LOCALIZATIONS
W. G. DWYER
1. Introduction
The aim of this paper is to describe the concept of localization, as it
usually comes up in topology, and give some examples of it. Many of
the results we will describe are due to Bousfield.
1.1. Localization. We start with the following very simple situation:
o C is a category, and
o E C is a subcategory.
The morphisms in E are called equivalences, and they are maps which
for one reason or another we want to treat as honorary isomorphisms.
The pair (C, E) is called a localization context.
1.2. Definition. An object X of C is said to be E-local (or just local if
E is understood) if any morphism f : A ! B in E induces a bijection
~=
f* : Hom C(B, X) -! Hom C(A, X) .
Roughly speaking, X is local if equivalent objects of C cannot be
distinguished by mapping them into X.
1.3. Definition. A map ffl : A ! X in C is said to be an E-localization
of A (or just a localization of A) if X is local and ffl is an equivalence.
The pair (C, E) is said to have good localizations if every object of C
has a localization.
Usually we refer to X as the localization of A, and leave the map
A ! X understood. Suppose that X is a localization of A, Y is a
localization of B, and f : A ! B is a map. It is easy to see that there
is a unique map f0 : X ! Y such that the diagram
fflA
A --- ! X
? ?
f?y f0?y
fflB
B --- ! Y
____________
Date: April 28, 2003.
1
2 W. G. DWYER
commutes. With a little argument, this shows that any two localiza-
tions of A are canonically isomorphic; it also shows that if (C, E) has
good localizations, there is a localization functor LE which assigns to
every object A a functorial localization LE(A). If f is a morphism in
E, then LE(f) is an isomorphism, and it is easy to see that the re-
verse implication holds if E satisfies the two out of three property, i.e.,
if whenever f and g are composable morphisms of C and two of f, g,
and gf belongs to E, then the third belongs to E as well. The intuition
is that LE(A) captures exactly the information in A which is invariant
under the notion of equivalence provided by E.
That's the general machinery. There is one main theorem. Let E-1 C
be the category obtained from C by formally inverting the arrows of E;
the functor C ! E-1 C is universal among functors C ! D which send
all of the arrows of E to isomorphisms. (There may be set theoretic
difficulties in forming E-1 C, but the following theorem can be inter-
preted as saying that these difficulties don't come up in the cases we
consider.)
1.4. Theorem. Suppose that (C, E) has good localizations. Let Loc E(C)
denote the full subcategory of C given by the local objects. Then the
composite functor
LocE (C) ! C ! E-1 C
is an equivalence of categories.
Proof. Since the localization functor LE : C ! Loc E(C) sends all of the
morphisms in E to isomorphisms in Loc E(C), it extends to a functor
L0E: E-1 C ! Loc E(C). We leave it to the reader to verify that, up to
natural equivalence, the functor L0Eis inverse to the functor appearing
in the statement of the theorem.
1.5. A slight twist. Sometimes the localization context (C, E) is de-
scribed by specifying the collection of local objects. In this case the
equivalences E are determined by working backwards from 1.2: a map
f : A ! B is an equivalence if for each local object X, the map
f* : Hom (B, X) ! Hom (A, X) is a bijection.
Next we briefly discuss a variant of the localization idea.
1.6. Colocalization. Colocalization is localization in the opposite cat-
egory, but it's worthwhile to be a little more explicit than this, if only
to establish some notation. Suppose that (C, E) is a pair of categories
as above; again, the morphisms in E are called equivalences.. An object
X of C is said to be colocal if any equivalence f : A ! B induces a
bijection
~=
f* : Hom C(X, A) -! Hom C(X, B) .
LOCALIZATIONS 3
Roughly speaking, X is colocal if equivalent objects of C cannot be
distinguished by mapping X into them. If A is an object of C, a map
j : X ! A is said to be a colocalization of A if X is colocal and j is an
equivalence. The pair (C, E) is said to have good colocalizations if every
object of C has a colocalization. As before, any two colocalizations of
A are canonically isomorphic, and, if (C, E) has good colocalizations,
there is a colocalization functor CE which assigns to every A a functorial
colocalization CE(A) ! A. If f in C is an equivalence then CE(f) is an
isomorphism, and the converse holds if E satisfies the two-out-of-three
condition. Let Coloc E(C) denote the full subcategory of C given by the
colocal objects; if (C, E) has good colocalizations, then the composite
functor
ColocE (C) ! C ! E-1 C
is an equivalence of categories.
1.7. Remark. Note that if (C, E) has both good localizations and good
colocalizations, then the category Coloc E(C) of E-colocal objects is
equivalent to the category Loc E(C) of E-local objects, since both are
equivalent to E-1 C [12 , 2.6]. The equivalences in question are given by
the restricting the localization functor to Coloc E(C) and restricting the
colocalization functor to Loc E(C).
1.8. Organization of the paper. Section 2 describes some algebraic
(co-)localization constructions, including local homology and local co-
homology for commutative rings. Section 3 discusses homological local-
ization for spaces and spectra, while x4 treats the more general idea of
localization with respect to a map. In a diagram setting, this kind of lo-
calization leads to Goodwillie calculus (4.10) and to motivic homotopy
theory (4.13). The next section describes colocalization with respect
to an object, and mentions a class of pairs (C, E) which have both good
localizations and good colocalizations (5.6). Section 6 investigates the
higher order structure of E-1 C when (C, E) has good localizations, and
x7 concludes with some remarks on how localizations and colocaliza-
tions are constructed.
1.9. On (not) working up to homotopy. In most of the examples
below, the ambient category C is the homotopy category associated
to some model category M. For instance, M might be the category
of spaces, the category of spectra, or the category of chain complexes
over a ring R, in which cases (given the right weak equivalences in M)
C = Ho (M) is the homotopy category of CW-complexes, the homotopy
category of spectra, or the derived category of R (2.2). Suppose in
general that M is a model category and that E is some subcategory
4 W. G. DWYER
of Ho (M), so that the localization functor LE, if it exists, assigns to
each object X of M a local object LE(X) in Ho (M) together with an
E-equivalence X ! LE(X) in Ho (M). In most of the interesting cases,
the localization functor LE : Ho (M) ! Ho (M) can be covered by a
more rigid functor ^LEdefined on M or at least on the subcategory Mc
of cofibrant objects in M. The functor ^LE takes values in M itself,
and lies in a commutative diagram
L^E
Mc --- ! M
? ?
? ?
y y
LE
Ho (M) --- ! Ho(M)
in which the vertical arrows are given by the natural map M !
Ho (M). In addition, for each cofibrant object X of M there is a
natural map X ! ^LE(X) which projects in Ho (M) to the localization
map X ! LE(X). There are many reasons to seek such a functor ^LE.
