A SHEAFTHEORETIC VIEW OF LOOP SPACES
MARK W. JOHNSON
Abstract.The context of enriched sheaf theory introduced in [15] provides
a convenient viewpoint for models of the stable homotopy category as well
as categories of finite loop spaces. Also, the languages of algebraic ge*
*ometry
and algebraic topology have been interacting quite heavily in recent yea*
*rs,
primarily due to the work of Voevodsky and that of Hopkins. Thus, the la*
*n
guage of Grothendieck topologies is becoming a necessary tool for the al*
*gebraic
topologist. The current document is intended to give a somewhat relaxed *
*in
troduction to this language of sheaves in a topological context, using f*
*amiliar
examples such as nfold loop spaces and pointed Gspaces. This language *
*also
includes the diagram categories of spectra from [19] as well as spectra *
*in the
sense of [17], which will be discussed in some detail.
1.Introduction
An excellent description of the formalism of enriched sheaf theory under cert*
*ain
technical assumptions on the base category is given in [5]. Unfortunately, the *
*cate
gory of topological spaces (which will refer to compactly generated, weak Hausd*
*orff
spaces with basepoints) does not satisfy these technical assumptions. Specifica*
*lly,
limits and colimits do not interact particularly well in topological spaces. T*
*hus,
the author was forced to develop a notion of enriched sheaves of topological sp*
*aces
in [15]. The main purpose was to pin down Peter May's suggestion that infinite
loop spaces might be viewed as something akin to sheaves of spaces. In fact, the
technical details of this statement will be one of the main results of the curr*
*ent
article.
Both the notion of a stack and a simplicial sheaf involve the notion of a site
or a Grothendieck topology. The importance of both stacks and simplicial sheaves
alone should justify some attempt to give a gradual introduction to the language
of sheaves, intended for topologists. Thus, using examples familiar to topologi*
*sts,
such as categories of nfold loop spaces and pointed Gspaces, the basic defini*
*tions
will be discussed.
The work of [19] deals with diagram categories of spectra. However, diagram
categories will be shown to be a particularly nice type of sheaf category. Thus*
*, their
work as well as the viewpoint of [17] is unified with that of nfold loop space*
*s and
pointed Gspaces in the language of sheaf categories. Unfortunately, there seem*
*s to
be significant uncertainty about the "fancy new models" for the stable homotopy
category such as symmetric or orthogonal spectra, (see [13] or [19]). The last *
*section
of this article will address the technical differences between these models.
____________
Date: January 29, 2001.
1991 Mathematics Subject Classification. Primary: 55P42; Secondary: 18F20, 1*
*8G55, 55P35.
Key words and phrases. loop spaces, spectra, Quillen closed model categories*
*, enriched sheaves.
1
2 M. W. JOHNSON
The article is organized as follows: section 2 recalls some basic information*
* about
topological diagram categories, section 3 introduces the notions of Grothendieck
topology and Grothendieck basis with a number of motivating examples. Section
4 defines the notion of sheaf on either a topology or a basis and culminates wi*
*th
the proof in our cases that sheaves on a basis agree with sheaves on the topolo*
*gy
generated by the basis. The focus shifts to stable homotopy theory in section 5,
where the example of spectra in the sense of [17] is shown to be a category of *
*sheaves
in the usual category of prespectra. Finally, in section 6 the different models*
* for
the stable homotopy category are discussed, with a comparison of their technical
properties.
2. Topological Diagrams
The language of enriched categories, while intimidating, becomes fairly famil
iar in topological contexts. Essentially, the question becomes one of consider*
*ing
topologies on morphism sets so that standard maps become continuous.
Once again, (pointed) topological spaces will mean o*, the category of com
pactly generated, weak Hausdorff spaces with basepoints. This category is closed
symmetric monoidal under the compactly generated smash product.
Definition 2.1.The category C is a (pointed) topological category provided the
morphism sets are equipped with topologies which make them topological spaces
such that the composition maps C(B; C) ^ C(A; B) ! C(A; C) are continuous in
these topologies.
Examples 2.2.
1. The most basic example of a topological category will be the category of
pointed spaces o*, with the Kelly functor applied to the compactopen topo*
*lo
gies on mapping spaces.
2. Any standard category may be made topological by giving the morphism
spaces discrete topologies and adding disjoint basepoints.
3. A full subcategory of a topological category remains topological, although
the question is a bit more delicate for subcategories where only a subset *
*of
maps are considered. This follows from the fact that subspaces may not be
compactly generated.
4. The category of pointed Gspaces for a compact Lie group G, as discussed in
2.8.2.
5. The category of nfold loops spaces and nfold loop maps, as indicated in
3.5.2.
All of the examples discussed throughout this article will be (pointed) topol*
*ogical
categories. The general viewpoint is that o* has become the "ground ring" in the
algebraic sense. Further discussion of this type of abstract viewpoint can be f*
*ound
in [16].
Definition 2.3.A functor between topological categories is a topological func
tor provided the induced maps of morphism spaces C(A; B) ! D(F (A); F (B)) are
continuous.
Similarly, natural transformations will be defined by assuming the usual com
mutative squares on morphism spaces consist of continuous maps.
LOOP SPACES AS SHEAVES 3
Warning 2.4.In general this disagrees with the usual meaning in an enriched cat
egory, however this will suffice for all of the examples considered in this art*
*icle.
A pair of topological functors F and G are topological adjoints provided the
usual bijection of morphism sets extends to a homeomorphism of spaces. That is,
C(X; G(Y )) must be naturally homeomorphic to D(F (X); Y ) rather than assuming
a natural bijection as would be the case for a standard adjunction. In particul*
*ar,
topological adjoints are standard adjoints.
