations on loop spaces, thus constructed delooping machine provides a con-
venient ground for proving uniqueness theorems of the kind of May and Thomason
[MT78 ], [Tho79 ]. Namely, given an operad, a PROP, or a semi-theory (i.e., a
machine of the type of Segal's -spaces, see [Bad03 ]), one can replace it by an
algebraic theory describing the same structure on spaces. On the other hand, it
is relatively easy to compare homotopy theories of objects described by various
algebraic theories. This implies Theorem 4.10 - a uniqueness result for "deloop*
*ing
theories".
Most of the arguments and constructions we use are formal and do not depend
on any special properties of loop spaces. Indeed, at least one implication of *
*the
statement of Theorem 1.1 holds when we replace Sn with an arbitrary pointedn
space A. If T A is an algebraic theory constructed analogously to T S above, th*
*en
for any mapping space Z = Map *(A, Y ), we can define a product preserving func*
*tor
eX:T A ! Spaces*such that eX(A) = Z. We do not expect that for an arbitrary
A also the opposite statement will be true, that is that any such functor will *
*come
from some space Map *(A, Y ). It should be true, however, that if for a given s*
*pace
A, the mapping spaces from A can be described as algebras over some operad,
PROP, semi-theory, algebraic theory, or using some other formalism employing
only finitary operations on a space, then they must be characterized by means of
the theory T A.
Another advantage of the proposed recognition principle is that the argument
seems to be more conceptual than in the previously known cases. For example,
we get an analogue (Corollary 4.9) of May's approximation theorem [May72 ] as a
simple consequence of, rather than a hard step towards the recognition principl*
*e.
This simplicity comes, no wonder, with a price tag attached: while the homol-
ogy of the little n-disks operad has a neat description as the operad describing
n-algebras, see F. CohenW[Coh76W, Coh88], even the rational homology of the cor*
*re-
sponding PROP Map *( lSn, kSn) is harder to come by, see the thesis [Chu04 ]
of the second author.n
The theory T S bears resemblance to the cacti operad of [SV04 ], which consis*
*ts
of (unpointed) continuous maps from a sphere Sn to a tree-like joint of spheres
Sn at finitely many points. This operad was invented as a bookkeeping device for
operations on free sphere spaces arising inWstring topology,nsee [CS99 ].
Also, the operadic part On := Map *(Sn, Sn) of T S has been described as
a ü niversal operad of n-fold loop spaces" by P. Salvatore in [Sal03]. As it w*
*as
also noted by Salvatore, while the space underlying an algebra over this operad*
* is
weakly equivalent to an n-fold loop space, in general a loop space will admit s*
*everal
actions of On. Therefore On-algebras can be seen as loop spaces equipped with
some extra structure.
Notation 1.2.
YET ANOTHER DELOOPING MACHINE 3
o Let Spaces*denote the category of pointed compactly generated (but not
necessarily Hausdorff) topological spaces. From the perspective of homo-
topy theory, there is no difference between this category and the catego*
*ry of
all pointed topological spaces. The category Spaces*has a model category
structure with the usual notions of weak equivalences, fibrations and co*
*fi-
brations, and it is Quillen equivalent to the category of pointed topolo*
*gical
spaces, see [Hov99 ]. The assumption that all spaces are compactly gener-
ated has the advantage that for any space X, the smash product functor
Y 7! Y ^ X is left adjoint to the mapping space functor Y 7! Map *(X, Y *
*).
This has some further useful consequences which we will invoke.
o If X is an unpointed space by X+ we will denote the space X with an
adjoined basepoint.
o All functors are assumed to be covariant.
o If C is a category, then Cop will denote the opposite category of C.
2.Algebraic theories and their algebras
Definition 2.1. An algebraic theory T is a category with objects T0, T1, . .t.*
*o-
gether with, for each n, a choice of morphims pn1, . .,.pnn2 MorT (Tn, T1) such*
* that
for any k, n the map
Yn Yn
pni:Mor T(Tk, Tn) ! MorT(Tk, T1)
i=1 i=1
is an isomorphism. In other words, the object Tn is an n-fold categorical produ*
*ct
of T1's, and pni's are the projection maps. In particular T0 is the terminal ob*
*ject
in T . We will also assume that it is an initial object. A morphism of algebr*
*aic
theories is a functor T ! T 0preserving the projection maps. We will consider
algebraic theories enriched over Spaces*; in particular, the sets of morphisms *
*will
be provided with a pointed topological space structure.