For instance, if M is the category of spaces there are ways to apply
^LEfibrewise in a fibration [18 , x1], while there is no good way to do
this with a functor like LE taking values in the homotopy category.
Strictly speaking, having a rigid localization functor is even necessary
to discuss something like the arithmetic square (3.6), since the homo-
topy pullback of a diagram in the homotopy category isn't functorially
determined.
1.10. Localized model categories. Even more is usually true [31 ] [12 , 9.9]
[9, 4.6]. Write M = (M, W), where W is the category of weak equiv-
alences in M, so that Ho (M) = W-1 M. (See x6 for an explanation
of why the pair (M, W) itself almost never has good localizations or
colocalizations.) Let p : M ! Ho (M) be the projection, E the subcat-
egory of Ho (M) that is providing the localization context, and W + E
the subcategory of M consisting of all maps f in M with p(f) 2 E.
If E contains all of the isomorphisms of Ho (M), which in practice is
always the case, then W W + E. Usually it is possible to produce a
model category structure on (M, W + E) with the same cofibrations as
in the original structure on (M, W), but with fewer fibrations. The ho-
motopy category Ho (M, W + E) = (W + E)-1M is then isomorphic to
E-1 Ho (M, W), and the rigid localization ^LE(X) is obtained by work-
ing in this new model structure and factoring the map X ! * into an
acyclic cofibration X ! ^LE(X) followed by a fibration ^LE(X) ! *.
LOCALIZATIONS 5
2. Algebra
In this section we discuss some localization and colocalization con-
structions in algebra and in homological algebra.
2.1. Classical localization for modules. Suppose that R is a com-
mutative ring and that S R is a multiplicative subset, i.e., a subset
of R which is closed under taking products. Let C be the category of
R-modules, and say that an R-module M is S-torsion if for each x 2 M
there is an element s 2 S such that sx = 0. Let E be the subcategory
of C containing all the objects of C as well as those maps f such that
ker(f) and coker(f) are S-torsion; it is easy to argue that the class of
maps of this sort is closed under composition. The local objects are
the modules M which are uniquely S-divisible, in the sense that for
each x 2 M and s 2 S there is a unique y 2 M with sy = x. The pair
(C, E) has good localizations, and the localization of a module M is its
classical algebraic localization S-1 M.
In general, the pair (C, E) does not have good colocalizations. Con-
sider for example the case in which R = Z and S is the multiplicative
set of nonzero integers. If T is a torsion abelian group, then the unique
map T ! 0 is an E-equivalence, and it follows that if M is a colocal
abelian group then Hom (M, T ) = 0 for every torsion abelian group T .
It is easy to check that this holds only if M = 0. But there are special
cases; if S contains only the unit in R, for instance, then every module
is colocal and the colocalization functor is the identity.
2.2. An aside on derived categories. Suppose that R is a ring.
Recall that the bounded derived category Db(R) is the category whose
objects are chain complexes of R-modules which vanish in sufficiently
low degrees and whose morphisms are derived chain homotopy classes
of maps. If X and Y are two chain complexes, Hom Db(R)(X, Y ) is com-
puted by finding a projective resolution X0of X (a bounded below chain
complex of projective modules which maps to X by an isomorphism
on homology) and then computing ordinary chain homotopy classes
of maps X0 ! Y . The (unbounded) derived category D(R) is defined
similarly, except that the chain complexes involved are not bounded
below, and it is necessary to take some extra care with the definition of
projective resolution [44 ]. If f : X ! Y is a map in the derived cate-
gory, we let Cone (f) denote the chain complex mapping cone (cofibre)
of f [46 , 1.2.8].
2.3. Classical localization for D(R). Suppose that R is a commuta-
tive ring and that S R is a multiplicative subset. Let C be the derived
category D(R), and E the subcategory containing all maps f such that
6 W. G. DWYER
H* Cone (f) is S-torsion. Then a chain complex X is E-local if and only
if the homology groups Hi(X) are uniquely S-divisible. The pair (C, E)
has good localizations; the localization of X is the chain complex S-1 X
obtained by algebraically localizing X dimension by dimension. Since
the algebraic localization process is exact, HiLE(X) = S-1 Hi(X). In
general, (C, E) does not have good colocalizations.
In contrast to the case above of classical localization, mod p local-
ization for modules is very different from mod p localization for objects
of the derived category.
2.4. Mod p localization for modules. Let C be the category of
abelian groups, p a prime number, and E the subcategory of C contain-
ing all maps f such that Z=p f is an isomorphism. Then an abelian
group is local if and only if it is a module over Z=p. The pair (C, E)
has good localizations, and the localization of an abelian group X is
Z=p X. The only colocal object is the trivial abelian group, and so
(C, E) does not have good colocalizations..
2.5. Mod p localization for D(Z). Let C = D(Z) be the derived
category of abelian groups, p a prime number, and E the subcategory
of maps X ! Y with the property that Z=p h X ! Z=p h Y is an
isomorphism. Here Z=p h X is the derived tensor product, i.e., the
graded tensor product of X with a projective resolution P of Z=p. To
be very explicit, a map X ! Y is in E if there is a projective resolution
X0 of X and a chain complex map X0 ! Y representing f such that
the induced map P X0 ! P Y gives an isomorphism on homology.
It turns out that the pair (C, E) has both good localizations and good
colocalizations. We will describe these in turn.
For any abelian group M, the connecting homomorphism from the
exact sequence
(2.6) 0 ! Z ! Z[1=p] ! Z=p1 ! 0
induces a map M ! Ext (Z=p1 , M), and M is said to be Ext-p-complete
if this map is an isomorphism [13 , VI]. It turns out that a chain complex
is E-local if and only if each of its homology groups is Ext-p-complete.
Let Hom h(X, Y ) denote the derived chain complex of maps from X to
Y , constructed by taking a projective resolution of X or an injective
resolution of Y and forming the usual mapping chain complex [46 ,
2.7.4]. The exact sequence 2.6 gives a map -1Z=p1 ! Z in the
derived category, and the localization of an object X is the induced
map
X ~= Hom h(Z, X) ! Hom h( -1Z=p1 , X).
LOCALIZATIONS 7
In particular, for each i there is an exact sequence
0 ! Ext (Z=p1 , HiX) ! HiLE(X) ! Hom (Z=p1 , Hi-1X) ! 0 .
If the homology groups of X are finitely generated, then HiLE(X) is
the ordinary p-completion of HiX, i.e.
HiLE(X) ~=Zp HiX ,
where Zp is the ring of p-adic integers.
A chain complex is colocal if and only if each of its homology groups
is a p-primary torsion group. The colocalization of X is given by the
map
-1Z=p1 h X ! Z h X ~= X
induced as above by 2.6. In particular, there are exact sequences
0 ! Z=p1 Hi+1X ! HiCE(X) ! Tor(Z=p1 , HiX) ! 0 .