This leads one to wonder how much more structure a standard adjoint pair must
preserve in order to qualify as a topological adjoint pair. The most convenient
answer is given in section II.6 of [3] and involves socalled tensors and coten*
*sors.
The statement is that a standard adjoint pair is a topological adjoint pair if *
*the left
adjoint preserves tensors, or equivalently, if the right adjoint preserves cote*
*nsors.
A topological category C is tensored over o* provided there exist natural obj*
*ects
XM 2 C for each X 2 C and M 2 o* with the property that the space C(XM; Y )
is naturally homeomorphic to o*(M; C(X; Y )) for arbitrary Y 2 C. Another way
to describe the situation is that (X?; C(X; ?)) becomes a topological adjoint p*
*air
between o* and C, which is also natural in the argument X.
There is a dual notion of cotensor, or hom object, which requires natural hom*
*e
omorphisms o*(M; C(X; Y )) C(X; hom(M; Y )) for any M 2 o* and X; Y 2 C.
Notice in particular that, for M 2 o*, the functors ? M and hom (M; ?) form
a topological adjoint pair between C and itself whenever C is both tensored and
cotensored over o*.
Examples 2.5.
1. The category o* is tensored and cotensored over itself, where the tensor is
defined as the compactlygenerated smash product and the cotensor is the
compactlygenerated mapping space.
2. The category of loop spaces is cotensored over spaces, with hom(M; Y ) aga*
*in
defined as the compactlygenerated mapping space. This is a loop space
because of the exponential law for o*. In fact, this category is also tens*
*ored
by later results, in a much more complicated way.
3. The usual notion of loops on a spectrum is simply hom (S1; Y ).
Next is the main definition of this section.
Definition 2.6.Suppose C is a small topological category. Then the category of
diagrams indexed by C will denote the category of topological functors from C
to o*, with topological natural transformations as morphisms.
Remark 2.7.Small in this context refers to the object class of C being a set.
Examples 2.8.
1. Choose C1 to be the category with one object, whose morphism space is
S0. Then the category of diagrams indexed by C1 is o* itself. To see thi*
*s,
notice a topological functor in this case consists of a space together wit*
*h a
continuous map S0 ! o*(X; X). However, since this must be a pointed map
and functors must preserve identities the map must be the adjoint of the
identity X ^ S0 X ! X.
2. More generally, suppose G is a topological group and let G denote the cate*
*gory
with one object whose morphism space is G+ . (Here G+ denotes G with
a disjoint basepoint added.) The composition law should be given by the
4 M. W. JOHNSON
group multiplication via G+ ^ G+ (G x G)+ ! G+ . Then the category of
diagrams indexed by G is the category of pointed left Gspaces. The catego*
*ry
of diagrams indexed on the opposite of G is the category of pointed right
Gspaces.
3. Let E be the topological category whose objects are the nonnegative intege*
*rs
so that E(n; n) = (n)+ and all other morphism spaces are the basepoint.
The composition law E(n; n)^E(n; n) ! E(n; n) is given by the multiplicati*
*on
in n. Then the category of diagrams indexed by E consists of sequences of
spaces {Xn} such that Xn is a pointed nspace and morphisms are sequences
of maps that are nequivariant at the nth entry.
4. Let A be the topological category whose objects are the nonnegative intege*
*rs
so that
z__mn_"_____
A(m; n) = S1 ^ . .^.S1
with composition given by associativity, unit or zero maps (if m < n, Smn
is a basepoint, while S0 is intended for m=n). Then the category of diagra*
*ms
indexed on the opposite of A is called prespectra (on the canonical indexi*
*ng
sequence) in [17] or often naive spectra in other sources. To see this, co*
*nsider
z___mn_"_____
the pointed continuous map S1 ^ . .^.S1! o*(Xn; Xm ) which is adjoint to
the usual structure map mn Xn ! Xm .
5. Let be a topological category with objects the nonnegative integers. Clea*
*rly,
there is an action of the symmetric group mn on A(m; n) by permuting
smash factors. This extends naturally to an action of the larger group m
on (m )+ ^mn Smn . One defines the spaces (m; n) to be (m )+ ^mn
Smn . The associativity and unit maps which induce the composition law in
A extend by naturality of these extended actions to make a small topologi*
*cal
category. The category of diagrams indexed on the opposite of is then the
"intersection" of the previous two examples. That is, objects of this new
diagram category are prespectra such that the nth entry has a n action,
while the structure maps become equivariant in an appropriate sense. This
category is called symmetric spectra and is discussed in the later portion*
*s of
[13] (the early portions are devoted to the simplicial version, which is q*
*uite
similar) or in [19]. The main technical point is that symmetric spectra fo*
*rm
a closed symmetric monoidal category, i.e. carry a "good" smash product on
the pointset level.
6. This is the "coordinatefree" version of the previous example. Let U denot*
*e a
countabledimensional real innerproduct space. Let O denote a topological
category whose objects are the finite linear subspaces of U. Given objects
V and W , define SWV to be the onepoint compactification of W ?V , the
orthogonal complement of V in W . Note W ?V is defined to be empty (hence
SWV is the basepoint) if V * W and contains only zero (hence SWV
is S0) if W = V . Clearly, the orthogonal group O(W  V ) acts naturally
on SWV . However, as in the previous example one defines O(W; V ) to be
O(W )+ ^O(WV )SWV with composition law given by extension of the nat
ural associativity, unit or zero maps SWW0 ^ SW0V ! SWV . Thus, O
becomes a small topological category and the category of diagrams indexed
LOOP SPACES AS SHEAVES 5
on the opposite of O is referred to as orthogonal spectra in [19]. As with*
* sym
metric spectra, the category of orthogonal spectra carries a closed symmet*
*ric
monoidal structure (or "good" smash product). The increased complexity of
the (nonfinite) orthogonal groups is justified by the coordinatefree nat*
*ure
of this construction as well as the fact that stable weak equivalences wil*
*l be
unambiguously defined (unlike in the case of symmetric spectra).
