A BROWN REPRESENTABILITY THEOREM VIA COHERENT
FUNCTORS
HENNING KRAUSE
Abstract.We discuss the Brown Representability Theorem for triangulated *
*cate-
gories having arbitrary coproducts.
In this paper we discuss the Brown Representability Theorem for triangulated *
*cate-
gories having arbitrary coproducts. This theorem is an extremely useful tool an*
*d various
versions appear in the literature. All of them require a set of objects which g*
*enerate the
category in some appropriate sense. Depending on the proof, there are essential*
*ly two
types: the first type is based on the analogue of iterated attaching of cells w*
*hich is used
in the topological case; the second type is based on solution sets and applies *
*a variant
of Freyd's Adjoint Functor Theorem.
Motivated by recent work of Neeman [7] and Franke [2], we prove a new theorem*
* of the
first type (Theorem A) and add, as an application, a Brown Representability The*
*orem
for covariant functors (Theorem B). The final Theorem C establishes a filtratio*
*n of a
triangulated category which clarifies the relation between results of the first*
* and the
second type.
The main theorem
Let T be a triangulated category and suppose that T has arbitrary coproducts.
Definition 1. A set of objects S0 perfectly generates T if the following holds:
(G1) an object X 2 T is zero provided that (S; X) = 0 for all S 2 S0;
(G2) for every countable set of maps Xi! Yi in T the induced map
a a
(S; Xi) -! (S; Yi)
i i
is surjective for all S 2 S0 provided that (S; Xi) ! (S; Yi) is surjectiv*
*e for all i
and S 2 S0.
Here, (X; Y ) denotes the maps from X to Y . For example, (G2) holds if every*
* S 2 S0
is small, that is, the functor (S; -) preserves arbitrary coproducts. Therefor*
*e the fol-
lowing result generalizes the classical Brown Representability Theorem for tria*
*ngulated
categories having a set of small generators [5].
Theorem A. Let T be a triangulated category with arbitrary coproducts, and *
*suppose
that T is perfectly generated by a set of objects. Then a functor F :T op ! A*
*b is
representable if and only if F is cohomological and sends coproducts in T to p*
*roducts.
In [7], Neeman proves a Brown Representability Theorem for triangulated categ*
*ories
which are well generated. A well generated triangulated category has a set of *
*perfect
generators so that one gets a quick proof for Neeman's result.
Examples of perfectly generated categories arise very naturally from triangul*
*ated cat-
egories having a set of small generators. Take for instance the stable homotopy*
* category
1
2 HENNING KRAUSE
of CW-spectra or the unbounded derived category of modules over an associative *
*ring.
Let T0 be a set of objects and let T be the smallest full triangulated subcateg*
*ory which
is closed under coproducts and contains T0. Then T is perfectly generated by a*
* set of
objects [7].
The proof of Theorem A is based on a reformulation of condition (G2) which is*
* given
in Lemma 3. Let us start with some preparations. We fix an additive category *
*T .
Following Auslander [1], a functor F :T op! Ab into the category of abelian gro*
*ups is
called coherent if there exists an exact sequence
(-; X) -! (-; Y ) -! F -! 0:
The natural transformations between two coherent functors form a set, and the c*
*oherent
functors T op! Ab form an additive category with cokernels which we denote by b*
*T(see
also [3] for this concept). A basic tool is the Yoneda functor
T - ! bT; X 7! (-; X):
Given an additive functor f :S ! T , we denote by f* :bS! bTthe right exact fun*
*ctor
which sends (-; X) to (-; fX).
Lemma 1. Let T be an additive category.
(1) If T has weak kernels, then bTis an abelian category.
(2) If T has arbitrary coproducts, then bT has arbitrary coproducts and the *
*Yoneda
functor preserves all coproducts.
Proof.Recall that a map X ! Y is a weak kernel for Y ! Z if the induced sequence
(-; X) -! (-; Y ) -! (-; Z)
is exact. The proof of (1) and (2) is straightforward. Note that for every fa*
*mily of
functors Fi having a presentation
(-;OEi)
(-; Xi) - ! (-; Yi) -! Fi- ! 0
`
the coproduct F = iFi has a presentation
a (-;` OEi) a
(-; Xi) -! (-; Yi) -! F -! 0:
i i
__
|_*
*_|
Given a class S of objects in an additive category T , we denote by Add S the*
* closure
of S in T under all coproducts and direct factors.
Lemma 2. Let T be an additive category with arbitrary coproducts and weak ke*
*rnels.
Let S0 be a set of objects in T and denote by f :S ! T the inclusion for S = *
*Add S0.
(1) S has weak kernels and bSis an abelian category.