These results can be proved with relatively simple formal arguments
[20 ]. The same arguments cover the following more general situation.
2.7. Local homology and cohomology. Suppose that R is a com-
mutative ring and that I R is an ideal generated by a finite number
r1, . .,.rn of elements. Let C = D(R) be the derived category of R-
modules, and E the subcategory of maps X ! Y with the property
that R=I h X ! R=I h Y is an isomorphism (2.5). Then the pair
(C, E) has both good localizations and good colocalizations.
We will now describe these explicitly. Given a map M ! N of
R-modules, let denote the chain complex which is trivial
except for M in degree 0 and N in degree -1, with the indicated map
as differential, and let K denote the tensor product chain complex
K = i< R ! R[1=ri] > .
Note that if R = Z and I = (p), then K is isomorphic in D(Z) to
-1Z=p1 . There is an obvious map K ! R which sends all the nega-
tive degree components of K to zero (here we are thinking of R as the
chain complex consisting of the module R concentrated in degree 0).
Then the localization of a chain complex X is the induced map
X ! Hom h(K, X)
and the colocalization is the induced map
K h X ! X .
The localization functor amounts to local homology at the ideal I [28 ]
[29 ] [1], and the colocalization functor to local cohomology at I [30 ].
For instance, if X is the chain complex consisting of the module M
concentrated in degree 0, then HiLE(X) is the local homology group
8 W. G. DWYER
HIi(M), and H-iCE(X) is the local cohomology group HiI(M). If R
is noetherian and the homology groups of X are finitely generated
over R, then HiLE(X) is the I-adic completion of HiX [28 ]. A chain
complex X is local if and only if for each integer i, the natural map
HiX ! HI0(HiX) is an isomorphism; X is colocal if and only if for each
i and each element x 2 Hi(X), there exists an integer k with Ikx = 0
[20 , x6].
3. Homological localization in topology
In this section we describe homological localization constructions for
spaces and spectra, some of them parallel to the algebraic construc-
tions from x2. The basic existence theorem is due to Bousfield [4],
[6]. In the following statement, the homotopy category of spaces means
W-1 M, where M is the category of topological spaces and W is the
subcategory of weak homotopy equivalences. This is equivalent to the
ordinary (geometric) homotopy category of CW-complexes, or to the
usual homotopy category obatained from simplicial sets.
3.1. Theorem. Suppose that C is the homotopy category of spaces or
the homotopy category of spectra. Let A be a spectrum, and E (A) the
collection of maps f in C such that A*(f) = ß*(A ^ f) is an isomor-
phism. Then (C, E ) has good localizations.
3.2. Remark. The notation E (A) is explained by the fact that the
smash product ^ is a kind of tensor product of spectra.
In the above situation, we write LA for the localization functor; if
A is the Eilenberg-MacLane spectrum HR, so that A* = H*(-; R) is
ordinary homology with coefficients in R, we write LR . Two spectra
A, B are said to be Bousfield equivalent if they give rise to the same
localization functor on the category of spectra; this happens if and only
if E (A) = E (B).
3.3. Classical localization for spaces. Here C is the homotopy cat-
egory of spaces, R is a subring of Q (i.e., a ring obtained from Z by
inverting some set of primes), A is the Eilenberg-MacLane spectrum
HR, and A* = H*(-; R). If X is one-connected there are isomorphisms
ßiLR (X) ~= R ßi(X), so that X is local if and only the homotopy
groups of X are modules over R. The same principle works if X is
nilpotent, as long as R ß1(X) is interpreted correctly [4] [13 ].
For non-simply connected spaces, LR (X) can be mysterious even if
R = Z; for instance, LZ(S1 _ S1) is unknown. If the commutator
subgroup of ß1X is perfect and X satisfies some other mild conditions,
then LZX is equivalent to the Quillen plus construction on X.
LOCALIZATIONS 9
3.4. Mod p localization for spaces. Here C is the homotopy category
of spaces, p is a prime, A = HZ=p, and A* = H*(-; Z=p). If X is one-
connected, then X is local if and only if the homotopy groups of X are
Ext-p-complete (2.5). More generally, if X is one-connected there are
short exact sequences [4] [13 ]
(3.5)
0 ! Ext (Z=p1 , ßiX) ! ßiLZ=p(X) ! Hom (Z=p1 , ßi-1X) ! 0 .
There are similar formulas for nilpotent spaces. In particular, if X is
one-connected and the homotopy groups of X are finitely generated,
there are isomorphisms ßiLZ=p(X) ~=Zp ßi(X). For this reason, the
localization LZ=p is sometimes referred to as p-completion.
3.6. The arithmetic square. A good feature of the arithmetic lo-
calization functors described above is that there is a way to rebuild a
space X, at least if X is nilpotent, from its various localizations. This
is provided by the arithmetic square (cf. 1.9).
3.7. Theorem. [45 ] [16 ] Suppose that X is a nilpotent space. Then
there is a homotopy fibre square
Q
X --- ! p LZ=p(X)
? ?
? ?
y y
Q
LQ (X) --- ! LQ ( p LZ=pX)
The bottom row is obtained by applying LQ to the top row. This
square can be viewed as describing how X is determined by p-adic data
for each prime p (upper right), rational data (lower left), and coherence
information over the rationals (arrows terminating at lower right).
3.8. Arithmetic localizations of spectra. Let C be the homotopy
category of spectra, R a subring of Q, and p a prime. If X is bounded
below in the sense that ßiX = 0 for i << 0, then ßiLR (X) is isomorphic
to R ßiX, and ßiLZ=p(X) lies in an exact sequence of the form 3.5,
so the above results apply as stated to the subcategory of C consist-
ing of objects which are bounded below. For spectra which are not
bounded below, these ordinary homology localizations are more com-
plicated. For instance, let KU be the periodic complex K-theory spec-
trum. Then the map H*(KU; Z) ! H*(KU, Q) isQan isomorphism, and
it follows that the map LZ(KU) ! LQ (KU) = i 2iHQ is an equiv-
alence. In particular, ß*LZ(KU) 6= Z ß*(KU). The situation can be
repaired by replacing the coefficient Eilenberg-MacLane spectrum HR
by the Moore spectrum M(R) = LR (S0), and similarly replacing HZ=p
by the Moore spectrum M(p) = S0 [p e1. Then for any spectrum X,
10 W. G. DWYER
ßiLM(R) (X) is isomorphic to R ßiX, and ßiLM(p)(X) lies in an exact
sequence of the form 3.5.