Given an algebraic theory T , a T -algebra Xe is a product preserving functor
eX:T ! Spaces*. A morphism of T -algebras is a natural transformation of func-
tors.
We will say that a space X admits a T -algebra structure, if there is a T -al*
*gebra
eXand a homeomorphism eX(T1) ~=X.
For an algebraic theory T , by AlgT we will denote the category of all T -alg*
*ebras
and their morphisms.
Example 2.2. For any pointed space A 2 Spaces* we can define an algebraic
theory TA enriched over Spaces*by setting
Mor TA(Tm , Tn) := Map *(Am , An).
Thus, TA is isomorphic to the full subcategory of Spaces*generated by the spaces
An for n 0.
For any Y 2 Spaces*, we can consider a product preserving functor
TA ! Spaces*, Tn 7! Map *(Y, An).
This shows that any mapping space Map *(Y, A) has a canonical structure of a
TA -algebra.
4 B. BADZIOCH, K. CHUNG, AND A. A. VORONOV
Example 2.3. Let A be again a pointed space, and let T A be a category with
objects T0, T1, . .a.nd morphisms
` `
Mor T A(Tm , Tn) = Map *( A, A).
n m
In other words, T A is isomorphic to the oppositeWof the full subcategory of Sp*
*aces*
generated by the finite wedges of A. Since nA is an n-fold coproduct of A in
Spaces*, Tn is an n-fold categorical product of T1's in T A. It follows that T *
*A is an
algebraic theory. For Y 2 Spaces*, we can define a functor
`
T A ! Spaces*, Tn 7! Map *( A, Y ).
n
Therefore the mapping space Map *(A, Y ) has a canonical structure of a T A-alg*
*ebra.
In particular, ifnA = Sn we get that any n-fold loop space canonically defines *
*an
algebra over T S .
2.4. A special instance0of an algebraic theory T Ais obtained when we take A = *
*S0.
The category T S is equivalent to the opposite of the category of finite pointe*
*d sets.
One can check that the forgetful functor
S0
UT S0:AlgT ! Spaces*, UT S0(Xe) = eX(T1),
gives an equivalence of categories.0Also, for any algebraic theory T there is a*
* unique
map of algebraic theories IT :T S ! T . If UT :AlgT ! Spaces*is the forgetful
functor, UT (Xe) = Xe(T1), then we have UT = UT S0O IT* where IT*: AlgT !
0
AlgT S is the functor induced by IT .
3. Tensor product of functors
Definition 3.1. Let C be a small topological category,oi.e.,pa small category e*
*n-
riched over Spaces*, and F 2 SpacesC*, G 2 SpacesC*. The tensor product F C G
is the colimit
W j1 W
F C G := colim (c,d)2CxCMor(c, d) ^ F (c) ^_G(d)//_//_c2CF (c) ^.G(c)
j2
The map j1 is the wedge of the maps ev^ id:(Mor (c, d) ^ F (c)) ^ G(d) ! F (d) ^
G(d), where evis the evaluation map, and j2 is similarly induced by the evaluat*
*ion
maps ev: Mor(c, d) ^ G(d) ! G(c).
The most important - from our perspective - property of the tensor product is
given by the following
op
Proposition 3.2. Let C be a small topological category and G 2 SpacesC*. Con-
sider the functor
Map *(G, -): Spaces*! SpacesC*, Z 7! Map *(G, Z).
The left adjoint of Map *(G, -) exists and is given by
- C G: SpacesC*! Spaces*, F 7! F C G.
For a proof see, e.g., [ML98 ].
YET ANOTHER DELOOPING MACHINE 5
3.3. Assume now that we have two small categories C and D enriched over Spaces*
and two functors F :C xD ! Spaces*and G: Cop ! Spaces*. For every d 2 D, the
functor F defines F (d): C ! Spaces*by F (d)(c) = F (c, d). Applying the tensor
product construction, we obtain a new functor F C G: D ! Spaces*such that
(F C G)(d) = F (d) C G. Since smash product in Spaces*commutes with taking
colimits, for any H :Dop ! Spaces*we have a natural isomorphism
H Dop(F C G) ~=(H DopF ) C G 2 Spaces*.
