THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL
ALGEBRA
MARK HOVEY
Abstract.The object of this paper is to prove that the standard categori*
*es
in which homotopy theory is done, such as topological spaces, simplicial*
* sets,
chain complexes of abelian groups, and any of the various good models for
spectra, are all homotopically self-contained. The left half of this sta*
*te-
ment essentially means that any functor that looks like it could be a te*
*nsor
product (or product, or smash product) with a fixed object is in fact su*
*ch a
tensor product, up to homotopy. The right half says any functor that loo*
*ks
like it could be Hom into a fixed object is so, up to homotopy. More pre*
*cisely,
suppose we have a closed symmetric monoidal category (resp. Quillen model
category) M. Then the functor TN :M -!M that takes M to M N is
an M-functor and a left adjoint. The same is true if N is an E-E0-bimodu*
*le,
where E and E0are monoids in M, and TN :Mod-E -!Mod-E0is defined
by TN (M) = M E N. Define a closed symmetric monoidal category (resp.
model category) to be left self-contained (resp. homotopically left self-
contained) if every functor F :Mod-E -!Mod-E0that is an M-functor and
a left adjoint (resp. and a left Quillen functor) is naturally isomorphi*
*c (resp.
naturally weakly equivalent) to TN for some N. The classical Eilenberg-W*
*atts
theorem in algebra then just says that the category Ab of abelian groups*
* is
left self-contained, so we are generalizing that theorem.
Introduction
The object of this paper is to extend the Eilenberg-Watts theorem [Eil60, Wat*
*60]
to situations such as topological spaces, chain complexes, or symmetric spectra*
*, in
which one is interested in objects and functors not up to isomorphism, but up
to some notion of weak equivalence. We first recall the standard Eilenberg-Watts
theorem.
Theorem 0.1. Let R and S be rings. If F : Mod-R -!Mod- S is additive and a
left adjoint, then F R is an R-S-bimodule and there is a natural isomorphism
X R F R -!F X.
Here, and throughout the paper, all modules are right modules unless otherwise
specified.
This is not the usual formulation of the Eilenberg-Watts theorem, which we now
recall.
Theorem 0.2. Let R and S be rings if F : Mod-R -! Mod-S is additive, right
exact, and preserves direct sums, then F R is an R-S-bimodule and there is a na*
*tural
isomorphism
X R F R -!F X.
____________
Date: October 21, 2009.
1
2 MARK HOVEY
These two theorems are equivalent, however, since a right exact additive func*
*tor
automatically preserves coequalizers, and then the adjoint functor theorem impl*
*ies
that any colimit-preserving functor between such nice categories is a left adjo*
*int.
We take the position that if M is a closed symmetric monoidal category, E and
E0 are monoids in M, and F : Mod-E -! Mod- E0 is a left adjoint and an M-
functor (the analogue of additive), then F is a sort of generalized E-E0-bimodu*
*le.
We therefore make the following definition. Given monoids E and E0in a closed
symmetric monoidal category M and an E-E0-bimodule N, define TN : Mod- E -!
Mod- E0by TN (M) = M E N. We will define this more precisely in the first sect*
*ion
below. Similarly, for an E E0-module N, define SN : Mod- E -!(Mod- E0)op by
SN (M) = Hom E(M, N).
Definition 0.3. Suppose M is a closed symmetric monoidal category. We say that
M is left self-contained if, for every pair of monoids E, E0 in M and every M-
functor F : Mod-E -!Mod- E0that is a left adjoint, there is an E-E0-bimodule N
and a natural isomorphism TN ~=F . Similarly, M is right self-contained if, for
every pair of monoids E, E0in M and every M-functor F : Mod-E -!(Mod- E0)op
that is a left adjoint, there is an E E0-module N and a natural isomorphism
F ~=SN . We say that M is self-contained if it is both left and right self-cont*
*ained.
The Eilenberg-Watts theorem stated above is then the assertion that Ab is left
self-contained. In fact, Eilenberg [Eil60] also proved that Ab is right self-co*
*ntained.
It would certainly be interesting to go on from here to try to find out when a
closed symmetric monoidal category is self-contained, but the author is primari*
*ly
interested in homotopy theory, so we go in a different direction.
Instead, we assume that we can also do homotopy theory in M. So we as-
sume that M is a model category in the sense of Quillen [Qui67]. The key idea
in model categories is that isomorphism is not the equivalence relation one car*
*es
about. Instead, there is a notion of weak equivalence. Formally inverting the w*
*eak
equivalences gives the homotopy category Ho M. We would like functors F on M
to induce functors LF on Ho M in a natural way, but this is generally true only*
* for
left Quillen functors F (for this and other model category terminology, see [Hi*
*r03]
or [Hov99 ]).
Thus we make the following definition.
Definition 0.4. Suppose M is a closed symmetric monoidal model category. We
say that M is homotopically left self-contained if, for every pair of monoids
E, E0 in M and every left Quillen M-functor F : Mod-E -!Mod- E0, there is an
E-E0-bimodule N and a natural isomorphism LTN -! LF of functors on HoMod- E.
Similarly, F is homotopically right self-contained if, for every pair of monoids
E, E0 in M and every left Quillen M-functor F : Mod-E -! (Mod- E0)op, there
is an E E0-module N and a natural isomorphism LF -! LSN of functors on
Ho Mod-E. And M is homotopically self-contained if it is both left and right
homotopically self-contained.
For this definition to make precise sense, we need to assume enough about M
to be sure that Mod- E inherits a model structure from that of M for all monoids
E, where the weak equivalences (resp. fibrations) in Ho Mod-E are maps that are
weak equivalences (resp. fibrations) when thought of as maps of M.
The main result of this paper is then the following theorem, proved in Sectio*
*n 6.
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 3
Theorem 0.5. The following closed symmetric monoidal model categories are ho-
motopically self-contained.
(1) (Compactly generated, weak Hausdorff )topological spaces.
(2) Simplicial sets.
(3) Chain complexes of abelian groups.
(4) Symmetric spectra
(5) Orthogonal spectra
(6) S-modules.
The last three are all models of stable homotopy theory. S-modules were in-
troduced in [EKMM97 ], orthogonal spectra in [MMSS01 ], and symmetric spectra
in [HSS00 ].
To approach Theorem 0.5, we recall the proof of the Eilenberg-Watts theorem.
The first step is to prove that, if F : Mod-R -!Mod- S is an additive functor, *
*then
F R is an R-S-bimodule, and there is a natural transformation
M E F E -!F M.
This step is completely general. The following proposition is proved as Propos*
*i-
tion 1.1, where we will define any unfamiliar terms.
Proposition 0.6. Suppose M is a closed symmetric monoidal category, E and E0
are monoids in M, and
F : Mod-E -!Mod- E0
is an M-functor. Then F E is an E-E0-bimodule, and there is a natural transfor-
mation
o :X E F E -!F X.
Given o, the proof of the usual Eilenberg-Watts theorem now proceeds not-
ing that o is an isomorphism when M = R, both sides preserve colimits, and R
generates Mod- R under colimits. There are partial generalizations of this to t*
*he
general case. The nicest we have is Theorem 2.3, which asserts that o is a natu*
*ral
isomorphism when F is a strict M-functor (defined in Section 2) and preserves
coequalizers.
However, our main interest is in homotopical algebra, where it is a mistake to
ask whether a natural transformation is an isomorphism for all X. Instead, we a*
*sk
whether it is a weak equivalence for all cofibrant X; equivalently, we ask when*
* the
derived natural transformation of o is an isomorphism of functors on Ho Mod- E.