3.9. Chromatic localizations of spectra. We will refer to ideas from
[40 ]. Again, C is the homotopy category of spectra. Pick a prime p, and
take all objects of C to be localized at p in the sense (3.8) that their
homotopy groups are modules over Z(p). Let K(n) be the n'th Morava
K-theory. The n'th chromatic localization functor Ln is defined to be
localization with respect to the wedge K(0) _ . ._.K(n). It is easy to
see that for any spectrum X there is a tower
. . .! Ln(X) ! Ln-1(X) ! . .!.L0(X) ,
and the chromatic convergence theorem [40 , 7.5.7] guarantees that if
X is a finite suspension spectrum the homotopy limit of this tower is
X. The chromatic approach to ß*S0 involves studying how the stable
homotopy ring is built up in the tower {ß*Ln(S0)}. Closely related to
Ln is the localization functor LK(n); in fact, LK(n)(X) determines the
difference between Ln(X) and Ln-1(X), in the sense that there is a
homotopy fibre square
Ln(X) --- ! LK(n)(X)
? ?
? ?
y y
Ln-1(X) --- ! Ln-1LK(n)(X)
rather parallel to 3.6. The spectrum K(0) is Bousfield equivalent to
HQ, and so a spectrum X is K(0)-local if and only if the homotopy
groups of X are uniquely p-divisible, or, in other words, if and only if
the degree p map dp : S0 ! S0 induces bijections [Si, X] ! [Si, X],
i 2 Z. The telescope conjecture can be interpreted as asserting that
there is a similar characterization of spectra which are local with respect
to K(0) _ . ._.K(n).
3.10. Conjecture. (Telescope Conjecture) Suppose that F = F (n) is a
finite complex of type n, and that v : aF ! F is a vn-self-map. Then
a spectrum X is local with respect to Ln (i.e., Ln(X) ~ X) if and only
if composition with v induces bijections
[ iF, X] ! [ a+iF, X], i 2 Z.
This is true for n = 0 (above) and n = 1, but the best current
evidence suggests that it is false in general [35 ].
There are a few calculations. For any X, Ln(X) is equivalent to
Ln(S0) ^ X. The spectrum L0(S0) is the Eilenberg-MacLane spectrum
HQ; L1(S0) is a periodic version of the im J spectrum [6] [39 ]. The
LOCALIZATIONS 11
spectrum L2(S0) is in a sense known for p > 3 [43 ], and partially known
for p = 2 or p = 3 [42 ] [25 ].
3.11. Chromatic localizations of spaces. All of the above chro-
matic localization functors can be applied to spaces as well as to spec-
tra. Again, there are some calculations. Mahowald and Thompson
have computed L1(X) when X is an odd sphere [36 ], and Bousfield has
done the same when X is an infinite loop space [7] or a finite H-space
[10 ]. If X is an infinite loop space with associated spectrum B1 X,
Bousfield has shown that in many cases LnX agrees except in low di-
mensions with 1 Ln(B1 X) [12 ]. Kuhn has found a way to construct
LK(n)(B1 X) in terms of X [34 ], but there's no close relationship be-
tween LK(n)(X) and 1 LK(n)(B1 X). In fact, surprisingly enough, for
most spaces the functor LK(n) agrees up to ordinary p-completion with
the functor Ln [11 ].
4. Localization with respect to a map
Suppose that C is a category with some notion of mapping object
Map (X, Y ) for any two objects X, Y in C. Here are some examples:
o C is the homotopy category of spaces, and Map (X, Y ) = Map h(X, Y ),
the derived space of maps from X to Y , i.e. the space obtained
by replacing X by a weakly equivalence CW-complex and tak-
ing the usual mapping space.
o C is the homotopy category of spectra, and Map (X, Y ) is the
derived mapping spectrum.
o C is the derived category of a ring R, and Map (X, Y ) = Hom h(X, Y )
is the derived chain complex of maps X ! Y (2.5).
Given f : A ! B in C, say that an object X of C is f-local if compo-
sition with f induces an equivalence
f* : Map (B, X) ~ Map (A, X) .
The meaning of "equivalence" depends on the category in which the
mapping objects lie; for spaces or spectra, equivalence would usually
mean weak homotopy equivalence, and for chain complexes it would
mean homology isomorphism. Let E(f) be the category of all maps
g : U ! V in C such that g induces an equivalence Map (V, X) !
Map (U, X) for every f-local object X.
4.1. Definition. In the above situation, if (C, E(f)) has good local-
izations, the localization functor is denoted Lf, and called localization
with respect to f.
12 W. G. DWYER
4.2. Remark. This is a case of working backwards (1.5): we first define
the local objects, and then use them to get the equivalences E(f). The
class E(f) is the smallest class of maps in C which contains the map f
and has a closure property which we leave it to the reader to formulate.
The following theorem of Bousfield and Dror Farjoun is parallel to
3.1
4.3. Theorem. [5] [18 ] If C is the homotopy category of spaces, the
homotopy category of spectra, or the derived category of a ring R, then
for any map f the pair (C, E(f)) has good localizations.
Theorem 4.3 applies in much greater generality [5] [31 ]; for instance,
it applies if C is a homotopy category of diagrams of spaces (4.8).
Virtually all of the localization functors we have looked at so far can
be expressed as localization Lf with respect to some appropriately
chosen map f. In fact, in the category of spaces any homotopically
idempotent functor E with a natural transformation Id ! E can be
expressed as Lf for some map, if you are willing to adopt some exotic
axioms for set theory [14 ].
4.4. Remark. If C is the homotopy category of spectra, there is another
possible interpretation for the mapping object Map (X, Y ); rather than
the derived mapping spectrum, one could take the zero space in the
corresponding infinite loop spectrum, or, more or less equivalently, the
(-1)-connective cover of the derived mapping spectrum. Theorem 4.3
remains true with this interpretation of mapping object, but the local-
ization functors change significantly and have more in common with
their unstable counterparts. Bousfield uses this notion of localization
in [11 , 2.2]. Colocalization functors also change under the connective
interpretation of äM p ": the phrase ä nd its desuspensions" would
have to be deleted from 5.3, and 5.4 would cease in general to be a
fibration sequence. For us, though, äM p " in the context of spectra
will always mean the entire derived mapping spectrum.
4.5. Postnikov sections. Suppose that C is the homotopy category
of spaces and that f is the unique map Sn ! *. Then a space X
is f-local if and only if for each basepoint x 2 X and each i n,
ßi(X, x) is trivial. The class E(f) is the class of all maps g : X ! Y
which for each basepoint x 2 X and each i < n induce a bijection
ßi(X, x) ! ßi(Y, g(x)). The localization Lf(X) is the Postnikov section
Pn-1X.