(2) The assignment F 7! F |S induces an exact functor f*: bT-! bS.
(3) The functor f* :bS-! bTis a left adjoint for f*.
(4) f*O f* ~=id and f* induces an equivalence bT=Ker f*! bS.
Proof.First observe that for every X 2 T , there exists an approximation X0 ! X*
* such
that X0 2 S and (S; X0)`! (S; X) is surjective`for all S 2 S. This follows from*
* Yoneda's
lemma if we take X0 = S2S0XS where XS = (S;X)S.
A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS 3
(1) To prove that bSis abelian it is sufficient to show that every map in S h*
*as a weak
kernel. To obtain a weak kernel of a map Y ! Z in S, take the composite of a we*
*ak
kernel X ! Y in T and an approximation X0 ! X.
(2) We need to check that for F 2 bTthe restriction F |S belongs to bS. It is*
* sufficient
to prove this for F = (-; Y ). To obtain a presentation, let X ! Y 0be a weak k*
*ernel
of an approximation Y 0! Y . The composite X0 ! Y 0with an approximation X0 ! X
gives an exact sequence
(-; X0)|S -! (-; Y 0)|S -! F |S -! 0:
Clearly, F 7! F |S is exact.
(3) Let F 2 bSand G 2 bT. Suppose first that F = (-; X). Then
(f*F; G) = ((-; fX); G) ~=G(fX) ~=(F; f*G):
This implies the adjointness isomorphism for an arbitrary F since f* is right e*
*xact.
(4) We have (f*O f*)(-; X) = (-; X) for all X 2 S, and f*O f* ~= idfollows si*
*nce __
f*O f* is right exact. For the rest we refer to Proposition III.5 in [4]. *
* |__|
Lemma 3. Let T be a triangulated category with arbitrary coproducts. Let S0 *
*be a set
of objects in T and let S = Add S0. Then the functor
h: T - ! bS; X 7! (-; X)|S;
is cohomological. It preserves countable coproducts if and only if (G2) holds f*
*or S0.
Proof.We apply Lemma 2. To this end write h as composite
h: T - ! bT-f*!bS:
The Yoneda functor is cohomological and f* is exact. Therefore h is cohomologic*
*al. It
is clear that h preserves coproducts if and only if f* preserves coproducts. W*
*e know
that f*: bT! bSinduces an equivalence bT=Ker f*! bS, and it is not hard to see *
*that f*
preserves coproducts`if and only if Ker f* is closed under taking coproducts. W*
*e fix a
coproduct F = iFi in bTand for each Fi a presentation
(-;OEi)
(-; Xi) - ! (-; Yi) -! Fi- ! 0:
Now suppose that Fi2 Kerf* for all i. Thus (S; OEi) is surjective for all S 2 S*
* and all i.
We have F 2 Kerf* if and only if the induced map
a a
(S; Xi) -! (S; Yi)
i i
is surjective for all S 2 S. Clearly, it is sufficient to have this for all S *
*2 S0, and_we
conclude that h preserves countable coproducts if and only if (G2) holds for S0*
*. |__|
Proof of the Theorem A.We fix a perfectly generating set S0 of objects in T an*
*d put
S = Add S0. Replacing S0 by {nS | n 2 Z; S 2 S0}, we may assume that (S0) = S0.
Let F :T op! Ab be a cohomological functor which sends coproducts in T to produ*
*cts.
We construct inductively a sequence
X0 -OE0!X1 -OE1!X2 -OE2!. . .
S
of maps in T and a set of maps ssi:(-; Xi) ! F for i 0 as follows.`Let U = S*
*2S0F S.
Each x 2 U corresponds to an element in F Sx and we put X0 = x2USx. We get an
4 HENNING KRAUSE
element in Y
F Sx ~=F X0;
x2U
and using Yoneda's lemma, this gives a map ss0: (-; X0) ! F . Suppose weShave a*
*lready
constructed ssi:(-;`Xi) ! F for some i 0. Let Ki= Ker ssi and let Ui= S2S0Ki*
*S.
We define Ti = x2UiSx and apply again Yoneda's lemma to obtain a map Ti ! Xi.