Given a topological functor F : C ! D, precomposition defines a topological
funtor from the category of diagrams indexed on D to that indexed on C. In fact,
this precomposition functor will have both a left and a right topological adjoi*
*nt by
the usual Kan extension formulae.
For each nonnegative integer n, there is an obvious topological functor from *
*C1
(of example 1) to A (of example 4) sending the unique object of C1 to n and act*
*ing as
the identity on morphism spaces. The comments above imply that precomposition
by this functor will have both a left and a right topological adjoint. However,
consideration of examples 1 and 4 implies this precomposition functor is isomor*
*phic
to the evaluation at the nth entry of a prespectrum. Hence, one has both left a*
*nd
right topological adjoints to evaluation which implies both limits and colimits*
* are
defined entrywise in prespectra. This case is general in the sense that left an*
*d right
adjoints to evaluation can be produced in this manner for any diagram category.
In fact, one can describe these adjoints for the current situation, while they
can become quite complicated for larger indexing categories. The left adjoint *
*to
evaluation at n is usually labeled the nth desuspension of a space. The kth ent*
*ry
of Ln(X) is knX, where negative suspensions indicate a basepoint. The right
adjoint is less wellknown, primarily because it yields stably trivial objects.*
* Given
M 2 o*, define a diagram Rn(M) indexed on the opposite of A with the kth entry
of Rn(M) defined as nkM (which should be taken to mean a basepoint when
k > n). This is clearly natural and yields the expected right adjoint to evalua*
*tion
at n without much difficulty.
In the case of example 5, the entries of Ln(M) look like (k)+ ^kn Skn ^ M.
An entry of Rn(M) is given by (map (Skn; map((n)+ ; X)))kn , or the space of
knequivariant maps from Skn to the cofree pointed nspace on M. A similar
modification involving orthogonal rather than symmetric groups yields formulae *
*for
the adjoints in the case of example 6. One should notice that even the nth space
of Rn(M) is not M, but rather the cofree space associated to M, map ((n)+ ; M).
Only the zeroth entry yields a standard loop space on M, namely nM.
By now the reader will have noticed the later examples were all described as
categories of diagrams indexed on the opposite of a certain category. This is i*
*n line
with the following definition.
Definition 2.9.Suppose C is a small topological category. Then the category of
(enriched) presheaves on C is the category of diagrams indexed on the opposite
of C.
Thus, from the examples above, presheaves on A yields the category of prespec
tra, presheaves on yields the category of symmetric spectra, presheaves on O
yields the category of orthogonal spectra and presheaves on G yields the catego*
*ry
of pointed right Gspaces.
Clearly, for Z 2 C the formula HZ(Y ) = C(Y; Z) defines a diagram HZ or
C(?; Z) indexed on the opposite of C, hence a presheaf. Presheaves of the form
C(?; Z) are commonly referred to as representable presheaves. In fact, the Yone*
*da
6 M. W. JOHNSON
embedding which sends Z to C(?; Z) is an equivalence of C with the full subcate*
*gory
of representable presheaves. This equivalence follows from the fact that natura*
*lity
implies any morphism is completely determined by where it sends the identity map
on the target object. (The reason for defining presheaves in terms of diagrams
indexed on an opposite category is to make this a covariant equivalence.)
3. Grothendieck Topologies and Bases
Unfortunately, the definition of a Grothendieck topology is quite daunting to
many, primarily because of the generality inherent in the definition. A series*
* of
examples will be discussed to motivate the definitions. A Grothendieck topology
may be viewed as a possible answer to the question, "which subobjects of a repr*
*e
sentable presheaf might be viewed as `filling up' the representable in some nat*
*ural
sense ?" The reader should keep in mind that the motivation is closely tied to *
*the
notion of sheaves to be introduced in the next section.
The approach to Grothendieck topologies taken here, following [4], will depend
upon the notion of a subobject of a presheaf. Ordinarily, category theorists de*
*fine
subobjects as (isomorphism classes of) monomorphisms into a fixed object.
However, it is more natural in topological situations to consider subsets equ*
*ipped
with the subspace topology.
Warning 3.1.Recall that the naive subspace topology may not be compactly gen
erated, hence "the subspace topology" refers to applying the Kelly functor k to*
* this
naive construction. If the naive subspace is either closed or open, the functor*
* k is
not required.
Thus, a subobject of a presheaf X will consist of another presheaf Y together
with a morphism of presheaves Y ! X so that each evaluation functor sends this
morphism to the inclusion of a subspace (in this compactly generated sense). Th*
*us,
for example, taking Y to be X with finer topologies on the entries in such a wa*
*y that
Y remains a presheaf would NOT yield a subobject of X in this sense. However, it
should be clear that the constant diagram on the basepoint is a subobject of ev*
*ery
presheaf.
Because the axioms are often confusing at first glance, the reader should keep
in mind that they bear some resemblance to familiar properties of open covers in
basic topology. The first axiom is essentially the statement that the identity *
*map
is an open cover, the second that restriction to a subspace preserves open cove*
*rs,
and the third reflects transitivity of open covers. (See [2] for a different ve*
*rsion of
a Grothendieck topology which reflects this analogy more closely.)
A sieve refers to a subobject of a representable functor in a presheaf catego*
*ry.
Suppose r : R ! C(?; C) is a sieve and D 2 C. Then a point x 2 RD gives a point
r(x) 2 C(D; C), hence a corresponding morphism xr : D ! C.