For this to make sense, we need to assume enough about M so that we get model
structures on Mod- E and Mod- E0, and we need to assume F is a left Quillen
functor as mentioned above. We then need a general theorem about when the
derived natural transformation Lo of a natural transformation o of left Quillen
functors is an isomorphism. The author thinks that such a theorem should have
been proved before, but knows of no published reference. The following theorem *
*is
a combination of Theorem 4.1 and Theorem 4.4.
Theorem 0.7. Suppose C and D are model categories, F, G: C -!D are left Quillen
functors, and o :F -! G is a natural transformation. Suppose that one of the two
following conditions hold.
(1) C and D are stable, and there is a class G of objects of Ho C such that
the localizing subcategory generated by G is all of Ho C and (Lo)(X) is *
*an
isomorphism for all X 2 G.
4 MARK HOVEY
(2) C is cofibrantly generated such that the domains of the generating cofib*
*ra-
tions of C are cofibrant, and (Lo)(X) is an isomorphism whenever X is a
domain or codomain of one of the generating cofibrations.
Then Lo is a natural isomorphism of functors on Ho C.
We can then combine Theorem 0.7 with Proposition 0.6 to get versions of the
Eilenberg-Watts theorem in homotopical algebra. Here are two of them, proved as
Theorem 5.1 and Theorem 5.4. They use some terms we will define later.
Theorem 0.8. Suppose M is a strongly cofibrantly generated, symmetric monoidal
model category. Let E and E0 be monoids in M, and F : Mod-E -!Mod- E0 be a
left Quillen M-functor. Suppose one of the hypotheses below holds.
(1) The domains of the generating cofibrations of M are cofibrant, and the
composite
A F QE -!F (A QE) -!F (A E)
is a weak equivalence when A is a domain or codomain of one of the gen-
erating cofibrations of M; or
(2) M is stable and monogenic with a cofibrant unit.
Then the natural transformation
QX E F QE -!X E F E -!F QX
is a natural isomorphism of functors on Ho Mod-E.
Theorem 0.5, asserting that standard model categories are homotopically self-
contained, now follows. Although we do not prove Theorem 0.5 in this way, a
good way to think about it is that o is obviously an isomorphism for X = E, both
the functors LTN and LF preserve suspensions and homotopy colimits, and in the
standard model categories, E generates all of Ho Mod- E under the operations of
homotopy colimits and suspensions.
We point out that Keller's work on DG-categories implies in particular that c*
*hain
complexes of abelian groups are homotopically self-contained [Kel94, Section 6.*
*4],
in a stronger form than the one we give. We can recover Keller's full result by*
* our
methods using the special features of the model structure on chain complexes of
abelian groups.
Note that the usual Eilenberg-Watts theorem is closely related to Morita theo*
*ry.
After all, if you have an equivalence of additive categories F : Mod-R -!Mod- S,
it satisfies the hypotheses of the Eilenberg-Watts theorem, so must be given by
tensoring with a bimodule. This is the beginning of Morita theory. Similarly, o*
*ur
versions of the Eilenberg-Watts theorem are related to the Morita theory of ring
spectra due to Schwede and Shipley [Sch04].
The original motivation for this work was the important, yet disturbing, pape*
*r of
Christensen, Keller, and Neeman [CKN01 ], where they proved that not all homol*
*ogy
theories on the derived category D(R) of a sufficiently complicated ordinary ri*
*ng R
are representable. This followed work of Beligiannis [Bel00], who proved that n*
*ot
every morphism between representable homology functors on D(R) is representable.
Note that there is no problem with representability of cohomology functors and
morphisms between them. This was rather a blow, and the author is not certain
the field has adequately adjusted to it yet. This paper began by trying to find*
* a
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 5
property of a homology functor h that would ensure that it is representable. Th*
*is
question is discussed in the last section of the paper.
The author owes a debt of thanks to his former student Manny Lopez, who first
introduced him to the Eilenberg-Watts theorem. He also wishes to express his de*
*bt
to Dan Christensen, Bernhard Keller, and Amnon Neeman, both for writing the
paper [CKN01 ] and for their comments on an early draft.
We should also mention Neeman's paper [Nee98]. This paper considers rep-
resentability for covariant exact functors on triangulated categories that pres*
*erve
products. The author thinks this needs further investigation, even in the abeli*
*an
category setting, although the situation is so different that such an investiga*
*tion
would not reasonably fit into this paper.
Contents
Introduction 1
1. M-functors 5
2. A strict analog of the Eilenberg-Watts theorem 8
3. Monoidal model categories 9
4. Natural transformations of Quillen functors 12
5. The Eilenberg-Watts theorem 14
6. Examples 17
7. Brown representability 21
References 25
1. M-functors
In this section, M is simply a closed symmetric monoidal category with all fi*
*nite
colimits and limits. We denote the monoidal product by , the closed structure
by Hom , and the unit by S. The purpose of this section is to introduce enough
background to prove the following proposition.
Proposition 1.1. Suppose M is a closed symmetric monoidal category with finite
colimits and limits, and E and E0 are monoids in M. If F : Mod-E -!Mod- E0 is
an M-functor, then F E is an E-E0-bimodule, and there is a natural transformati*
*on
o :X E F E -!F X
of E0-modules that is an isomorphism when X = E. Similarly, if F is a contravar*
*i-
ant M-functor, then F E is an E E0-module, and there is a natural transformation
o :F X -!Hom E (X, F E)
of E0-modules that is an isomorphism when X = E.
For this proposition to make sense, we recall that a monoid E in a monoidal
category M is the object E equipped with a unit map S -!E and a multiplication
map E E -! E that is unital and associative. In this case, a (right) module X
over E is an object X equipped with an action map X E -!E that is associative
and unital. The category Mod- E is the category of such right modules and module
maps, which are of course maps in M compatible with the actions.
6 MARK HOVEY
The category Mod- E then becomes a closed (left) module category over M,
which means it is tensored, cotensored, and enriched over M. In more detail, the
tensor of K 2 M and X 2 Mod- E, is the object K X 2 Mod- E, where E acts
only on X. The cotensor of K 2 M and X 2 Mod- E is Hom (K, X) 2 Mod- E.
The action of E is defined by
Hom (K, X) E ~=Hom (K, X) Hom (S, E) -! Hom (K, X E) -!Hom (K, X),
where the last map is induced by the action on X (recall that S is the unit). T*
*he
enrichment Hom E(X, Y ) 2 M for X, Y 2 Mod- E is defined as the equalizer of the
evident two maps
Hom (X, Y ) -!Hom (X E, Y ),
one of which uses the action on X and the other of which uses the action on Y .*
* We
have the usual adjunction isomorphisms
Mod- E(K X, Y ) ~=Mod- E(X, Hom (K, Y )) ~=M(K, Hom E(X, Y )).
Now, if X is a right E-module and Y is a left E-module, we can form the tensor
product X E Y 2 M as the coequalizer of the two maps
X E Y -! X Y.
We can also define E-E0-bimodules in the usual way. In this case, the action map
X E0-! X
must be a map of left E-modules, which is equivalent to the left E-action being*
* a
map of right E0-modules. If Y is an E-E0-bimodule (with the evident definition),
then X E Y is in Mod- E0, because the action map
X Y E0-! X Y
descends through the coequalizer. Similarly, if Y is both an E-module and an E0-
module in compatible fashion, which is equivalent to Y being an E E0-module,
then Hom E(X, Y ) is naturally an E0-module.