3.4. Our main interest lies in the following instances of these constructions:
1) For A 2 Spaces*, let T A be the algebraic theory defined in Example 2.3.
Consider the functor A
A :Spaces*!WSpacesT*
given by A (Y )(Tk) := Map *( kA, Y ). By Proposition 3.2, A has a left adjo*
*int
A W W
BA :SpacesT* ! Spaces*, given by BA (F ) =WF T A oA.WHere o A denotes the
functor from (T A)op to Spaces*such that o A(Tk) = k A. Note that A (Y ) pre-
*
* A
serves products, and so A takes values in the full subcategory AlgT A SpacesT*
**.
Thus, we get an adjoint pair (BAW, A ) of functors between AlgT Aand Spaces*.
2) For A 2 Spaces*, let End( oA) denote the functor T Ax (T A)op ! Spaces*
defined by ` ` `
End( A)(Tk, Tl) := Map *( A, A).
o k l
W
Using the canonical map IT A:T S0! T A (see 2.4), we can view End( oA) as a
functor on the category T Ax (T S0)op. For Y 2 Spaces*, define
` 0 A
FT A(Y ) := End( A) (T S0)op S (Y ) 2 SpacesT*.
o
One can check that FT A(Y ) preserves products, i.e., defines a T A-algebra. Th*
*us
we get a functor
A
FT A:Spaces*! AlgT , Y 7! FT A(Y ),
which is left adjoint to the forgetful functor
A
UT A:AlgT ! Spaces*, UT A(Xe) = eX(T1).
We will call FT A the free T A-algebra functor and FT A(Y ) the free T A-algebra
generated by Y .
3) Consider again an algebraic theory T A and let op be the simplicial categ*
*ory.
Let eXo:T Ax op ! Spaces*be a simplicial T A-algebra. Let [o]+ : ! Spaces*
denote the pointed cosimplicial space [n] 7! [n]+ . In this case the tensor pr*
*oduct
eXo op [o]+ =: |Xeo| gives the geometric realization of Xeo. Since realizati*
*on
preserves products in Spaces*, we see that |Xeo| is a T A-algebra.
3.5. Notice that the isomorphism of Section 3.3 shows that for a pointed simpli*
*cial
space Yo we have |FT AYo| ~=FT A|Yo|, and that similarly for a simplicial T A-a*
*lgebra
eXowe get |BA eXo| ~=BA |Xeo|.
3.6. Finally, consider the functors A and UT Aof Section 3.4. The composition
UT AO A :Spaces* ! Spaces* is given by UT AO A (Y ) = Map *(A, Y ). As a
result its left adjoint BA O FT Ais the smash product BA O FT A(Y ) = Y ^ A. Th*
*is
observation indicates that the algebraic theory T A may be suitable for describ*
*ing
6 B. BADZIOCH, K. CHUNG, AND A. A. VORONOV
mapping spaces from A, at least in some cases. Indeed, a simple computation
shows that for a finite pointed set Z, we have Mor AlgT(AMap*(A, Z ^ A), eX) ~=
Map *(Z, UT A(Xe)). Thus, by the adjointness of FT Aand UT A, we get
Lemma 3.7. For any pointed finite set Z, we have a canonical isomorphism
FT AZ ~=Map *(A, Z ^ A)
of T A-algebras.
Combining this isomorphism with the equality BA (FT A(Z)) = Z ^ A, we see that
BA acts as a classifying space for Map *(A, Z ^ A). Our goalnwill be to show th*
*at
when we take A = Sn, this construction works for any T S - algebra.
4. Model categories and Quillen equivalences
Our strategy of approaching Theorem 1.1 will be to reformulate it in the lang*
*uage
of model categories and prove it in this form. Below we describe model category
structures we will encounter in this process. As it was the case so far, most o*
*f our
setup will apply to mapping spaces Map *(A, Y ) from an arbitrary space A, and
only in the proof of Theorem 4.8, we will specialize to A = Sn.