Definition 3.2.Suppose C is a small topological category. Then a Grothendieck
topology on C is the choice, for every object C 2 C of a family >(C) of sieves *
*in
C(?; C) (often called the covering sieves) which satisfy the following axioms:
o each C(?; C) is in >(C) for C 2 C;
o given a covering sieve r : R ! C(?; C) and a morphism f 2 C(D; C), the sie*
*ve
f1 (R) must cover C(?; D), where f1 (R) is defined by pulling back R over
LOOP SPACES AS SHEAVES 7
the morphism f;
f1 (R)______//R
 
 
fflffl fflffl
C(?; D)_f*_//C(?; C)
o suppose s : S ! C(?; C) is a covering sieve, and r : R ! C(?; C) is a siev*
*e such
that f1 (R) covers C(?; D) for each morphism f of the form xs, correspond*
*ing
to x 2 SD . Then the sieve R must be a covering sieve in C(?; C) as well.
As usual, there is also a notion of basis for a Grothendieck topology. In fa*
*ct,
the most natural description of the example of spectra in this framework is as a
category of sheaves on a basis. (See 5.2.) The examples will be discussed after*
* the
definition of a basis.
In order to define a basis, one needs appropriate notions of image of a morph*
*ism
of presheaves and union of such images. Since morphisms are actually natural
transformations, the image of a natural transformation may be defined entrywise
and will remain a presheaf.
For a union of images, simply take the subspace topology on the underlying set
images, equipped with the "structure maps" of the target object.
Definition 3.3.An (enriched) Grothendieck basis on C will consist of a family
of sieves satisfying the following axioms:
o each identity 1 : C(?; C) ! C(?; C) is a basis cover;
o given a basis cover s : S ! C(?; C) and a morphism g : D ! C, there exists
some basis cover r : R ! C(?; D) which is a subobject of g1(S), i.e. which
factors through g1(S) ! C(?; D);
R


fflffl
g1(S)_______//_S
 
 
fflffl fflffl
C(?; D)_g*_//C(?; C)
o suppose s : S ! C(?; C) is a basis cover and {Rx ! C(?; Dx)} is a family of
basisScovers indexed over elements x 2 SDx. Then the union of their images
x2S xs(Rx) must be a basis cover of C(?; C).
The topology associated to a basis will consist of the sieves r : R ! C(?; C)*
* which
factor some inclusion of a basis cover. The following shows this process actua*
*lly
yields a Grothendieck topology.
Proposition 3.4.Any basis induces a Grothendieck topology, with R ! C(?; C) a
cover in the topology precisely when there is an element of the basis R0 such t*
*hat
R0is a subobject of R.
Proof.Since pullbacks preserve subobjects by definition, the first two axioms f*
*or
a Grothendieck topology are obvious from those for a basis. It is not immediate*
*ly
obvious how to apply the third axiom for a basis in order to verify that for a
8 M. W. JOHNSON
topology. Thus, given a covering sieve S from the topology and an arbitrary sie*
*ve
R ! C(?; C), choose a subobject S0of S which is a basis cover. Given any morphi*
*sm
f : D ! C of the form xs, one assumes that f1 (R) is a cover of D. Hence, there
exists a covering sieve in the basis,SSx, which is a subobject of f1 (R). Now,*
* the
third axiom for the basis implies x2S0f(Sx) is a coverSof C. However, since t*
*he
topology on R is inherited from C(?; C), the union x2S0f(Sx) is a subobject of*
* R __
by construction. This implies R is a cover in the topology by definition. *
* __
Examples 3.5.
1. The category C1 from 2.8.1 supports only two topologies. The first consist*
*s of
both the basepoint and the identity as covers. The second consists of only*
* the
identity as a cover. These examples are completely general and play the ro*
*le of
the discrete and indiscrete topologies. By declaring every sieve to be a c*
*over,
the axioms are clearly satisfied for an arbitrary indexing category. Late*
*r,
it will become clear that the only sheaf on this topology is the basepoint.
Similarly, declaring only isomorphisms to be covering sieves one exhibits a
topology on an arbitrary indexing category. Here the condition of being a
sheaf will become vacuous, so that presheaf categories will be examples of
sheaf categories using this indiscrete topology.
2. Let An denote the category with two objects 0 and n, and morphism spaces
given by An(n; 0) = Sn, An(0; n) the basepoint and S0 as endomorphism
spaces, with topological unit maps for composition. This is a small topolo*
*gical
category, hence yields a topological presheaf category Pn. By analogy with
2.8.4, this category Pn consists of pairs of spaces X0 and Xn together with
a continuous map nX0 ! Xn. One may form a basis for a topology here
in a manner quite similar to the previous example. The symbol Sn will
denote the representable on the object n, i.e. the object written n* ! S0.
Similarly, S0 will denote the representable on the object 0, i.e. the obj*
*ect
written nS0 ! Sn (where the map is the expected isomorphism). In line
with later notation, Sn Sn corresponds to the object n* ! Sn. There
is an obvious inclusion of Sn Sn as a subobject of nS0 ! Sn which will
be indicated as in : Sn Sn ! S0 (the subspace is clearly closed). The
claim is that in together with the identities form a basis for a topology.*
* As
before, the first axiom is trivial. For the second axiom, one must consid*
*er
several cases of pulling back over different morphisms. In any case, pulli*
*ng
back a monomorphism over the zero map will yield an isomorphism, which is
a cover by assumption. Pulling back over identities is clearly going to yi*
*eld
nothing new, hence one need only consider pulling back over some nonzero
morphism Sn ! S0. Notice the 0 entry of the preimage f1 (Sn Sn)
is the basepoint in any case, as a subobject of Sn0 = *. Since in is an
isomorphism at entry n, the n entry of the preimage will be isomorphic to *
*the
n entry of Sn which is simply S0. In other words, f1 (Sn Sn) Sn
which is a cover by assumption.