Finally, if F : Mod-E -!Mod- E0is a functor, we say that F is an M-functor if
F is compatible with (one of, and hence all of) the tensor, cotensor, and enric*
*hment
over M. That is, if F is an M-functor, then there are natural maps
K F X -!F (K X), F (Hom (K, X)) -!Hom (K, F X)
and Hom E(X, Y ) -!Hom E0(F X, F Y )
satisfying all the properties one would expect. If F is contravariant, then the*
* roles
of the tensor and cotensor are reversed, so F is an M-functor when there are na*
*tural
maps
F (K X) -!Hom (K, F X), K F X -!F (Hom (K, X))
and Hom E(X, Y ) -!Hom E0(F Y, F X)
satisfying all the properties one would expect. As an example of how to get the*
*se
maps from the enrichment
Hom E (X, Y ) -!Hom E0(F Y, F X),
the adjoint to the identity of K X is a map
K -!Hom E (X, K X).
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 7
Composing this with the map
Hom E (X, K X) -!Hom E0(F (K X), F X)
and taking the adjoint gives us the map
F (K X) -!Hom (K, F X).
With this background in hand, we can now prove Proposition 1.1.
Proof.We begin with the covariant case. For any monoid E, we have an isomor-
phism of monoids
E ~=Hom (S, E) ~=Hom E(E, E).
If we had elements, we would say this takes x 2 E to the map E -!E that is left
multiplication by x. Using the fact that F is an M-functor, we get a map
Hom E(E, E) -!Hom E0(F E, F E)
of monoids. The adjoint
E F E -!F E
in Mod- E0 to the composite map
E -!Hom E0(F E, F E)
maps F E an E-E0-bimodule, though there are many details for the conscientious
reader to check.
Similarly, the map
X -!Hom E (E, X) -!Hom E0(F E, F X)
has adjoint the desired natural transformation
o :X E F E -!F X
of E0-modules. There are even more details to check here. In particular, a prio*
*ri
the adjoint to
OEX :X -!Hom E0(F E, F X)
is just a map
X F E -!F X
of right E0-modules. However, OEX is in fact a map of right E-modules, using the
left E-module structure on F E to make the target of OEX a right E-module. This
means that the adjoint descends through the relevant coequalizer diagram to give
the desired map
o :X E F E -!F X.
We now assume that F is contravariant. We get the right E-module structure
on the E0-module F E via the adjoint to the map of monoids
Eop -!Hom E(E, E)op -!Hom E0(F E, F E).
Here, if X is a monoid, Xop is the monoid with the reversed multiplication. Sim*
*i-
larly, we have the map
X ~=Hom E(E, X) -!Hom E0(F X, F E),
which is in fact a map of right E-modules. This has adjoint a map
F X -!Hom (X, F E),
which in fact factors through Hom E(X, F E), giving us the desired natural tran*
*s-
formation.
8 MARK HOVEY
2.A strict analog of the Eilenberg-Watts theorem
We are most interested in analogues of the Eilenberg-Watts theorem that com-
bine homotopy theory and algebra. But we can also prove a purely categorical
version of the Eilenberg-Watts theorem, and we do so in this section.
We begin with a lemma about the structure of modules over a monoid E in
a symmetric monoidal category M. Define an E-module to be extended if it is
isomorphic to one of the form X E, for some X 2 M.
Lemma 2.1. Suppose M is a symmetric monoidal category with all coequalizers,
and E is a monoid in M. Then any E-module is a coequalizer in Mod- E of a
parallel pair of morphisms P1 ' P0 between extended E-modules.
Proof.Given an E-module X, the action map p: X E -! X is a map of E-
modules when we give X E the free E-module structure. This map is an M-split
epimorphism by the map i: X -!X E induced by the unit of E. Hence X is the
coequalizer in M of ip: X E -!X E and the identity of X E. It follows that
X is the coequalizer in Mod- E of
1 ~: X E E -!X E,
where ~ denotes the multiplication in E, and the composite
X E E p-1-!X E i-1-!X E E 1-~--!X E.
This requires a little argument, using the fact that colimits in Mod- E can be
calculated in M.
We then get a general version of the Eilenberg-Watts theorem, whose proof is
straightforward enough, given Lemma 2.1, to leave to the reader.
Theorem 2.2. Let M be a closed symmetric monoidal category with all finite
colimits and limits, E, E0 be monoids in M, and F : Mod-E -! Mod- E0 be an
M-functor that preserves coequalizers. Suppose the natural map
oX :X E F E -!F X
is an isomorphism for all extended E-modules X. Then it is a natural isomorphism
for all E-modules X. Similarly, if F : Mod-E -! Mod-E0 is a contravariant M-
functor that takes coequalizers to equalizers, and the natural map
oX :F X -!Hom E (X, F E)
is an isomorphism for all extended E-modules X, then it is an isomorphism for a*
*ll
E-modules X.
Note that Theorem 2.2 does not imply the usual Eilenberg-Watts theorem di-
rectly, though the method of proof does do so. Indeed, when M is abelian groups,
every E-module is a coequalizer of a map of free E-modules (not just extended
ones), so it suffices to know that F is right exact (which is equivalent to pre*
*serving
coequalizers in this case) and oX is an isomorphism on free E-modules. For this,
we need F to preserve direct sums, and then we recover the usual Eilenberg-Watts
theorem.
There is a special case in which oX is automatically an isomorphism on extend*
*ed
E-modules. An M-functor F : Mod-E -! Mod-E0 is called a strict M-functor
if the structure map
K F X -!F (K X)
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 9
is an isomorphism for all K 2 M and X 2 Mod- E. In the contravariant case, F is
a strict M-functor if the structure map
F (K X) -!Hom (K, F X)
is an isomorphism for all K 2 M and X 2 Mod- E.
Theorem 2.3. Suppose M is a closed symmetric monoidal category with all finite
colimits and limits, E, E0 are monoids in M, and F : Mod-E -! Mod- E0 is a
strict M-functor that preserves coequalizers. Then the natural map
oX :X E F E -!F X
is an isomorphism for all X. Similarly, if F : Mod-E -!Mod- E0is a contravariant
strict M-functor that takes coequalizers to equalizers, then the natural map
F X -!Hom E (X, F E)
is an isomorphism for all X.
Proof.It suffices to check that oX is an isomorphism on all extended E-modules.
But if X 2 M,
F (X E) ~=X F E ~=(X E) E F E
since F is a strict M-functor. The contravariant case is similar.
3.Monoidal model categories
Since we are most interested in versions of the Eilenberg-Watts theorem "up
to homotopy", we now need to introduce the relevant homotopical algebra. For
this, we will of course need to assume knowledge of model categories, for which
see [Hir03, Part 2] or [Hov99 ].
Our base category M will be both closed symmetric monoidal and have a model
structure. Obviously, we will need to assume some compatibility between the mod*
*el
structure and the monoidal structure on M. This is well understood in the theory
of model categories, and we adopt the following definition,
Definition 3.1. (1)Suppose M is a monoidal category, and f :A -! B and
g :C -! D are maps in M. The pushout product of f and g, written
f g, is the map
(A D) qA C (B C) -!B D
from the pushout of A D and B C over A C, to B D.
(2) Now suppose M is a symmetric monoidal category equipped with a model
structure. We say that M is a symmetric monoidal model category
if the following conditions hold:
(a) If f and g are cofibrations, then f g is a cofibration.
(b) If f is a cofibration and g is a trivial cofibration, then f g is a *
*trivial
cofibration.
This definition is different from [Hov99 , Definition 4.2.6], where a unit co*
*ndition
was added to ensure that the homotopy category of a symmetric monoidal model
category has a unit. It automatically holds when the unit is cofibrant. This un*
*it
condition is not needed for some of our versions of the Eilenberg-Watts theorem,
and when it is needed, it is much easier to assume the unit is cofibrant, so we*
* omit
it.
10 MARK HOVEY
But we also need the category Mod- E of modules over a monoid E in M to
be a model category, in a way that is compatible with M. We therefore make the
following definition.