4.6. Homology localization. If C is the homotopy category of spaces
or spectra and A is a chosen spectrum, then the homology localization
LOCALIZATIONS 13
functor LA (x3) is equivalent to Lf for some huge non-canonical A*-
equivalence f : U ! V . For`instance, if A = HZ=p, then f can be taken
to be the disjoint union fff, where fffruns over all mod p homology
isomorphisms Uff! Vffbetween countable CW-complexes (take them
to be subspaces of R1 , so that the collection {fff} forms a set). In
fact, Bousfield essentially constructs LA by implicitly identifying it as
Lf [6]. It was one of Bousfield's great insights that is possible to work
with this idea without having a particularly explicit hold on the map
f associated to A.
4.7. Nullification. Let C be the homotopy category of spectra, W
an object of C, and f : W ! * the unique map from W to the con-
tractible spectrum. The localization functor Lf is called nullification
with respect to W and assigns to X a universal map X ! PW (X) with
Map (W, PW (X)) ~ *. Localization with respect to f : U ! V is the
same as nullification with respect to the cofibre of f. In particular, the
homology localization functor LA (4.6) can be expressed as nullification
PW with respect to some huge spectrum W with A*(W ) = 0. The n'th
finite localization functor or telescopic localization functor functor Lfn
is nullification with respect a finite spectrum F (n + 1) of type n + 1
[37 ]. The telescope conjecture (3.10) is equivalent to the conjecture
that Ln = Lfn, or to the conjecture that a spectrum X is Ln-local if
and only if Map (F (n + 1), X) ~ *. This is clear from 3.10, since the
cofibre of a vn-self-map of F (n) is a finite complex of type n + 1.
Unstably, the situation is more complicated. Let C be the homotopy
category of spaces, W an object of C, and f : W ! * the unique map
from W to a point. Again the localization Lf is called nullification with
respect to W and denoted PW . The notation PW comes both from the
fact that nullification was originally called periodization [8], and from
the fact that for W = Sn, PW is the Postnikov functor Pn (4.5). It
is no longer true that localization with respect to a map f is nullifi-
cation with respect to the cofibre W of f, although there is a natural
transformation PW ! Lf [9, 4.3]. In particular, unstable homology
localization functors can not in general be identified as nullification
functors. Bousfield and Dror-Farjoun have studied PW extensively [8],
[18 ]. One of the main results is that the nullification functors pre-
serve fibrations up to an error term which is frequently quite small.
Bousfield has used this to prove the remarkable theorem [8, 11.5] that
nullification with respect to an finite complex of type n + 1 preserves
the (unstable) vi-periodic homotopy for i n.
4.8. An aside on diagrams. Suppose that D is a small category. A
D-space, or diagram of spaces with the shape of D, is a functor from D
14 W. G. DWYER
to spaces. These functors form a category, in which the morphisms are
the natural transformations. Say that a morphism X ! Y is a weak
equivalence if for each object d 2 D the induced map X(d) ! Y (d)
is a weak homotopy equivalence of spaces. There is a model category
structure on D-spaces with these weak equivalences [23 ] [17 ] [19 , x2];
in the simplest structure like this, the fibrations are the maps X ! Y
which for each d 2 D give a Serre fibration X(d) ! Y (d). We will
refer to the associated homotopy category as the homotopy category
of D-spaces.
For each d 2 D there is a "free D-space based at d" denoted F and
given by F (x) = Hom (d, x); here we are treating the set Hom (d, x)
as a discrete space. Essentially by Yoneda's lemma, for any D-space
X the derived mapping space Map h(F , X) is canonically weakly
equivalent to X(d). Note that f : d ! d0 gives a morphism f* :
F ! F , such that Map h(f*, X) can be identified up to weak
equivalence with X(f) : X(d) ! X(d0).
4.9. Inverting morphisms. Let C be the homotopy category of D-
spaces, where D is a small category, and suppose that u : d ! d0 is
a map in D. Let f denote u* : F ! F . Then a D-space X
is f-local if and only if X(u) : X(d) ! X(d0) is a weak homotopy
equivalence. It is hard to describe the localization functor Lf in simple
terms.
Here's a related example, which we offer without proof,`in which it is
possible to describe Lf. Let f be the disjoint union u u* indexed by
all of the morphisms u of D. Then a D-space X is f-local if and only
if for each morphism u of D, X(u) is a weak homotopy equivalence.
The category of f-local objects is in a homotopical sense equivalent
to the category of fibrations over the classifying space BD. For an
arbitrary D-space X, Lf(X) is a D-space whose value at each object
d is equivalent to the homotopy fibre, over the vertex corresponding to
d, of the natural map
hocolimD X ! hocolim D* = BD .
In this case Lf is a kind of averaging or integration functor.
4.10. Goodwillie calculus. We begin with something along the lines
of 4.9 but a little more general. Suppose that D is a small category,
P another (usually much smaller) category, and j : P ! D a functor.
Assume that P has an initial object OE, and let P \OE be the subcategory
obtained by deleting the initial object. Given a D-space X, we can
compose with j to get a P-space X0 and ask whether the natural map
(4.11) X0(OE) ! holimP\ffiX0
LOCALIZATIONS 15
is a weak equivalence. It is easy to construct a map L(j) of D-spaces
such that the answer to this question is "yes" if and only if X is L(j)-
local. To be explicit, there is a functor F 0from Pop to D-spaces which
sends x 2 P to the free diagram F . Then L(j) is the natural
map
L(j) : hocolim P\ffiF 0! F 0(OE) .
The homotopy colimit here is calculated in the category of D-spaces, in
other words, objectwise in the category of functors from D to spaces.
The point to keep in mind is that Map h converts homotopy colimits in
the first variable into homotopy limits of mapping spaces.
Now let D be the category of finite complexes; we choose some small
model for this which contains all finite complexes up to homotopy, and
is big enough to contain products, homotopy pushouts, etc. Let C be
the homotopy category of D-spaces. A D-space X is said to be a homo-
topy functor if X(f) is a weak equivalence of spaces whenever f : d ! d0
is a weak equivalence of CW-complexes. Let f1 be the obvious map of
D-spaces with the property that X is a homotopy functor if and only
if X is f1-local (4.9). Let P be the poset of subsets of {1, . .,.n}, so
that P is a category with the empty set as initial object. An n-cube of
finite complexes is a functor j : P ! D, and such an n-cube is said to
be strongly homotopy cocartesian if each of its two-dimensional`faces is
a homotopy pushout diagram (see [27 ]). Let f2 be the map j L(j),
where j ranges over all strongly homotopy cocartesian n-cubes. A D-
space X is f2-local if and only if it carries each strongly homotopy
cartesian n-cube into a homotopy cartesian diagram, or, in the`termi-
nology of [27 ], if and only if X is n-excisive. Let f = f1 f2. Then X
is f-local if and only if X is an n-excisive homotopy functor. The con-
struction which assigns to a homotopy functor X the n-excisive functor
Lf(X) gives the n'th stage PnX in the Goodwillie tower for X [26 ].