Finally, for the covering sieve S in the third axiom notice that the only
choices are Sn Sn or S0 over S0 and Sn over Sn . From the existence
of identity maps, it should be clear that the axiom is always satisfied fo*
*r all
S but Sn Sn ! S0. In this case, choose R ! S0 a sieve such that
f1 (R) covers Sn for f = xs and x 2 Sn. Since in is an isomorphism at
entry n, this implies Rn (S0)n (the only cover of Sn is an isomorphism).
LOOP SPACES AS SHEAVES 9
However, this implies R Sn Sn or R S0 since these are the only
such sieves. In particular, R is a covering sieve by definition, which ver*
*ifies
the third axiom.
4. Sheaves on a Site or a Basis
Let C denote a small topological category, together with a Grothendieck topol*
*ogy
> on C, where P is the presheaf category indexed on C. The pair (C; >) is refer*
*red
to as a Grothendieck site.
Definition 4.1.Suppose (C; >) is a Grothendieck site.
o A presheaf Y 2 P will be called a sheaf on this site provided each precomp*
*o
sition P(r; Y ) : P(C(?; C); Y ) ! P(R; Y ) by a covering sieve r : R ! C(*
*?; C)
is a homeomorphism.
o A separated presheaf on a site indicates P(r; Y ) is a monomorphism for
each covering sieve R.
o The topology > will be called a subcanonical topology provided each rep
resentable presheaf C(?; C) is a sheaf on this site.
One should keep in mind that the nature of monomorphisms implies that the
class of separated presheaves on a site will be closed under subobjects.
Now suppose B is a basis for a topology on P. The obvious variations of the
previous definitions, considering only basis covers yield notions of sheaves on*
* a
basis, etc.
Definition 4.2.A subcategory is called:
o reflective provided there exists a left adjoint to the inclusion of the su*
*bcat
egory, and the left adjoint is called the reflector.
o strongly reflective if the reflector preserves monomorphisms between sieve*
*s.
The following result is highly suggestive of the relation of sheaf categories*
* to
localizations.
Lemma 4.3. Suppose the subcategory of sheaves on a basis is reflective. Then t*
*he
reflector sends each inclusion of a covering sieve in the basis r0 : R0! C(?; C*
*) to
an isomorphism.
Proof.Naturality of the reflector yields the following commutative diagram of i*
*so
morphisms.
(r0)*
P(C(?; C); Y_)___________//P(R0; Y )
 
fflffl fflffl
S(S(C(?; C)); Y_)S(r0)*__//S(S(R0); Y )
Hence, the morphism S(r0) induces an isomorphism under precomposition against
every object in the subcategory. It is a brief exercise in diagram chasing_to_*
*see
S(r0) must then be an isomorphism of sheaves. __
In order to avoid the technically difficult construction of the associated sh*
*eaf
functor (see [15]), the following is phrased conditionally. The usual role of *
*the
associated sheaf functor is to be the strong reflector for the category of shea*
*ves.
10 M. W. JOHNSON
Proposition 4.4.Suppose the subcategory of sheaves on a basis B is strongly re
flective. Then it agrees with the category of sheaves on the topology generated*
* by
the basis.
Proof.It should be clear that any sheaf on the topology generated by a basis mu*
*st
be a sheaf on the basis by definition. Since both subcategories are full, it su*
*ffices
to show any sheaf on the basis is a sheaf in the topology. Thus, suppose Y is a
sheaf on the basis while r : R ! C(?; C) is the inclusion of a cover in the top*
*ology
generated by the basis. If the reflector S applied to r is an isomorphism, then*
* the
following diagram implies P(r; Y ) is an isomorphism.
*
P(C(?; C); Y_)____r______//P(R; Y )
 
fflffl fflffl
S(S(C(?; C)); Y_)_S(r)*__//_S(S(R); Y )
However, by assumption there exists a commutative diagram
R0FF
 FFr0FF
 FFF
fflffl##
R _____//C(?; C)
of subobjects where r0 : R0 ! C(?; C) is a cover in the basis. Applying S to th*
*is
diagram then yields a factorization of an isomorphism as two monomorphisms (by
the assumption of a strong reflector), which must then be isomorphisms for_form*
*al
reasons. __
In fact, the associated sheaf functor often allows one to recover the topolog*
*y as
those r : R ! C(?; C) with S(r) an isomorphism, see [5], [18] (or [15]) for det*
*ails.
Examples 4.5.
1. Consider the category Pn together with the basis for a topology introduced
in 3.5.2. By construction the diagram
i*n
Pn(S0; Y )_________//_Pn(Sn Sn; Y )
 
fflffl fflffl
Y0 ___________________//_nYn
commutes, where the bottom map is adjoint to nY0 ! Yn. Since both
vertical maps are homeomorphisms, requiring Y to be a sheaf in the topology
discussed in 3.5.2 is equivalent to demanding that Y0 is homeomorphic to
nYn via the adjoint structure map. Hence, the category of sheaves on this
basis is equivalent to the category of nfold loop spaces. Notice 4.4 then
implies the category of sheaves on the topology generated by this basis is
another description of nfold loop spaces. (The strong reflector is given*
* by
the functor sending nX0 ! Xn to n(nXn) ! Xn.)
2. There is a somewhat more natural variation on the previous example which
is closer to 2.8.4. Thus, let An0consist of the full subcategory of A who*
*se
objects are the nonnegative integers between 0 and n (inclusive). This may*
* be
LOOP SPACES AS SHEAVES 11
equipped with a basis for a topology consisting of all inclusions Sk Skm*
* !