For any algebraic theory T , the category of T -algebras AlgT has a model cat-
egory structure with weak equivalences and fibrations defined objectwise, i.e.,*
* via
the forgetful functor UT , [SV91 ]. For a CW-complex A 2 Spaces*, let RA Spaces*
denote the category of pointed spaces together with the following choices of cl*
*asses
of morphisms:
- a map f :Y ! Z is a weak equivalence in RA Spaces*, if f*: Map *(A, Y ) !
Map*(A, Z) is a weak equivalence of mapping spaces;
- a map f is a fibration if it is a Serre fibration;
- a map f is a cofibration if it has the left lifting property with respec*
*t to all
fibrations which are weak equivalences in RA Spaces*.
Proposition 4.1. The category RA Spaces*is a model category.
Proof.The statement follows from a general result on the existence of right loc*
*al-
izations of model categories, see [Hir03, 5.1, p. 65].
Note that for A = S0, this defines the standard model category structure on
Spaces*.
In order to avoid confusing RA Spaces*with Spaces*, we will call weak equiva-
lences (respectively, fibrations and cofibrations) in RA Spaces*A-local equival*
*ences
(respectively, fibrations and cofibrations). Notice that a map f :Y ! Z is an
Sn-local equivalence, iff it induces isomorphisms f*: ßq(Y ) ! ßq(Z) for q n.
4.2. A cofibrant resolutionAof a T A-algebra. Directly from the definition of t*
*he
model structure on AlgT , it follows that every T A-algebra is a fibrant objec*
*t. The
structure of cofibrant algebras is more complicated (see [SV91 ]). For an arbit*
*rary
algebra eX2 AlgT A, one can however describe its cofibrant replacement as follo*
*ws.
Recall the adjoint pair
FT A:Spaces* ____//_AlgToA:UToA_
of Section 3.4.2.
YET ANOTHER DELOOPING MACHINE 7
Proposition 4.3. For any CW-complex A 2 Spaces*, the functors
FT A:Spaces* ____//_AlgToA:UToA_
form a Quillen pair.
Proof.The functor UT Asends weak equivalences and fibrations in AlgT Ato weak
equivalences and fibrations in Spaces*, respectively, thus the conclusion follo*
*ws.
Next, consider the adjoint functors
| . |: SSets*__//_Spaces*:Singooo_
between the categories of pointed spaces and pointed simplicial sets, where Sin*
*go
is the singularization functor and | . | is geometric realization. We will deno*
*te by
FT0A:SSets* ! AlgT Athe composition of | . | and FT A, and by U0T A:AlgT A!
SSets*the functor obtained by composing UT Awith Singo. The functors FT0A, U0T A
form again a Quillen pair. Therefore forAany T A-algebra eX, they define a simp*
*licial
object FT0AU0ToAeXin the category AlgT which has the algebra (FT0AU0TA)(k+1)eX
in its k-th simplicial dimension. Its face and degeneracy maps are defined usin*
*g the
counit and the unit of adjunction, respectively (compare [May72 , Chapter 9]). *
*Let
|FT0AU0T AoeX| denote the objectwise geometric realization of FT0AU0ToAeX.
Lemma 4.4. |FT0AU0T AoeX| is a T A-algebra.
Proof.Clearly, |FT0AU0T AoeX| is a functor from T A to Spaces*. Also, since we
are working in the category of compactly generated spaces, realization preserves
products, and so |FT0AU0T AoeX| is a T A-algebra.
Similarly to [Bad02 , 3.5, p. 903], we get
Lemma 4.5. For any eX2 AlgT Athere is a canonical weak equivalence
|FT0AU0T AoeX| ! eX.
The above lemma remains to be true, if we replace the functors FT0Aand U0T A
with FT Aand UT A, respectively. What we will use in the sequel (see Step 3 of *
*the
proof of Theorem 4.8) though is that the free algebras (FT0AU0T A)nXe are gener*
*ated
by spaces obtained as realizations of simplicial sets. The algebra |FT0AU0T Aoe*
*X| can
be taken as a cofibrant replacement of eX, since we have
Lemma 4.6. For any eX2 AlgT Athe algebra |FT0AU0T AoeX| is a cofibrant object in
AlgT A.
Proof.This is a consequence of [SV91 ], which describes the structure of cofibr*
*ant
objects in the model category AlgT .