Definition 3.2. Suppose M is a closed symmetric monoidal model category. We
say that M is strongly cofibrantly generated if there are sets I of cofibrations
in M and J of trivial cofibrations in M such that, for every monoid E in M, the
sets I E and J E cofibrantly generate a model structure on Mod- E where the
weak equivalences are maps of E-modules that are weak equivalences in M.
This definition implies that the maps of I E are small with respect to (I *
* E)-
cell, and similarly for J E, as this is part of the definition of cofibrantly *
*generated.
Note also that a map p is a fibration (resp. trivial fibration) in Mod- E if an*
*d only
if it has the right lifting property with respect to the maps of J E (resp. I*
* E),
which is equivalent to p being a fibration (resp. trivial fibration) in M. Th*
*is
definition also implies that the functor that takes X 2 M to X E 2 Mod- E is a
left Quillen functor.
We note that pretty much every closed symmetric monoidal model category that
is commonly studied is strongly cofibrantly generated. The most useful theorem
along these lines is the following, which is a paraphrase of [SS00, Theorem 4.1*
*].
Theorem 3.3 (Schwede-Shipley). Suppose M is a cofibrantly generated, closed
symmetric monoidal model category. If M satisfies the monoid axiom of [SS00,
Definition 3.3] and every object of M is small, then M is strongly cofibrantly *
*gen-
erated.
In fact, one does not need every object of M to be small. Not every topo-
logical space is small, but the category of (compactly generated weak Hausdorff)
topological spaces is still strongly cofibrantly generated.
One advantage of the strongly cofibrantly generated hypothesis is the followi*
*ng
proposition.
Proposition 3.4. Suppose M is a strongly cofibrantly generated, closed symmetric
monoidal model category, and E is a monoid in M. If f is a cofibration in M and*
* g
is a cofibration in Mod- E, then the pushout product f g is a cofibration in Mo*
*d- E,
which is a trivial cofibration if either f or g is so. In particular, if A is c*
*ofibrant in
Mod- E, and f is a cofibration in M, then f A is a cofibration in Mod- E, and
is a trivial cofibration if f is so.
This proposition means that Mod- E is an M-model category, in the language
of [Hov99 , Definition 4.2.18], except that we again have omitted a unit condit*
*ion.
Proof.The last sentence follows from the rest of the proposition by taking g to*
* be
the map 0 -! A. It suffices to check the statement about f g when f and g are
generating cofibrations or trivial cofibrations [Hov99 , Corollary 4.2.5]. In t*
*his case,
g will be of the form h E for a map h in either I or J, and f will be in eith*
*er I
or J. But then
f (h E) ~=(f g) E
so the result follows from the fact that tensoring with E is a left Quillen fun*
*ctor.
The reader might now reasonably expect us to assert that the tensor product
X E Y : Mod- E x (E-E0-Bimod) -!Mod- E0
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 11
is also a Quillen bifunctor, so that the pushout product of cofibrations is a c*
*ofibra-
tion and so on. However, this seems to require E to be cofibrant in M, and in a*
*ny
case is not the most natural thing for us to consider since we know nothing abo*
*ut
the structure of F E as a bimodule.
Instead, we have the following proposition.
Proposition 3.5. Suppose M is a strongly cofibrantly generated, closed symmetric
monoidal model category, E and E0are monoids in M, and A is an E-E0-bimodule
that is cofibrant as a right E0-module. Then the functor
X 7! X E A: Mod- E -!Mod- E0
is a left Quillen functor, with right adjoint Y 7! Hom E0(A, Y ). Similarly, if*
* A is
an E E0-module that is fibrant as an object of M, then the functor
X 7! Hom E(X, A): Mod- E -!Mod- E0
is a contravariant left Quillen functor, with right adjoint Y 7! Hom E0(Y, A).
Proof.We begin with the covariant case, and leave to the reader the check that
Hom E0(A, -) is indeed right adjoint to our functor. Note that we use the left
E-module structure on A to make Hom E0(A, -) a right E-module.
We need to show that if f is a cofibration or trivial cofibration in Mod- E, *
*then
f E A is a cofibration or trivial cofibration in Mod- E0. The proof is similar*
* in both
cases, so we just work with cofibrations. Let I be a set of generating cofibrat*
*ions in
M so that I E is a set of generating cofibrations in Mod- E. Then any cofibrat*
*ion
in Mod- E is a retract of a transfinite composition of pushouts of maps of I *
*E.
Since the tensor product is a left adjoint, it preserves retracts (of course), *
*transfinite
compositions, and pushouts. So it suffices to show that if f 2 I, then
(f E) E A
is a cofibration in Mod- E0. But of course
(f E) E A ~=f A,
so this is ensured by the fact that A is cofibrant as a right E0-module and Pro*
*po-
sition 3.4.
The contravariant case is mostly similar. Again, we leave to the reader the p*
*roof
of the adjointness relation, where one must take into account the fact that the
functors are contravariant. We must then show that if f is a cofibration (resp.
trivial cofibration) in Mod- E, then Hom E(f, A) is a fibration (resp. trivial *
*fibra-
tion) in Mod- E0, or equivalently, in M. As before, the two cases are similar, *
*so we
only do the cofibration case. Every cofibration in Mod- E is a retract of a tra*
*ns-
finite composition of pushouts of maps of I E, where I is the set of generati*
*ng
cofibrations of M. Since Hom E(-, A) is a contravariant left adjoint, it preser*
*ves
retracts, converts transfinite compositions to inverse transfinite compositions*
*, and
converts pushouts to pullbacks. Since retracts, inverse transfinite compositio*
*ns,
and pullbacks of fibrations are fibrations, it suffices to check that
Hom E(g E, A) = Hom (g, A)
is a fibration for all g 2 I. But this is true because A is fibrant in M.
It now follows easily that the natural transformation of Proposition 1.1 desc*
*ends
to the homotopy category, at least if we replace E by a cofibrant approximation
12 MARK HOVEY
QE before applying F . More precisely, recall that in a model category M, we can
choose a functor Q: M -!M such that QX is cofibrant for all X, and a natural
trivial fibration QX -!X.
Corollary 3.6. Suppose M is a strongly cofibrantly generated, closed symmetric
monoidal model category, E and E0are monoids in M, and F : Mod-E -!Mod- E0
is a left Quillen M-functor. Then the natural transformation o of Proposition 1*
*.1
has a derived natural transformation
Lo :QX E F QE -!F QX = (LF )(X)
of functors on Ho Mod- E. Similarly, if F : Mod-E -!Mod- E0 is a contravariant
left Quillen M-functor, then the natural transformation o of Proposition 1.1 ha*
*s a
derived natural transformation
Lo :(LF )(X) = F QX -!Hom E (QX, F QE)
of functors on Ho Mod-E.
In applications, it is sometimes useful to note that this corollary only requ*
*ires
F to be left Quillen as a functor to Mod- E0 with some model structure where
the weak equivalences are the maps of Mod- E0 which are weak equivalences in M.
Indeed, the model structure on Mod- E0 is never used in the proof of this corol*
*lary
except that Ho Mod-E0is the target of the functors, and Ho Mod-E0depends only
on the weak equivalences of Mod- E0.
We also note that this corollary would remain true if we used QF E instead
of F QE, but that would go against one of the basic principles of model category
theory, to cofibrantly replace objects BEFORE applying left Quillen functors.
4. Natural transformations of Quillen functors
The object of this section is to find conditions under which the derived natu*
*ral
transformation
Lo :LF -! LG
of a natural transformation
o :F -! G
of left Quillen functors is a natural isomorphism.
Theorem 4.1. Let C and D be model categories, F, G: C -! D be left Quillen
functors, and o :F -! G be a natural transformation. Suppose C is cofibrantly
generated so that the domains of the generating cofibrations are cofibrant, and*
* oX
is a weak equivalence for X any domain or codomain of a generating cofibration.