Sm with n k m 0. (The case n = 1 will be proven below for the
interested reader.) As in the previous example, the category of sheaves on
this basis is equivalent to the category of nfold loop spaces as well as *
*being
strongly reflective. Consider the functor which takes an arbitrary preshea*
*f Y
to the diagram whose kth entry is nkYn. This construction clearly yields a
sheaf and a simple exercise verifies the universal property of a reflector*
*, hence
4.4 applies.
Combining the previous two examples, one can see that more than one site may
be used to model the same category of sheaves up to topological equivalence. Th*
*is
suggests that one attempt to find presheaf categories which are closer to a giv*
*en
category of sheaves. Since colimits in the category of sheaves are formed by ap*
*plying
an associated sheaf functor to the colimit in a presheaf category, one would li*
*ke to
minimize the damage done by this associated sheaf functor in order to produce
tractable models for such colimits.
The technique is based on a relatively simple idea. Given a sheaf category S
described in terms of a site (C; >), form a new indexing category E which consi*
*sts
of the full subcategory of "representable sheaves", i.e. the associated sheave*
*s of
any representable functors in the presheaf category. Thus, the objects of E are*
* in
11 correspondence with the objects of C, but the morphism spaces of E are, in
general, richer than those of C. In particular, there is a natural inclusion fu*
*nctor
J : E ! S which may be used to construct an (topological) adjoint pair between
PE and S. The left adjoint L : PE ! S will be defined as the left (topological)
Kan extension of J over the Yoneda embedding E ! PE. The right adjoint is then
given by precomposition with J, i.e. R(X)E = S(J(E); X) for each X 2 S and
E 2 E. Notice, this is completely analogous to the construction of the geometric
realization/singular set pair between simplicial sets and spaces, hence L is ge*
*nerally
called the realization functor and R its associated singular functor.
In fact, the functor R is an equivalence onto its image, because every elemen*
*t of
S is an (indexed) colimit of elements in the image of the functor J. The questi*
*on
then becomes one of constructing a Grothendieck topology for PE where the image
of R is the category of sheaves on this topology. This construction is possible*
* by
exploiting the fact that RL is left adjoint to the inclusion of the image of R.*
* In
line with the comment following 4.4, this adjunction usually allows one to build
a topology consisting of those inclusions r : R ! E(?; E) which are sent to an
isomorphism by RL.
Examples 4.6.
1. Consider the category of nfold loop spaces as described in example 4.5.1
above. This approach suggests defining a new indexing category En with 0
and n as objects, and morphism spaces given by as in example 3.5.2 other
than the fact that En(0; 0) should be the space of topological endomorphis*
*ms
of Sn rather than simply S0. Hence the objects of PEn will be those elemen*
*ts
X of Pn which, in addition, have a natural action of this endomorphism spa*
*ce
(considered as a pointed topological monoid) on X0. Notice the inclusion w*
*ill
not be full, since all morphisms in PEn must satisfy an additional equivar*
*iance
condition. As expected from the discussion above, this diagram category mo*
*re
nearly approximates nfold loop spaces than the category Pn.
12 M. W. JOHNSON
2. Similarly, one can improve upon the indexing category considered in example
4.5.2 in this manner. The resulting diagram category consists of elements *
*of
the former diagram category which support an action of the topological end*
*o
morphisms of Snk on the kth entry, as well as various additional structu*
*re
maps. For example, there must be a structure map nnkXk ! X0 which
would correspond to n on the usual structure map nkXk ! Xn when
X0 nXn. Once again, the inclusion will not be a full functor as more
equivariance conditions are imposed in the new category.
These examples are suggestive of the usual practice of imposing an action of *
*the
endomorphisms operad in the study of infinite loop spaces.
5.Spectra as Sheaves
The motivating example for the author's work in the first chapter of [15] is
summarized in the following few results.
Lemma 5.1. On the category PA , the collection of morphisms Sk Skm ! Sm
with k m 0 form a basis for a topology.
Proof.First, the case k = m is included to ensure that each identity is a basis
cover. For the second axiom, choose any morphism f : Sl ! Sm and consider the
preimage f1 (Sk Skm ). If f is the zero map, the fact that Sk Skm ! Sm
is a monomorphism implies this preimage is isomorphic to Sl itself. However,
each nonzero f is itself a monomorphism. This implies that for entries below
k, the preimage is the basepoint. Since the inclusion Sk Skm ! Sm is
an isomorphism for entries greater than or equal to k, the preimage will also be
isomorphic to Sl for entries greater than or equal to k. Because the structure
maps (and topologies since all inclusions are closed embeddings) are inherited *
*as a
subobject of Sl, this uniquely determines the preimage as Sk Skl for k > l
or Sl itself if k l. In either case, the preimage is also a basis element.
The third axiom is accessible primarily because the composition law in the ca*
*t
egory A consists of isomorphisms. Hence, every morphism Sl ! Sm factors
through each intermediate Sj for l j m. Also, the entries of any Sk Skm
consist of either the zero map or all possible maps. Thus, taking the union of *
*all
possible images of a family of Sj Sjlwill yield some Sk Skm sitting_inside
Sm . __
Corollary 5.2.The category of spectra (on the canonical indexing category, in t*
*he
sense of [17]) is the category of sheaves on the basis given by 5.1, as well as*
* being
strongly reflective.