Next, let A 2 Spaces*. Recall (Section 3.4.1) that we have an adjoint pair of
functors (BA , A ). Moreover the following holds:
Proposition 4.7. For any CW-complex A 2 Spaces*, the functors
BA :AlgT A_____//RAoSpaces*:oA_
form a Quillen pair.
8 B. BADZIOCH, K. CHUNG, AND A. A. VORONOV
Proof.The functor A sends A-local equivalencesAand A-local fibrations to weak
equivalences and fibrations in AlgT , respectively which yields the statement *
*fol-
lows.
Our main result, Theorem 1.1, can now be restated more precisely as follows:
Theorem 4.8. For n 0 the Quillen pair
n_____// n
Bn :AlgT S oo___RSn Spaces*: ,
where Bn := BSn and n := Sn, is a Quillen equivalence. In particular, the two
functors induce an equivalence of the homotopy categories.
Corollary 4.9n(Approximation theorem). For any CW-complex X 2 Spaces*, the
following T S -algebras are weakly equivalent:
FnX ~-! n nX,
where FnX denotes the free T Sn-algebra FT SnX on X and nX = Sn ^ X is
the reduced suspension. Moreover, these equivalences establish an equivalence *
*of
monads Fn ~ n n on the category of CW-complexes.
Let us first deduce Theorem 1.1 and Corollary 4.9 from Theorem 4.8.
Proof of Theorem 1.1.Let Xe be any T Sn-algebra, and let Xe ~! eXcbe its cofi-
brant replacement. Like any other object in RSn Spaces*, BnXec is fibrant and
therefore Theorem 4.8 implies that the adjoint Xec ! nBnXecnof the identity
isomorphism BnXec '-!BnXec is a weak equivalence of T S -algebras. Therefore
eX(T1) ' nBnXec(T1), and we indeed recover the statement of Theorem 1.1.
Proof of Corollary 4.9.Byn[SV91 ] the free Fn-algebra generated by a CW-complex
X is cofibrant in AlgT S . The space BnFnX is fibrant, as any object of RSn Spa*
*ces*.
Then the isomorphism BnFnX id-!BnFnX implies by Theorem 4.8 that the adjoint
FnX ! nBnFnX is a weak equivalence. On the other hand, BnFnX = nX by
3.6. Thus, we get a weak equivalence FnX -~! n nX. It defines an equivalence
of monads, because of the naturality of the construction.
Proof of Theorem 4.8.It is enough to show that for every cofibrant T Sn-algebra
eX, the unit jXe:Xe ! nBnXe of the adjunction (Bn, n) is a weak equivalence
n Sn
in AlgT S . Indeed, for eX2 AlgT , Y 2 Spaces*, and f :eX! nY , we have a
commutative diagram
jfX
eXFF___// nBnXe
FFF | n [
f FFF##Fffflffl||
nY,
where f[ is the adjoint tonf. Assume that eXis cofibrant. By assumption jXeis a
weak equivalence in AlgT S . If f is also a weak equivalence, then so is nf[. *
*In
particular the map
nf[(T1): n(BnXe) = ( nBnXe)(T1) ! ( nY )(T1) = nY
YET ANOTHER DELOOPING MACHINE 9
is a weak equivalence of spaces, or, in other words, f[ is an Sn-local weak equ*
*iva-
lence.
Conversely, if f[ is an Sn-local equivalence, then nf[ is an objectwise weak
equivalence, and so is f. n
The proof of the fact that for a cofibrant eX2 AlgT S , the map jXeis a weak
equivalence follows from a bootstrap argument below.
1) Let eX= Fn(Z), where Z is an arbitrary pointed discrete space. Since Fn is a
left adjoint functor, it commutes with colimits. Therefore, since Z is the coli*
*mit of
the poset of finite subsets Y of Z containing the basepoint, we get:
Fn(Z) = colimY ZFn(Y ) = colimY ZMap*(Sn, Y ^ Sn).
The second equality follows from 3.7. Furthermore, since Sn is a compact space,
we have colimY ZMap*(Sn,nY ^ Sn) = Map *(Sn, Z ^ Sn). Therefore, the map jXe
is an isomorphism of T S -algebras by 3.6.