Then Lo :LF -! LG is a natural isomorphism.
Proof.It suffices to show that oY is a weak equivalence for all cofibrant Y . D*
*enote
the generating cofibrations of C by I. Any cofibrant Y is a retract of an objec*
*t X
such that the map 0 -! X is a transfinite composition of pushouts of maps of I.
It then suffices to prove that oX is a weak equivalence, which we do by transfi*
*nite
induction. The base case is clear since F and G preserve the initial object. Fo*
*r the
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 13
successor ordinal step, we have a pushout diagram
A --f--! B
?? ?
y ?y
Xff ----! Xff+1
where f is a map of I, and oXffis a weak equivalence. This gives us two pushout
squares of cofibrant objects
F A ----! F B
?? ?
y ?y
F Xff----! F Xff+1
and
GA ----! GB
?? ?
y ?y
GXff ----! GXff+1
in which the top horizontal map is a cofibration. The natural transformation o
defines a map from the top square to the bottom one, which is a weak equivalence
on every corner except possibly the right bottom square. The cube lemma [Hov99 ,
Lemma 5.2.6] then implies o is a weak equivalence on the bottom right corner as
well.
For the limit ordinal step of the induction, we have a transfinite composition
Xfi= lim-ff,
and is the smallest localizing subcategory containing G.
Theorem 4.4. Let C and D be model categories, F, G: C -! D be left Quillen
functors, and o :F -! G be a natural transformation. Suppose C and D are stable,
and there is a class G of objects of Ho C such that loc = Ho C and (Lo)X is *
*an
isomorphism for X 2 G. Then Lo :LF -! LG is a natural isomorphism.
Proof.Because F and G are a left Quillen functors, LF and LG preserve exact
triangles (see [Hov99 , Section 6.4]), coproducts, and suspensions. Hence the c*
*ollec-
tion of all X such that (Lo)X is an isomorphism is a localizing subcategory. Si*
*nce
it contains G, it contains all of Ho C.
5. The Eilenberg-Watts theorem
We can now combine the results of the last two sections to prove homotopical
versions of the Eilenberg-Watts theorem.
Theorem 5.1. Suppose M is a strongly cofibrantly generated, symmetric monoidal
model category in which the domains of the generating cofibrations are cofibran*
*t.
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 15
Let E and E0 be monoids in M, and F : Mod-E -! Mod- E0 be a left Quillen
M-functor. Suppose that the composite
A F QE -!F (A QE) -!F (A E)
is a weak equivalence when A is a domain or codomain of one of the generating
cofibrations of M. Then there is a natural isomorphism
QX E F QE -!F QX = (LF )(X)
of functors on Ho Mod- E. Similarly, if F : Mod-E -!Mod- E0 is a contravariant
left Quillen M-functor such that the composite
F (A E) -!Hom (A, F E) -!Hom (A, F QE)
is a weak equivalence when A is a domain or codomain of one of the generating
cofibrations of M, then there is a natural isomorphism
(LF )(X) = F QX -!Hom E (QX, F QE)
of functors on Ho Mod-E.
Note that in the composite
A F QE -!F (A QE) -!F (A E)
the first map is the structure map of the M-functor F , and the second map is
induced by the weak equivalence QE -! E. A similar remark holds in the con-
travariant case.
We also note that the model structure on Mod- E0 that we use is again im-
material in the proof, as long as its weak equivalences are the maps that are w*
*eak
equivalences in M. In practice, this means that the condition that F be left Qu*
*illen
is not as restrictive as it might appear at first glance. We also point out tha*
*t in
case F is a strict M-functor, we should use Theorem 2.3 to analyze F rather than
Theorem 5.1.
Proof.Begin with the covariant case. Proposition 3.5 tells us that
o :X E F QE -!F X
is a natural transformation of left Quillen functors. Now apply Theorem 4.1. The
contravariant case is similar, where we think of
o :F X -!Hom E (X, F QE)
as a natural transformation of left Quillen functors from Mod- E to (Mod- E0)op.
We get versions of Theorem 5.1 corresponding to each version of Theorem 4.1.
Corresponding to Theorem 4.2, we have the following theorem.
Theorem 5.2. Suppose the hypotheses of Theorem 5.1 hold true, except that rather
than assuming the domains of the generating cofibrations of M are cofibrant, we
instead assume that Mod- E0 is left proper in the covariant case and right prop*
*er in
the contravariant case. Then the conclusions of Theorem 5.1 remain true.
The reason for the right proper hypothesis in the contravariant case is that *
*we
have left Quillen functors from Mod- E to (Mod- E0)op, so we need (Mod- E0)op to
be left proper.
Note that Theorem 5.2 will work with any left proper model structure on Mod- *
*E0
in which the weak equivalences are the maps that are weak equivalences in M.
16 MARK HOVEY
Also, the model structure on Mod- E0that we get from the fact that M is strongly
cofibrantly generated will be left proper if M is so and E0 is cofibrant in M, *
*for
in this case every cofibration in Mod- E0 is in particular a cofibration in M. *
*This
model structure on Mod- E0is right proper whenever M is, since weak equivalences
and fibrations are detected in M.
Corresponding to Corollary 4.3, we have the following theorem.
Theorem 5.3. Suppose M is a strongly cofibrantly generated, stable, symmetric
monoidal model category where the unit is cofibrant. Let E and E0 be monoids in
M, and F : Mod-E -! Mod- E0 be a left Quillen M-functor. Suppose that the
structure map
A F E -!F (A E)
is a weak equivalence when A is a cokernel of one of the generating cofibration*
*s of
M. Then there is a natural isomorphism
QX E F E -!F QX = (LF )(X)
of functors on Ho Mod- E. Similarly, if F : Mod-E -!Mod- E0 is a contravariant
left Quillen M-functor such that the structure map
F (A E) -!Hom (A, F E)
is a weak equivalence when A is a cokernel of one of the generating cofibration*
*s of
M, then there is a natural isomorphism
(LF )(X) = F QX -!Hom E (QX, F E)
of functors on Ho Mod-E.
Proof.Since the unit S is cofibrant in M, E is cofibrant in Mod- E. This is the
reason we do not need QE anywhere. The main point of the proof is that if M is
stable with cofibrant unit, so is Mod- E0. Indeed, the suspension functor in Ho*
* M
or on Ho Mod-E0is the total derived functor of S1 -, where S1 is the suspensi*
*on
of the unit of M (this is where we are assuming the unit is cofibrant). If the
suspension functor is an equivalence on Ho M, there is an object S-1 2 Ho M so
that tensoring with S-1 is the inverse of suspension. But then we can use S-1
to define an inverse of the suspension on Ho Mod- E0 as well. Thus we can apply
Corollary 4.3.
Finally, we get the most direct analogue to the original Eilenberg-Watts the-
orem as a corollary to Theorem 4.4. Here we need to assume M is stable with
cofibrant unit and Ho M is monogenic. This means that the unit S is a compact
weak generator of the triangulated category Ho M. Recall that an object X of a
triangulated category T is called compact if T (X, -)* preserves coproducts. The
object X is called a weak generator if the functor T (X, -)* is faithful on ob-
jects, so that Y = 0 in T if and only if T (X, Y )* = 0. As we will discuss in *
*the
next section, many of the most common stable symmetric monoidal categories are
monogenic. The relevance of the monogenic condition to Theorem 4.4 is that if X
is a compact weak generator of a triangulated category T with all coproducts, t*
*hen
loc = T [HPS97 , Theorem 2.3.2].