Proof.As in example 4.5.1, considering diagrams of the form
i*n
Pn(Sn ; Y_)________//_Pn(Sk Skn; Y )
 
fflffl fflffl
Yn ___________________//_knYk
implies the sheaf condition is equivalent to saying each adjoint structure map *
*is a
homeomorphism. The fact that the category of spectra is strongly reflective is *
*the_
focus of the first section of the appendix to [17]. *
*__
LOOP SPACES AS SHEAVES 13
Definition 5.3.A cofinal sieve will refer to a sieve r : R ! Sn such that each
evaluation rj is an isomorphism for some choice of k and all j k.
(Compare this with the definition of cofinal given in [1].) The term cofinal
topology was suggested by R. Bruner.
Theorem 5.4. The collection of cofinal sieves form a Grothendieck topology on
PA , the cofinal topology. Furthermore, the category of spectra (on the canoni*
*cal
indexing sequence) is equivalent to the category of sheaves on the cofinal topo*
*logy.
Proof.The previous corollary reduces the proof of the theorem to the identifica*
*tion
of the cofinal sieves as those sieves containing an element of the basis from 5*
*.1.
Since the inclusion ikn: Sk Skn ! Sn is an isomorphism above entry k and
the inclusion of a basepoint below, any cofinal sieve contains some Sk Skn by
definition.
Conversely, suppose a sieve r : R ! Sn contains some Sk Skn. Then
evaluation at each entry j k leads to a factorization of an isomorphism as a
pair of monomorphisms (Sk Skn)j ! Rj ! Snj. However, this implies both
maps are actually homeomorphisms, hence rj is a homeomorphism for all j__k as
desired. __
6.Building the Stable Homotopy Category
The reader will notice that all of the current topological models for the sta*
*ble
homotopy category, with the exception of the Smodules of [7], have been placed
in the framework of enriched sheaf categories. As presheaf (or diagram) categor*
*ies,
prespectra, symmetric and orthogonal spectra are all categories of sheaves on "*
*in
discrete" topologies. However, some form of localization is required in order *
*to
produce a model for the stable category from any of these as a starting point. *
*The
category of spectra has just been exhibited as a proper sheaf category, or cate*
*gorical
localization of prespectra. A similar homotopytheoretic localization also yiel*
*ds the
stable category (see Theorem 6.8 below), in a manner more familiar to students *
*of
Adams's model from [1]. Analogous homotopytheoretic localizations of symmet
ric and orthogonal spectra produce models for the stable category. However, this
structure on orthogonal spectra may be produced more easily by direct comparison
with the stable structure on prespectra, as in Theorem 6.10 below. The intent of
this section is to contrast the technical properties of these models.
The language of homotopy theory used here will be that of Quillen's homotopic*
*al
algebra [20]. Those unfamiliar with this language may want to glance at the rec*
*ent
books [9] or [11]. For full details of the existence of the model structures de*
*scribed
below see a more general case in [14] or [12] or the author's original viewpoin*
*t in
[15].
Recall that a model structure consists of choosing three classes of maps: weak
equivalences, cofibrations and fibrations, which must satisfy a series of axiom*
*s gen
eralizing standard notions such as the homotopy extension property. The followi*
*ng
definition describes one standard method of building new model categories from
old ones, originally due to Quillen.
Definition 6.1.Suppose C is a model category while L : C ! D and R : D ! C
form an adjoint pair. Then the model structure is said to lift over the adjoint
pair provided D is a model category with a morphism h 2 D a weak equivalence or
fibration precisely when R(h) 2 D is a weak equivalence or fibration, respectiv*
*ely.
14 M. W. JOHNSON
Remark 6.2.
1. In practice, this technique is usually applied to cofibrantly generated mo*
*del
categories. Then an additional condition is that the generating cells in D*
* are
the set of morphisms L(g) with g a generating cell in C.
2. The generalization of this technique to include a set of adjoint pairs is *
*de
scribed in [14] and implies the existence of the strict structures describ*
*ed
below.
Examples 6.3.
1. The usual model structure on pointed spaces is lifted from unpointed space*
*s in
this sense, where the adjoint pair is adding a disjoint basepoint or forge*
*tting
the basepoint. This explains why basepoints other than the fixed basepoint
must be considered in defining a weak homotopy equivalence of pointed spac*
*es.
2. The usual structure on spaces may be lifted to Gspaces via the free func
tor/forgetful functor pair. However, this is generally not an important st*
*ruc
ture there, as it essentially ignores the orbit data. The usual structure*
* on
Gspaces comes from a similar trick involving a set of adjoint pairs, or l*
*ifting
a set of times and intersecting the various structures. See [14] for detai*
*ls of
this approach.
One useful point about lifting model structures is that it simplifies the que*
*stion
of whether the adjoint pair is a Quillen equivalence. This is exploited in the *
*proof
of Theorem ortstable below.
Definition 6.4.Suppose C and D are both model categories with (L; R) an adjoint
pair and L : C ! D.
1. The pair is called a (strong) Quillen pair if L preserves cofibrations and*
* R
preserves fibrations.
2. A Quillen pair is called a Quillen equivalence if for each cofibrant X 2 C
and fibrant Y 2 D the morphism L(X) ! Y is a weak equivalence in D
precisely when its adjoint map X ! R(Y ) is a weak equivalence in C.
Remark 6.5.The condition of being a Quillen pair is sufficient to imply the adj*
*unc
tion descends to an adjunction on the associated homotopy categories. A Quillen
pair descends to an adjoint equivalence of homotopy categories if and only if i*
*t is a
Quillen equivalence.
The following is generally referred to as the "strict structure", in line wit*
*h the
notation of [6]. The adjectives in the statement are technically important but *
*the
casual reader may want to disregard them.
Proposition 6.6.The categories of prespectra, symmetric spectra and orthogonal
spectra each carry a cofibrantly generated, proper, topological model structure*
* with
fibrations and weak equivalences defined entrywise.