2) Let Zo be a pointed simplicial set, and let eX= Fn0(Zo), where Fn0= FT0Sn. We
have by 3.5
Fn0(Zo) = Fn(|Zo|) ~=|FnZo|,
where FnZo denotes the simplicial T Sn-algebra obtained by applying Fn in each
simplicial dimension of Zo. By Step 1 for every k 0, we have an isomorphism
jk: Fn(Zk) ! nBnFn(Zk), assembling into a simplicial map by naturality. Thus,
the map
|jo|: eX! | nBnFn(Zo)|
is also an isomorphism. Next, notice that by 3.6, we have BnFn(Zk) = Zk^ Sn, so
it is an (n - 1)-connected space. Therefore (see [May72 , Theorem 12.3]), we ha*
*ve a
natural weak equivalence | nBnFn(Zo)| ' n|BnFn(Zo)|. (A technical condition
of properness of BnFn(Zo), needed for applying May's theorem, is satisfied here,
as Zo is discrete and Bn and Fn are admissible functors, see [May72 , Definitio*
*ns
11.2 and A.7].) Combining this with the isomorphism |BnFn(Zo)| ~=Bn|Fn(Zo)|
we get a weak equivalence
| nBnFn(Zo)| ' nBn|Fn(Zo)| ~= nBnXe
It follows that jXeis a weak equivalence.
3) Let Xe be any T Sn-algebra and Fn0U0noeXits simplicial resolution as in Sec-
tion 4.2, where U0n= U0T Sn. Note that, in every simplicial dimension k, the al*
*gebra
(Fn0U0n)kXe is of the form considered in Step 2. It follows that for k 0, we *
*have a
weak equivalence
(1) jk: (Fn0U0n)kXe-~! nBn(Fn0U0n)kXe.
To see that the map
|jo|: |Fn0U0noeX| ! | nBnFn0U0noeX|
is also a weak equivalence, we can use a result of May [May72 , Theorem 11.13].
The assumption of strict properness [May72 , Definition 11.2] of the simplicial*
* spaces
Fn0U0noeXand nBnFn0U0noeX, needed for May's theorem, is not hard to verify, si*
*nce
all the functors Fn, Un, |Singo(.)|, Bn, and n are admissible in the sense of *
*[May72 ,
Definition A.7]. May also assumes that the realizations of the simplicial space*
*s are
connected H-spaces, which will not be satisfied in our case, in general. His re*
*sult
however readily generalizes to the case of simplicial spaces whose realizations*
* are
10 B. BADZIOCH, K. CHUNG, AND A. A. VORONOV
H-spaces with ß0's having a group structure, as it is the case for the simplici*
*al
spaces at hand for n 1. The H-space structure is not there for n = 0, but in *
*this
case, the statement of the theorem is trivial, anyway.
Using arguments similar to those employed in Step 2, we get from here that
j :|Fn0U0noeX| ! nBn|Fn0U0noeX|
is a weak equivalence.
4) Let eXbe any cofibrant algebra. We have a commutative diagram:
j
|Fn0U0noeX|_~//_ nBn|Fn0U0noeX|
h|~| |nBnh|
fflffl|jfX fflffl|
Xe___________// nBnXe,
where h is the weak equivalence of Lemma 4.5. The functor Bn is a left Quillenn
functor and as such it preserves weak equivalences between cofibrant T S -algeb*
*ras,
while n preserves all weak equivalences. Therefore nBnh is a weak equivalence,
and, as a consequence, so is jXe.
Theorem 4.10. Suppose T is an algebraic theory such that it
(1) acts on n-fold loops spaces nX by naturalnoperations ( nX)k ! ( nX)l,
i.e., admits a morphism OE : T ! T S , and
(2) via this action deloops n-fold loop spaces in the sense of Theorem 4.8, *
*i.e.,
n Sn OE*
the loop functor RA Spaces*--! AlgT -! AlgT establishes a Quillen
equivalence.
Then OE : T ! T Snis a weak equivalence of topological theories.
This theorem is, in fact, an obvious corollary of a uniqueness theorem [Bad03*
* ,
Theorem 1.6] (theories considered in [Bad03 ] are enriched over simplicial sets*
*, but
the proof of this result holds for topological theories with little changes).
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School of Mathematics, University of Minnesota, Minneapolis, MN 55455
E-mail address: badzioch@math.umn.edu
E-mail address: krchung@math.umn.edu
E-mail address: voronov@math.umn.edu