Theorem 5.4. Suppose M is a strongly cofibrantly generated, stable, monogenic,
symmetric monoidal model category where the unit is cofibrant. Let E and E0 be
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 17
monoids in M, and F : Mod-E -! Mod- E0 be a left Quillen M-functor. Then
there is a natural isomorphism
QX E F E -!F QX = (LF )(X)
of functors on Ho Mod- E. Similarly, if F : Mod-E -!Mod- E0 is a contravariant
left Quillen M-functor, then there is a natural isomorphism
(LF )(X) = F QX -!Hom E (QX, F E)
of functors on Ho Mod-E.
Proof.As discussed in the proof of Theorem 5.3, the fact that M is stable with
cofibrant unit S means that Mod- E is stable for all monoids E. We claim that E*
* is
in fact a compact weak generator of the category Ho Mod-E. For this, let U deno*
*te
the forgetful functor from Mod- E to M, so that U preserves and detects weak
equivalences. Since U preserves weak equivalences, its total right derived func*
*tor
RU is just U itself; that is, (RU)X = UX. Hence RU will also preserve coproduct*
*s.
Thus
a a a
Ho Mod-E(E, Xff) ~=Ho M(S, (RU)( Xff)) ~=Ho M(S, UXff)
~= M HoM(S, UXff) ~=M Ho Mod- E(E, Xff).
Thus E is compact. Similarly, if Ho Mod-E(E, X)* = 0, then Ho M(S, UX)* = 0,
so UX is weakly equivalent to 0 = U(0). Since U detects weak equivalences, X is
weakly equivalent to 0, so E is a weak generator. As mentioned above, we then g*
*et
that loc = Ho Mod-E by the proof of [HPS97 , Theorem 2.3.2] (the statement
of that theorem makes some irrelevant assumptions about a tensor product). Since
(Lo)E is an isomorphism, Theorem 4.4 completes the proof.
6.Examples
In this section, we prove Theorem 0.5 by applying our versions of the Eilenbe*
*rg-
Watts theorem to the standard model categories of symmetric spectra, chain com-
plexes, simplicial sets, and topological spaces.
We begin with the symmetric spectra of [HSS00 ], based on simplicial sets. Th*
*is is
a symmetric monoidal model category (under the smash product) whose homotopy
category is the standard stable homotopy category of algebraic topology, so it *
*is
stable and monogenic. It is cofibrantly generated, it satisfies the monoid axio*
*m, and
every object is small, so it is strongly cofibrantly generated by 3.3 of Schwed*
*e and
Shipley. The unit S is cofibrant in symmetric spectra, making symmetric spectra
easier for us to handle than the S-modules of [EKMM97 ]. A monoid in symmet-
ric spectra is frequently called a symmetric ring spectrum, and the homotopy
category of symmetric ring spectra is equivalent to any other homotopy category
of A1 ring spectra [MMSS01 ]. A model category that is enriched over symmetric
spectra is frequently called a spectral model category, and an enriched functor
is called a spectral functor.
Theorem 5.4 then gives an Eilenberg-Watts theorem for symmetric ring spectra.
Theorem 6.1. Symmetric spectra are homotopically self-contained.
There are similar theorems for symmetric spectra based on topological spaces
and for the orthogonal spectra of [MMSS01 ]. The only (slight) subtlety is tha*
*t no
nontrivial object of the category is small with respect to all maps (although e*
*ven
18 MARK HOVEY
that problem can be avoided by using the -generated spaces of Jeff Smith, which
are supposed to be a locally presentable category), so one has to take a little*
* care
in proving that the model categories in question are strongly cofibrantly gener*
*ated.
There is also a similar theorem for the S-modules of [EKMM97 ], even though
the unit S is not cofibrant in that case. The main issue here is we need enough
control over the unit to be sure that Mod- E is stable and that E is a small we*
*ak
generator of Ho Mod- E, for any S-algebra E. One cannot point directly to a
theorem in [EKMM97 ] that says this, but it does follow from the results Chap*
*ter III.
Proposition III.1.3 is especially relevant.
We now turn to chain complexes. Here the base symmetric monoidal model
category is the category of unbounded chain complexes of abelian groups Ch(Z),
with the projective model structure [Hov99 , Section 2.3]. This is a cofibrant*
*ly
generated model category satisfying the monoid axiom, in which every object is
small. It is therefore strongly cofibrantly generated by Theorem 3.3. It is a*
*lso
stable and monogenic, and the unit is cofibrant. Thus Theorem 5.4 applies. A
monoid in this category is a differential graded algebra and an enriched functo*
*r is
often called a DG-functor.
Theorem 6.2. Ch (Z) is homotopically self-contained.
In fact, Theorem 6.2 is actually a special case of [Kel94, Section 6.4], where
Keller proves that any DG-functor F : Mod-E -! Mod- E0 that commutes with
direct sums has QX E F E ~=(LF )(X). That is, he does not assume that F is left
Quillen. In the special case when E0 is an ordinary ring, we can recover Keller*
*'s
result, and this is a worthwhile exercise, as it illustrates that one can often*
* use the
methods of Theorem 5.4 to get tighter results than one would at first expect. T*
*he
proof below likely works for arbitrary DG-algebras as well.
We begin with a useful general lemma.
Lemma 6.3. Suppose M is a closed symmetric monoidal, strongly cofibrantly gen-
erated, model category in which the unit S is cofibrant, E, E0 are monoids in M,
and F : Mod-E -!Mod- E0 is an M-functor. Then F preserves the homotopy re-
lation on maps between cofibrant and fibrant objects, so preserves weak equival*
*ences
between cofibrant and fibrant objects.
Note that this means we could define a derived functor DF for any M-functor
F via (DF )(X) = F (QRX), where R denotes fibrant replacement. However, this
would be neither a left nor a right derived functor in general, and does not se*
*em to
have good properties without further assumptions on F .
Proof.Let I be a cylinder object in M for the unit S. Then I X is a cylinder
object for any cofibrant E-module X. In particular, if Y is a fibrant E-module,*
* and
f, g :X -!Y are homotopic, then there is a homotopy I X -!Y between f and
g. This corresponds to a map I -! Hom E(X, Y ) in M. Since F is an M-functor,
we get an induced map I :Hom E0(F X, F Y ), which is a homotopy between F f and
F g.
Theorem 6.4. [Keller] Suppose R and S are ordinary rings, and F :Ch (R) -!
Ch(S) is a DG-functor that commutes with arbitrary coproducts. Then there is a
natural isomorphism
Lo :QX E F E -!F QX = (LF )(X)
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 19
of functors on D(R). Similarly, if F :Ch (R) -! Ch(S) is a contravariant DG-
functor that converts coproducts to products,then there is a natural isomorphism
(LF )(X) = F QX -!Hom E (QX, F E)
of functors on D(R).
Proof.Lemma 6.3 tells us that a DG-functor preserves chain homotopy. Because ev-
ery object is fibrant, then, any DG-functor G has a left derived functor (LG)(X*
*) =
G(QX), where QX is a cofibrant (DG-projective) replacement for X (since weak
equivalences between DG-projective objects are chain homotopy equivalences). Th*
*is
left derived functor is automatically exact on D(R), as pointed out to the auth*
*or
by Keller. Indeed, we can use the injective model structure on D(S) [Hov99 , Th*
*eo-
rem 2.3.13], in which cofibrations are degreewise split monomorphisms. As a fun*
*ctor
to this model structure, a DG-functor like F automatically preserves cofibratio*
*ns,
and hence LF is exact. To see this, we show that F preserves degreewise split
monomorphisms. Indeed, if f :X -!Y is a degreewise split monomorphism, there
is a g 2 Hom R(Y, X)0 (which is, of course, not a cycle unless f is actually sp*
*lit) such
that gf is the identity. Applying F , we see that F f is also degreewise split.*
* Al-
together then, Lo is a natural transformation between exact coproduct-preserving
functors on D(R) from that is an isomorphism on R. It is therefore an isomorphi*
*sm
on the localizing subcategory generated by R, which is D(R).