4.12. Loop spaces and theories. Let denote the category of the
finite ordered sets {0, . .,.m}, m 0, and weakly monotone maps, so
that a op-space is a simplicial object in the category of spaces. Given
such a simplicial space X, write Xm = X({0, . .,.m}), and say that X
is very special if for each m the natural map
æ1 x . .x.æm : Xm ! X1 x . .x.X1
is a weak homotopy equivalence, where æi : Xm ! X1 is the map cor-
responding to the function {0, 1} ! {0, . .,.m} with æi(0) = 0 and
æi(1) = i. (When m = 0 the right hand side is the empty prod-
uct, which is the one-point space, and the condition reads that X0 is
weakly contractible.) This definition is due to Bousfield [8, 3.3]. As
16 W. G. DWYER
in 4.10, there is an evident map f of op-spaces such that X is very
special if and only if X is f-local. It follows from ideas of Segal [41 ,
App. B] that the homotopy theory of very special op-spaces is equiv-
alent to the homotopy theory of loop spaces. The equivalence takes
a very special op-space X to a loop space canonically weakly equiv-
alent to X1. In particular, the homotopy category of loop spaces can
be obtained as the localization of a diagram category. There is one
immediate consequence. If is a functor from spaces to spaces which
respects homotopy and preserves products up to weak equivalence, then
dimensionwise application of preserves the category of very special
op-spaces. It follows that if U is a loop space, the F (U) is canonically
weakly equivalent to a loop space [18 , x3]. It is possible to treat the
theory of n-fold loop spaces, 2 n 1, in a similar way [3].
It is also possible to go a bit further. For the rest of this para-
graph, take "space" to mean "simplicial set". Let A be the category
of algebraic objects of some fixed equational type, e.g., the category of
groups, abelian groups, groups of nilpotency class n, monoids, rings,
commutative rings, Lie algebras, . .L.et Fm be the free object of A on
m generators, and let TA be the opposite of the full subcategory of A
generated by the Fm . The category TA is the "theoryö f the objects
in A, in the sense that the morphisms of TA determine all of the ways
of combining elements of such an object by algebraic operations. If X
is a TA -space, write Xm = X(Fm ), and say that X is special if for each
m the natural map
æ1 x . .x.æm : Xm ! X1 x . .x.X1
is a weak homotopy equivalence, where æi : Xm ! X1 is the map
corresponding to the A-morphism F1 ! Fm which sends the generator
of F1 to the i'th generator of Fm . Again, there is an evident map f of
TA -spaces such that X is special if and only if X is f-local. Starting
with this observation, Badzioch [2] has shown that the homotopy theory
of special TA -spaces is equivalent to the homotopy theory of simplicial
objects in A; the equivalence takes a special TA -space X to a simplicial
object which is canonically weakly equivalent, as a simplicial set, to
X(F1). In particular, the homotopy category of simplicial objects in
A can be obtained as the localization of a diagram category. Again, it
follows that if is a functor from spaces to spaces which preserves weak
equivalences and preserves products up to weak equivalence, and U is
a simplicial object in A, then F (U) is canonically weakly equivalent to
a simplicial object in A.
If A is the category of groups, then the category of special TA -spaces
is closely related to the category of very special op-spaces [2, 1.6 ff].
LOCALIZATIONS 17
Rather than explain this in detail, we give the following example from
[2]. Consider the TAop-space B which assigns to a free group F the
pointed classifying space BF , where BF is constructed in some func-
torial way from F . Note that BFm is equivalent to a wedge of m circles.
Now for any pointed space Y there is a special TA -space Y given by
letting Y (F ) be the space of basepoint-preserving maps Map *(BF, Y );
the "special" property amounts to the observation that Map *(BFm , Y )
is equivalent to Map *(BF1, Y )m . Since Y (F1) is the loop space Y ,
we conclude from the main theorem of [2] that Y is naturally weakly
equivalent to a simplicial group.
4.13. Motivic homotopy theory. In this paragraph, "space" means
simplicial set. We follow [24 ]. Suppose that S is a noetherian scheme
of finite dimension, and D the category of smooth schemes over S. A
simplicial presheaf on D is just a Dop-space; the category of these is de-
noted Pre(D). Any scheme M over S gives an object of Pre(D), which
we will denote æM, according to the formula æM(U) = Hom (U, M).
The category Pre (D) has a homotopy theory as in 4.8 in which the
weak equivalences are the maps X ! Y which induce weak equiva-
lences X(U) ! Y (U) for each object U 2 D. Call this the sectionwise
homotopy theory; it turns out to be too rigid for most algebraic pur-
poses. If D is furnished with a Grothendieck topology T , then there
is a more flexible T -local homotopy theory [33 ] on Pre (D) in which a
weak equivalence is a map X ! Y which induces an isomorphism on
associated sheaves of homotopy groups for all appropriate choices of
basepoints. See [19 ] for a nice account of how, in a fashion reminiscent
of 4.10 above, this can be obtained as the localization of the sectionwise
theory with respect to a map f0. The map f0 is easy to describe. In
the language of [19 , x1], for each T -hypercover U of an object X in D
there is a map
fU : hocolim næUn ! æX ,
`
and f0 is just the disjoint union U fU , where U runs through a large
enough set of hypercovers [19 , x6]. Now specialize to the case in which
T is the Nisnevich topology (a choice explained by the desire to acco-
modate algebraic K-theory, which has certain patching properties with
respect to the Nisnevich topology), and further localize the T -local ho-
motopy theory on Pre(D) with respect to a map f00derived as in [24 ,
4.1] from æA1 ! *, where A1 is the affine line and * is the constant
one-point presheaf. What results is the A1-homotopy theory or motivic
homotopy theory of Voevodsky and Morel [24 , 4.11] [38 ]. Thus motivic
homotopy theory over S is the localization of the ordinary`homotopy
theory of Dop-spaces with respect to the map f = f0 f00.
18 W. G. DWYER
5. Colocalization with respect to an object
Suppose again that C is a category with some notion of mapping
object (x4), and that W is an object of C. Say that a map X ! Y
is a W -cellular equivalence if it induces an equivalence Map (W, X) !
Map (W, Y ), and let EMap (W ) denote the category of W -cellular equiv-
alences.
5.1. Definition. In the above situation, if the pair (C, EMap (W )) has
good colocalizations, the colocalization functor is written CellW and
called cellularization with respect to W .
The basic existence theorem is again due to Bousfield [5] and Dror
Farjoun [18 ].
5.2. Theorem. If C is the homotopy category of spaces, the homotopy
category of spectra, or the derived category of a ring R, then for any
object W in C the pair (C, EMap (W )) has good colocalizations.