The following is essentially a corollary, because the structure on prespectra*
* is
transported to the subcategory of spectra by lifting over the associated spectr*
*um
functor (and its right adjoint, the inclusion of spectra in prespectra).
Corollary 6.7.The category of spectra carries a cofibrantly generated, topologi*
*cal
model structure with fibrations and weak equivalences defined entrywise.
LOOP SPACES AS SHEAVES 15
There are two key technical elements, both due to Lewis in [17]. First, cofib*
*ra
tions in spectra will be entrywise closed embeddings. Second, sequential colimi*
*ts
over entrywise closed embeddings between spectra do not require an application *
*of
the associated spectrum functor. Thus, the small object argument may be applied,
despite the fact that evaluations do not preserve colimits, in general.
This is the first example so far of a model for the stable homotopy category.
Thus, a categorical localization of the category of prespectra yields such a mo*
*del
and one is lead to wonder if a homotopical localization might yield a model as *
*well.
Consider the map f in PA which is defined as the coproduct of all morphisms
Sk ^ Skn ! Sn with k n. There is a general approach to inverting a map in
the homotopy category of a model category, commonly called a localization of the
model category. (See [10].)
Theorem 6.8. The flocalization of the strict structure on prespectra is Quill*
*en
equivalent to the structure on spectra given by Corollary 6.7. Furthermore, the
flocalization of the strict structure yields the same homotopy theory as Adams*
*'s
model for the stable homotopy category.
The key point in the proof of this theorem is that the associated spectrum
functor acts as an flocal replacement functor, at least for cofibrant prespect*
*ra.
(See Lemma 4.3.5 in [15].) The comparison with Adams's model then follows almost
immediately from Theorem III.3.4 in [1].
There are natural forgetful functors from orthogonal to symmetric spectra and
from symmetric spectra to prespectra. Each of these may be written as an enrich*
*ed
precomposition, hence has both left and right adjoints via Kan extensions, which
are topological functors. In particular, there are L : PA ! P and L0: PA ! PO
each left adjoint to a forgetful functor, which preserve the entrywise smash pr*
*oduct
with a space (or tensor) and preserve representable functors.
The following is phrased differently than the approach in [13], but is essent*
*ially
equivalent.
Theorem 6.9. The functor L together with the forgetful functor forms a (topolo*
*g
ical) Quillen equivalence between the flocalization of the strict structure on*
* pre
spectra and the L(f)localization of the strict structure on symmetric spectra.
Unfortunately, the L(f)local equivalences are not determined by the usual st*
*able
homotopy groups, unless the objects in question are L(f)local. There was origi
nally much confusion about the definition of stable weak equivalences in symmet*
*ric
spectra, which seems relatively straightforward from this description. A functo*
*rial
way of dealing with the difference between stable weak equivalences and maps be
tween symmetric spectra inducing isomorphisms on stable homotopy groups has
been introduced in [22].
By contrast, the situation in orthogonal spectra is simpler, primarily becaus*
*e a
local replacement operation is not necessary.
Theorem 6.10. The flocalization of the strict structure on prespectra lifts o*
*ver
the adjoint pair consisting of L0 and the forgetful functor to yield a Quillen *
*equiv
alent (topological) structure on orthogonal spectra.
The key difference between L and L0stems from the fact that the connectivity
of orthogonal groups remains constant, while that of symmetric groups increases
in n. (Both of these left adjoints are built from free functors, whose connecti*
*vity
16 M. W. JOHNSON
depends on that of the group involved.) Hence, one may verify directly that L0(*
*f)
is a stable homotopy equivalence as a map of prespectra (see [19]) and the rest
follows formally. (See [15].)
In particular, it is true that a morphism g of orthogonal spectra is a L0(f)*
*local
equivalence precisely when g is a stable homotopy equivalence (considered as a *
*mor
phism of prespectra). This is one of the technical advantages of orthogonal spe*
*ctra,
which justifies the added complexity of dealing with the compact topological gr*
*oups
OV rather than the finite groups n.
The category of prespectra plays a pivotal role as the intermediate model that
all the other models may be compared with. In fact, this is not a surprise from*
* the
viewpoint of [16], where prespectra is shown to be initial among stable topolog*
*ical
model categories, in a certain sense. The BousfieldFriedlander category has be*
*en
shown to satisfy a similar property by [21] as a key step along the way to their
classification of simplicial models for the stable homotopy category.
The category of spectra is quite useful because stable homotopy equivalences *
*and
entrywise weak equivalences coincide. It is often quite useful to have all obj*
*ects
fibrant, which is also the case for spectra. For example, much of the recent wo*
*rk
on multiplicative stable homotopy by Goerss and Hopkins [8] has been done in th*
*is
framework.
Symmetric spectra is the simplest model which carries a symmetric monoidal
structure reflecting the smash product in the stable homotopy category, as desc*
*ribed
by Boardman. This yields pointset models for constructions such as THH and
function spectra. The hard question in this category is to determine whether a
morphism is a stable weak equivalence.
Finally, orthogonal spectra also carries a symmetric monoidal structure refle*
*ct
ing the smash product in the stable homotopy category, as described by Boardman.
The difficulty of determining stable weak equivalences is avoided, since they a*
*re pre
cisely the stable homotopy equivalences in the usual sense. However, the indexi*
*ng
category is somewhat more intimidating. Fortunately, most formulae seem to be d*
*e
termined in symmetric spectra and then translated into the appropriate orthogon*
*al
analog.
Thus, the symmetric spectra continue to play a vital role in the discussion of
orthogonal spectra, just as prespectra are technically vital to understand poin*
*tset
constructions in spectra.
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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
Email address: johnson.295@nd.edu