We now consider simplicial sets, which are of course not stable. An excellent*
* de-
scription of the model structure on the category SSet of simplicial sets can be*
* found
in [GJ99 ]; there is also a description in [Hov99 , Chapter 3]. We find that SS*
*et is
a strongly cofibrantly generated, closed symmetric monoidal (under the product)
model category in which every object is cofibrant. (The fact that every object *
*is
cofibrant makes the monoid axiom automatic, hence SSet is strongly cofibrantly
generated). We can therefore apply Theorem 5.1. The set I of generating cofibra-
tions consists of the maps @ [n] -! [n]. We note that the vertex n is a simplic*
*ial
deformation retract of [n] [Hov99 , Lemma 3.4.6], though no other vertex is.
Theorem 6.5. Simplicial sets are homotopically self-contaiend.
Pointed simplicial sets are also homotopically self-contained, and the proof *
*is
very similar.
Proof.We just prove the covariant case, as the contravariant case is similar. *
*In
view of Theorem 5.1, we have to show that
OEA :A x F E -!F (A x E)
is a weak equivalence for A = @ [n] and A = [n]. For A = [n], we have a
commutative diagram
[n] x F E----! F ( [n] x E)
?? ?
y ?y
* x F E ----! F (* x E)
where the vertical maps are induced by the simplicial homotopy equivalence [n]*
* -!
* that collapses [n] onto the vertex n. The same proof as that of Lemma 6.3
implies that F preserves simplicial homotopy equivalences. Thus the vertical ma*
*ps
20 MARK HOVEY
are simplicial homotopy equivalences, and the the bottom horizontal map is an
isomorphism, so the top map is a weak equivalence as required.
We prove that OE@ [n]is a weak equivalence by induction on n. The case n = 0
is trivial, and the case n = 1 is straightforward since @ [1] is the coproduct *
*of two
copies of [0], and F preserves coproducts. Now suppose that OE@ [n]is a weak
equivalence. We first show that OE [n]=@ [n]is a weak equivalence using the cube
lemma [Hov99 , Lemma 5.2.6]. Indeed, we have two pushout squares
@ [n] x F E----! [n] x F E
?? ?
y ?y
* x F E ----! [n]=@ [n] x F E
and
F (@ [n] x E)----! F ( [n] x E)
?? ?
y ?y
F (* x E) ----! F ( [n]=@ [n] x E)
of cofibrant objects, where the top horizontal maps are cofibrations. The map
OE defines a map from the first pushout square to the second, which is a weak
equivalence at every spot except the lower right corner. The cube lemma tells us
that it is also a weak equivalence at the lower right corner.
Now, there is a map g :@ [n+1] -! [n]=@ [n] of simplicial sets, which is a we*
*ak
equivalence. It is easier to explain this map geometrically. The geometric real*
*ization
of @ [n + 1] is a triangulation of the n-sphere, and the geometric realization *
*of
[n]=@ [n] is the usual CW description of an n-sphere, with one point and one
n-cell. What we want to do is to take one face of @ [n + 1] and spread it out o*
*ver
the whole n-cell, sending everything else in @ [n + 1] to the basepoint. This *
*is
obviously a homotopy equivalence. We can realize it simplicially by sending each
k-simplex of @ [n + 1] for k n except 123 . .n.to the simplex represented by k
0's, and sending 123 . .n.to the unique nondegenerate n simplex of [n]=@ [n].
We then get the commutative diagram below.
@ [n + 1] x F E----! F (@ [n + 1] x E)
? ?
gxFE ?y ?yF(gxE)
[n]=@ [n] x F E----! F ( [n]=@ [n] x E)
The bottom horizontal map is a weak equivalence, as we have seen. The vertical
maps are also weak equivalences, because the map g is a weak equivalence between
cofibrant objects, and all functors involved preserve those (in particular, F E*
* is cofi-
brant in Mod- E0since E is cofibrant in Mod- E, so the product with F E preserv*
*es
such weak equivalences). Hence the top horizontal map is a weak equivalence,
completing the induction step and the proof.
We now consider the case when our base model category is topological spaces.
Of course, we need a closed symmetric monoidal category of topological spaces;
for definiteness, we choose the compactly generated weak Hausdorff spaces used
in [EKMM97 ], and refer to this category as Top . The model structure on Top
is described in [Hov99 , Section 2.4], and it is strongly cofibrantly generated*
* with
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 21
generating cofibrations Sn-1 -! Dn for all n 0. A Top -functor is usually cal*
*led
a continuous functor.
Theorem 6.6. Topological spaces are homotopically self-contained.
Again, we find similarly that pointed topological spaces are homotopically se*
*lf-
contained as well.
We note that of course the most obvious topological monoid is a topological
group G, in which case we are talking about G-spaces. The model structure we are
using on G-spaces is the one in which the weak equivalences are equivariant maps
which are underlying weak equivalences. We would like to be using the complete
model structure, where a map f is a weak equivalence if and only the induced map
fH on H-fixed points is a weak equivalence for all subgroups H (in some family,
perhaps). Our methods would apply to this case, except for one point. We get the
natural transformation
X xG F G -!F X
but it is not clear that the left hand functor would ever be a left Quillen fun*
*ctor.
Proof.The proof is precisely analogous to that of Theorem 6.5. Theorem 5.1 tells
us that we have to show that
OEA :A x F E -!F (A x E)
is a weak equivalence for A = Sn-1 and A = Dn. The space Dn is contractible,
and continuous functors preserve homotopy by Lemma 6.3, so we can use the same
argument as in Theorem 6.5 for A = Dn. We can use induction on n for A = Sn-1,
just as in Theorem 6.5, and it is even easier, as Dn=Sn-1 is homeomorphic to
Sn.
7. Brown representability
In this section, we point out that our results are relevant to Brown represen*
*tabil-
ity of homology and cohomology theories. For simplicity, we stick to the stable*
* case.
Recall, then, that if T is a triangulated category, a homology functor is an
exact, coproduct-preserving functor h: T -! A to some abelian category A. We wi*
*ll
refer to the graded version h* of h, defined by hn(X) = h( nX), as the associat*
*ed
homology theory. When considering a homology theory, we need to consider the
isomorphisms hn(X) ~=hn+1( X) as part of the data. Similarly, a cohomology
functor is an exact, contravariant functor T -!A that converts coproducts to
products, and we have a similar induced cohomology theory. A cohomology functor
h is representable if there is a natural isomorphism
h(X) ~=T (X, Y )
for some object Y of T , and we say that Brown representability for cohomol-
ogy functors holds if every cohomology functor is representable. This is true in
considerable generality; see [Nee01, Proposition 8.4.2], for example.
Representability for homology functors is much more complicated. Even under-
standing what it means for a homology functor to be representable is not obviou*
*s.
Since we will be working with triangulated categories of the form Ho Mod- E, for
E a monoid in a strongly cofibrantly generated, closed symmetric monoidal stable
22 MARK HOVEY
monogenic model category M, the natural definition for us is that a homology
functor h is representable if there is a natural isomorphism
h(X) ~=Ho M(S, X LEY )
for some left E-module Y , where X LEY denotes the derived tensor product and
S denotes the unit of M. Note that this is much more subtle; for example, there*
* is
no reason to think that a morphism between representable homology theories must
be itself represeentable by a map between the representing objects.
Of course, homology functors, and natural transformations between them, on
the stable homotopy category are representable. The same is true for D(R) for
countable rings R. However, Christensen, Keller, and Neeman proved in [CKN01 ]
that there are rings R for which not every homology theory on D(R) is represent*
*able.
Before that, Beligiannis [Bel00] had proved that natural transformations between
representable homology functors on D(R) need not be representable.