Again, the theorem applies in much greater generality [31 ], and in
particular applies to any homotopy category of diagrams of spaces.
5.3. Stable Cellularization. Suppose that C is the homotopy cate-
gory of spectra or the derived category D(R), and that W 2 C. Then
for any X, CellW (X) lies in the smallest subcategory of spectra con-
taining W and its desuspensions and closed under weak equivalence
and homotopy colimit; this subcategory is called the localizing subcat-
egory generated by W , and the objects in it are said to be built from
W . In fact, CellW (X) is characterized by the fact that it is built from
W and admits a map CellW (X) ! X which lies in EMap (W ). There is
a cofibration sequence (4.7)
(5.4) CellW(X) ! X ! PW (X)
which indicates that cellularization and nullification with respect to W
capture complementary parts of X. The notation CellW (X) is meant
to suggest that this spectrum is built from W and its cones in the same
way a cell complex is built from disks and spheres.
5.5. Unstable cellularization. If C is the homotopy category of un-
pointed spaces, then for any nonempty W and any X, CellW (X) ! X
is an isomorphism [18 , 2.A.4]. Suppose that C is the homotopy category
of pointed spaces and that W 2 C. Then for any X, CellW (X) belongs
to the smallest class of spaces which contains W and is closed under
weak equivalence and pointed homotopy colimits; this is the closed class
of spaces determined by W [18 , 2.D], and again the objects in it are said
to be built from W . The space CellW (X) is characterized by the fact
LOCALIZATIONS 19
that it is built from W and admits a map CellW (X) ! X which lies
in EMap (W ). (Here EMap (W ) is defined using pointed mapping spaces
Map (X, Y ).) In the sequence
CellW(X) ! X ! PW (X)
the composite of the two maps is null, but this is not in general a fi-
bration sequence. Chach'olski [15 ] has studied the relationship between
CellW and PW in detail.
5.6. Localization/colocalization for the same E. In practice, it
is unusual for a pair (C, E) to have both good localizations and good
colocalizations. Suppose for instance that C is the homotopy category
of spectra. Then in localizing with respect to a map f, the equiva-
lences E(f) are detected by Map (-, L) for f-local objects L, while in
colocalizing with respect to an object W , the equivalences EMap (W ) are
detected by Map (W, -). It would appear to be hard to find f and W
such that E(f) = EMap (W ). But there is one case in which this does
happen. If W is a finite spectrum, then for any X
Map (W, X) ~ DW ^ X ,
where DW = Map (W, S0) is the Spanier-Whitehead dual of W . Thus
EMap (W ) = E (DW ) = E(f) for some huge f (4.6), and, if E de-
notes this class of equivalences, (C, E) has both good localizations and
colocalizations. This is particularly useful in chromatic situations [32 ].
Suppose that C is the Ln-local homotopy category of spectra, let W be a
finite complex of type n, and let E = EMap (W ) = E (DW ). Then (C, E)
has both good localizations and good colocalizations; the colocalization
functor is the monochromatic functor Mn, and the localization functor
is localization LK(n) with respect to the Morava K-theory K(n). As
always in this situation (1.7) the categories of local and colocal ob-
jects are equivalent [32 , 6.19]. There is a parallel phenomenon in the
telescopic case, where Ln is replaced Lfn[12 , 3.3].
Suppose that R is a commutative ring and that C = D(R) is its
derived category. Given W in C, let E (W ) denote the category of all
maps X ! Y with the property that W hRX ! W hRY is an isomor-
phism in C. Bousfield's arguments [6] show that (C, E (W )) has good
localizations. Similarly, (C, EMap (W )) has good colocalizations. Sup-
pose that I = is a finitely generated ideal in R. The fact
that (C, E (R=I)) has good colocalizations as well as good localizations
(see 2.7) is explained by the surprising fact that E (R=I) = EMap (R=I)
[20 , 6.5]. This is related to the example involving finite spectra de-
scribed above. In the notation of 2.7, let W be the chain complex
20 W. G. DWYER
given by
ri
W = i< R -! R > .
Then R=I and W can be built from one another, so that E (R=I) =
E (W ) and EMap (R=I) = EMap (W ). Since W is built from a finite
number of copies of R, for any X there is an equivalence
Map (W, X) ~ DW h X ,
where DW = Map (W, R) is the Spanier-Whitehead dual of W . This
gives EMap (W ) = E (DW ). But by inspection, DW is just a suspension
of W itself, so that E (DW ) = E (W ).
6. Higher invariants of localization
Suppose that (C, E) is a localization context. Theorem 1.4 indicates
that the process of localizing individual objects X of C, say by con-
structing LE(X), is closely connected to the process of öl calizing" C
itself by forming E-1 C. But forming E-1 C creates in general a lot of
higher order structure, and the question we want to ask here is how
this higher order structure is related to the localization functors.
The nature of this higher order structure is easy to understand. Sup-
pose that C has a single object, and that E consists of all of the mor-
phisms in C. Then C amounts to a monoid M, and forming E-1 C
involves forming the group completion M-1 M. In this case the higher
structure is the topological space BM; the component group ß0 BM
is M-1 M, but the other homotopy groups ßi BM, i > 0, and indeed
the whole homotopy type of the space BM, are higher invariants of
the group completion process. Another way to obtain these invariants
is to form a resolution R of M by a simplicial monoid which is free
in each degree, and apply the group completion process degreewise to
obtain a simplicial group R-1R. The geometric realization of R-1R is
then weakly equivalent to BM. This exhibits BM or R-1R as the
result of applying to M a kind of total derived functor of the group com-
pletion process. Given a general pair (C, E) of categories it is not hard
to construct a simplicial resolution (RC, RE) of (C, E) by free categories
(in this context, each category is free on a directed graph with vertex
set Obj (C)) and form the simplicial localization L(C, E) = R-1ERC [22 ].
For objects X, Y of C, Hom L(C,E)(X, Y ) is then a simplicial set with
ß0 Hom L(C,E)(X, Y ) ~=Hom E-1C(X, Y )
The homotopy types of these simplicial function complexes, taken to-
gether with the composition maps between them, embody the higher
order structure created in forming E-1 C.
LOCALIZATIONS 21
6.1. Proposition. Let (C, E) be a localization context. If (C, E) has
either good localizations or good colocalizations, then the function com-
plexes in the simplicially enriched category L(C, E) are homotopically
discrete, in the sense that each component is weakly contractible.
This can be interpreted as saying that if the localization functor LE
exists, then the higher order structure associated to E-1 C is trivial. As
will become clear below (see the proof of 6.1) this follows from the fact
that if LE exists then each map u : X ! Y in E-1 C, represented (1.4)
by u0: LE(X) ! LE(Y ), has a canonical zigzag
fflX u0 fflY
X -! LE(X) -! LE(Y ) - Y
of morphisms in C which represent it. Any other representing zigzag
can be naturally deformed to this canonical one, and so the space of
representing zigzags is contractible.