We can use our versions of the Eilenberg-Watts theorem to partially salvage
Brown representability for homology theories.
Definition 7.1. Suppose M is a strongly cofibrantly generated, stable, monogeni*
*c,
symmetric monoidal model category where the unit S is cofibrant. Let E be a
monoid in M, and h: Ho Mod- E -! Mod- A be a homology (resp. cohomology)
functor, where A is an ordinary ring. We say that h* has a strict model if ther*
*e is a
monoid E0in M with Ho M(S, E0)* ~=A* as rings, a left Quillen M-functor (resp.
a contravariant left Quillen M-functor) F : Mod-E -! Mod- E0, and a natural
isomorphism
ae: h(X) ~=Ho Mod-E0(E0, F QX) ~=Ho M(S, F QX)
of A-modules
Here, then, is our version of Brown representability, which follows immediate*
*ly
from Theorem 5.4.
Theorem 7.2. Suppose M is a strongly cofibrantly generated, stable, monogenic,
symmetric monoidal model category where the unit S is cofibrant. Let E be a mon*
*oid
in M, and h: Ho Mod- E -!Mod- A be a homology (resp. cohomology)theory with
a strict model F : Mod-E -!Mod- E0. Then h is representable. More precisely, in
the homology case, we have a natural isomorphism
h(X) ~=Ho Mod-E(E, QX E F E),
and in the cohomology case, we have a natural isomorphism
h(X) ~=Ho Mod-E(QX, F E).
We point out that, although the hypotheses in Theorem 7.2 are much stronger
than in usual forms of Brown representability, the conclusion is also stronger.*
* Typ-
ically, Brown representability theorems just say that a cohomology functor h is
representable by an object of Ho Mod-E, but this theorem says that the represen*
*t-
ing object is actually an E E0-module.
In practice, we usually do not need to assume quite so much to get a Brown re*
*p-
resentability theorem. We will illustrate this in the case of chain complexes. *
*The
main point is that every object is fibrant in Ch(Z). This means that, if F :Ch *
*(R) -!
Ch(S) is a DG-functor, then F preserves weak equivalences between cofibrant ob-
jects (see Lemma 6.3), and so has a left derived functor (LF )(X) = F (QX). We
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 23
remind the reader that we can think of suspension in D(R) as X = S1 QX,
where S1 is the complex which is Z in degree 1 and 0 elsewhere.
Definition 7.3. Suppose R and S are rings, and h*: D(R) -!Mod- S is a homol-
ogy (resp. cohomology) theory. We say that h* has a chain model if there is a
DG-functor F :Ch (R) -!Ch (S) and a natural isomorphism
ae: h*(X) ~=H*(F QX)
of S-modules that is compatible with the suspension. To explain this, we assume
h* is a homology theory and leave the evident modifications in the cohomology
case to the reader. To say that ae is compatible with the suspension means that*
* the
isomorphism hn(X) ~=hn+1( X) corresponds to the composite
Hn(F QX) ~=Hn+1(S1 F QX) -!Hn+1(F (S1 QX)) ~=Hn+1(F Q X),
where the last isomorphism comes from the fact that F preserves the homology
isomorphism
Q X = Q(S1 QX) -!S1 QX
between cofibrant objects. We remind the reader
Theorem 7.4. Suppose R and S are rings and h*: D(R) -! (Mod- S)* is a ho-
mology or cohomology theory with a chain model F :Ch (R) -!Ch (S). Then h* is
representable. More precisely, in the homology case, there is a natural isomorp*
*hism
h*(X) ~=H*(QX E F E)
and in the cohomology case there is a natural isomorphism
h*(X) ~=D(R)(QX, F E)*.
Again we remind the reader that the representing object for h* in the above
theorem is a complex of R-S-bimodules, and for h* it is a complex of R S-
modules. The usual Brown representability theorems, when they apply, would just
give an action of S on the representing object up to homotopy.
Proof.We assume h* is a homology theory, and leave the modifications in the
cohomology case to the reader. Since F is a DG-functor, we have a natural trans-
formation (by Proposition 1.1)
o :X E F E -!F X.
Since both the domain and target of this natural transformation preserve weak
equivalences between cofibrant objects, there is a derived natural transformati*
*on
Lo :QX E F E -!(LF )(X) = F (QX).
This natural transformation is an isomorphism when X = S0R. As in the proof
of Theorem 5.4, the localizing subcategory generated by S0R is all of D(R). If *
*we
knew F were a left Quillen functor, we could then use Theorem 4.4 to conclude
that Lo is an isomorphism for all X. Instead, we use the fact that h is a homol*
*ogy
theory to conclude that LF is exact and preserves coproducts and suspensions,
giving the desired result.
To see that LF preserves coproducts, we just note that the natural map
a a
(LF )Xi-! (LF )( Xi)
i i
24 MARK HOVEY
becomes an isomorphism on applying H*, and is therefore an isomorphism in D(S).
Because F is a DG-functor, there is a natural map
S1 QF QX -!F (S1 QX),
and our hypothesis that the isomorphism
h*(X) ~=H*(F QX)
is compatible with the suspension guarantees that this is a homology isomorphis*
*m,
so is an isomorphism in D(S). Hence LF commutes with the suspension.
To see that LF preserves exact triangles, we need to recall a little more abo*
*ut
exact triangles in D(R). Any such exact triangle comes from a cofibration f :X *
*-!
Y of cofibrant objects and is isomorphic in D(R) to the sequence
X f-!Y -! C(f) -! X
where Cyl(f) is the mapping cylinder of f and C(f) is the mapping cone. So
Cyl(f) is the pushout of Y f- X -i1-X-!I X, where I is the chain complex
previously mentioned, with Z in degree 1 and Z Z in degree 0. The map i1 hits
the "right endpoint" copy of Z in degree 0. There is a chain homotopy equivalen*
*ce
Cyl(f) -!Y corresponding to "stepping on the cylinder". Then Cf is the quotient
of the composite
X i0-1--!I X -!Cyl(f),
so that (Cf)n = Yn Xn-1, and the map Cf -! X is the quotient of Y -! C(f).
We will show that there is a commutative diagram
F X ----! Cyl(F f)----! C(F f)- ---! F X
flfl ? ? ?
fl ?y ?y ?y
F X ----! F (Cyl f)----! F (Cf)- ---! F ( X)
in which the vertical maps are homology isomorphisms. The rightmost vertical
map comes from the fact that F is a Ch(Z)-functor, and we have already seen that
this map is a homology isomorphism (for DG-projective X). The map Cyl(F f) -!
F (Cyl f) also exists because F is a Ch(Z) functor; it is the map F Y -! F (Cyl*
* f)
on F Y and the map
I F X -!F (I X) -!F Y
on I F X. Since Cyl(f) -!Y is a chain homotopy equivalence, so is F (Cyl f) -!
F Y . Of course, Cyl(F f) -!F Y is also a chain homotopy equivalence, from which
we conclude that Cyl(F f) -!F (Cyl f) is a homology isomorphism.
By taking quotients with some care, using the fact that F is a Ch (Z)-functor
again, we get an induced map C(F f) -!F (Cf) making the diagram above commute
on both sides. Now, the top row has a long exact sequence in homology, and the
fact that h* is a homology theory means that the bottom row also has a long exa*
*ct
sequence in homology. The 5-lemma then implies that the map C(F f) -!F (Cf) is
a homology isomorphism, completing the proof that LF is a coproduct-preserving
triangulated functor.
We point out that there is a similar theorem to Theorem 7.2 for morphisms be-
tween homology and cohomology functors with a strict model. If such a morphism
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA 25
is induced by an M-natural transformation of the strict model, then it is repre-
sentable as a morphism between the representing objects. There is also an analog
of Theorem 7.4 for morphisms.
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Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org