What can be done with a localization context (C, E) which does
not have good localizations or good colocalizations? This is really the
purview of homotopy theory, which in practice relies on the theory of
model categories to obtain access to the homotopy types of the func-
tion complexes in L(C, E) in cases in which these function complexes
are not homotopically discrete.
6.2. Example. Suppose that C is the category of topological spaces, and
E is the subcategory of weak homotopy equivalences. Then (C, E) does
not have good localizations or colocalizations; in fact, the only colocal
object is the empty space, and the only local objects are the empty
space and the one-point space. This is consistent with 6.1, since if X is
a CW-complex and Y is an arbitrary space the simplicial set of maps
X ! Y in L(C, W) is not usually homotopically discrete: its geometric
realization has the weak homotopy type of the ordinary topological
function space of maps X ! Y .
6.3. Remark. In most of the localization contexts (C, E) discussed in
this paper, C = Ho (M) = W-1 M, where (M, W) is the localization
context provided by a model category M and its subcategory W of
weak equivalences. If LE exists in this situation, then by 6.1 no in-
teresting higher order structure is created in passing from W-1 M to
(W + E)-1M (see 1.10 for the notation). This suggests that the higher
structure involved in (W + E)-1M should be the same as the higher
structure involved in W-1 M. In fact, in the situation sketched in 1.10
this is the case: the function complexes in L(M, W + E) are equivalent
to the function complexes in L(M, W) which involve objects of M that
are E-local in W-1 M.
22 W. G. DWYER
Proof of 6.1. We sketch a proof based on [21 ]. Assume that (C, E) has
good localizations, let C0 be the full subcategory of C given by the E-
local objects, and E0 = E \ C0. Let H = LH (C, E) be the hammock
localization of (C, E) and H0 = LH (C0, E0). By [21 , 2.2], it is enough
to show that Hom H(X, Y ) is homotopically discrete, and in fact [21 ,
3.3] it is enough to do this when X and Y are E-local objects. Note
that the complex Hom H0(X, Y ) is homotopically discrete, since all the
morphisms in E0 are invertible and localizing a category by inverting
morphisms which already have inverses does not introduce any higher
structure [22 , 5.3]. By [21 , 5.5], the complex Hom H(X, Y ) is the colimit
of simplicial sets m(X, Y ), where m is a word {C, E-1 } and m(X, Y )
is the nerve of a category [21 , 5.1] whose objects are zigzags of pattern
m connecting X to Y in C. Similarly, Hom H0(X, Y ) = colim m0(X, Y ).
There are inclusions m0(X, Y ) ! m(X, Y ). Applying the functor LE
to the zigzags and using the natural transformation Id ! LE gives a
deformation retraction m(X, Y ) ! m0(X, Y ), and so by [21 , 4.5, 5.4],
the colimit map
Hom H0(X, Y ) = colim m0(X, Y ) ~-!colim m(X, Y ) = Hom H(X, Y )
is a weak equivalence.
7. Constructing localizations and colocalizations
The localization map fflA : A ! LEA has two universal properties:
it is initial among maps A ! X with X local, and terminal among
equivalences A ! A0. To check the first statement, suppose that f :
A ! X is a map with X local. Since fflA is an equivalence, it induces a
bijection
~=
Hom C(LE(A), X) -! Hom C(A, X)
and so there is a unique map f0 : LE(A) ! X such that f0 . fflA = f.
This can be expressed by saying that fflA is an initial element of the
category in which an object is a map A ! X with X local, and a
morphism from A ! X to A ! X0 is a map X ! X0 which makes the
evident diagram commute. Similarly, fflA is a terminal element of the
analogous category in which an object is an equivalence A ! A0.
An initial element in a category is the limit of all of the objects
in the category (more precisely, the limit of the identity functor from
the category to itself), and a terminal element is the colimit of all the
objects. In the present situation this gives two parallel, more or less
LOCALIZATIONS 23
tautological, formulas for the localization LE(A).
8
< lim X (X local)
LE(A) = A!X 0
: colim A (A ! A0 an equivalence)
A!A0
where in each case the limit or colimit is taken over the indicated cate-
gory of maps. In general these formulas are not useful for constructing
LE(A); they just translate the problem of building LE(A) into the more
or less equivalent problem of whether the corresponding limit or colimit
exists. However, this analysis does suggest two approaches to obtain-
ing LE(A), which we will label the Bousfield-Kan approach and the
Bousfield approach.
7.1. The Bousfield-Kan approach. Find some natural collection of maps
fff: A ! Xffwith each Xffis local, and hope that limfffXffcomputes
LE(A).
7.2. The Bousfield approach. Find some natural collection of equiva-
lences gfi: A ! Afi, and hope that colimgfiAficomputes LE(A).
In situations with a topological flavor, it's natural to adjust the above
constructions to use homotopy limits and homotopy colimits instead
of limits and colimits. The Bousfield-Kan approach is used in [13 ]
to construct homology localizations of spaces with respect to ordinary
homology theories; the approach succeeds in many cases, but not in all.
The advantage of this approach is that the homotopy limit diagram
involved is very small (just a cosimplicial space) and easy to calculate
with. Mahowald and Thompson [36 ] and Bousfield [10 ] implicitly use
something like this approach to compute unstable L1-localizations of
odd spheres and H-spaces. Bousfield uses the second approach in [4],
[5], and [6] to construct arbitrary homology localizations as well as
localizations with respect to arbitrary maps. The technique for building
the nullification (4.7) PW (X) of a spectrum X with respect to a finite
spectrum W is especially simple. Let N(X) be the result of attaching
a cone C( iW ) to X for each map iW ! X, i 2 Z; PW (X) is then
the homotopy colimit of the sequence
X ! N(X) ! N2(X) ! N3(X) ! . ...
For larger W it would be necessary to continue the process transfinitely.
Dually, the colocalization map CE(A) ! A has two universal prop-
erties; it is terminal among maps X ! A with X colocal, and initial
among equivalences A0 ! A. This leads to two tautological formulas
24 W. G. DWYER
for CE(A):
8
< colim X (X colocal)
CE(A) = X!A 0
: lim A (A0! A an equivalence)
A0!A
As above, this suggests two approaches to constructing colocalizations.
The (homotopy) colimit approach is the one that is usually used [18 ]
[8]. We do not know of any effective way to build colocalizations with
limits, short of constructing a localization in terms of colimits, and
then expressing this in the opposite category!
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Department of Mathematics, 255 Hurley, University of Notre Dame,
Notre Dame IN 46556 USA
E-mail address: dwyer.1@nd.edu