We complete this to a triangle
Ti-AEi!Xi-OEi!Xi+1-Oi!Ti
and get an exact sequence
F (Ti) FOi-!F Xi+1-FOEi!F Xi-FAEi!F Ti
since F is cohomological. The construction implies (F AEi)ssi= 0 and this gives*
* an element
ssi+12 F Xi+1 such that (F OEi)ssi+1= ssi. Thus we have a factorization
(-;OEi) ssi+1
ssi:(-; Xi) - ! (-; Xi+1) -! F:
For each i 0 the map AEiinduces an epimorphism (-; Ti)|S ! Ki|S and we get the*
*refore
an exact sequence
(-;AEi)|S ssi|S
(-; Ti)|S - ! (-; Xi)|S -! F |S -! 0:
We obtain in bSfor each i 0 the following commutative diagram with exact rows
ssi|S
0 - ! Ki|S? -! (-;?Xi)|S -! F?|S -! 0
?y0 ?y ?
i y id
ssi+1|S
0 - ! Ki+1|S -! (-; Xi+1)|S -! F |S -! 0
where i= (-; OEi)|S. Each i has a factorization
ssi|S i
i:(-; Xi)|S -! F |S -! (-; Xi+1)|S
and therefore ssi+1|S Oi= id. This gives the following commutative diagram
(-; X1)|S? -!1 (-; X2)|S? - 2! (-;?X3)|S -!3 . . .
?yo ?yo ?yo
` id`0 ` id`0 ` id`0
F |S K1|S -! F |S K2|S - ! F |S K3|S -! . . .
and taking colimits in bS, we get an exact sequence
a id- ia (ssi|S)
(*) 0 -! (-; Xi)|S - ! (-; Xi)|S - ! F |S -! 0:
i i
Now consider the triangle
a id-OEia a
Xi -! Xi- ! X -! ( Xi)
i i i
and observe that Y a
(ssi) 2 F Xi~= F ( Xi)
i i
induces a map ss :(-; X) ! F . We apply the functor
T - ! bS; X 7! (-; X)|S;
A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS 5
which is cohomological and preserves countable coproducts by Lemma 3. This give*
*s an
exact sequence
a id- ia a id- i a
(-; Xi)|S - ! (-; Xi)|S -! (-; X)|S -! (-; Xi)|S -! (-; Xi)|S:
i i i i
We compare this sequence with (*). The map id - i is a monomorphism, since
(S) = S, and it follows that ss|S :(-; X)|S ! F |S is an isomorphism. Moreover,*
* the
subcategory of all Y 2 T such that ssY is an isomorphism is triangulated, cont*
*ains S0,
and is closed under arbitrary coproducts.
Now let T 0be the localizing subcategory of T which is generated by S0. Thu*
*s T 0
is the smallest triangulated subcategory of T which contains S0 and is closed *
*under
coproducts. We claim that T 0= T . To see this let Y 2 T and apply the construc*
*tion in
the first part of this proof to F = (-; Y ). The corresponding map ss :(-; X) !*
* (-; Y )
is induced by a map X ! Y since X 2 T 0. We complete this map to a triangle
W -! X -! Y - ! W
and use (G1) to obtain W = 0 = W since (S; X) ! (S; Y ) is an isomorphism for a*
*ll
S 2 S0, and (S0) = S0 by our assumption. Thus T 0= T and we obtain (-; X) ~=F i*
*n __
the first part of the proof. *
* |__|
Corollary. Let T be a triangulated category with arbitrary coproducts which is *
*perfectly
generated by a set of objects S0. Suppose that T 0is a full triangulated subcat*
*egory which
is closed under countable coproducts and contains all coproducts of objects in *
*S0. Then
T 0= T .
Thus perfect generators are strong generators in the sense of [2]. Note that *
*for every
cardinal fi > @0 a fi-perfect generating set in the sense of [7] is automatical*
*ly perfect as
in Definition 1.
Remark. If T has a set S0 of perfect generators, then the construction of eac*
*h X 2 T
implies that the functor X 7! (-; X)|S preserves arbitrary coproducts. Therefo*
*re the
following stronger condition holds for S0:
(G20) for every set of maps Xi! Yi in T the induced map
a a
(S; Xi) -! (S; Yi)
i i
is surjective for all S 2 S0 provided that (S; Xi) ! (S; Yi) is surjectiv*
*e for all i
and S 2 S0.
Brown representability for the dual
Let T be a triangulated category with arbitrary coproducts. In this section w*
*e prove a
Brown Representability Theorem for T op. The first result of this type is due t*
*o Neeman
[6] and requires T to be generated by a set of small objects. This has been gen*
*eralized
in [7]. The concept which is used here stresses the symmetry between T and T op.
Definition 2. A set S0 of objects is a set of symmetric generators for T if the*
* following
holds:
(G1) an object X 2 T is zero provided that (S; X) = 0 for all S 2 S0;
(G3) there exists a set T0 of objects in T such that for every map X ! Y in*
* T
the induced map (S; X) ! (S; Y ) is surjective for all S 2 S0 if and only*
* if
(Y; T ) ! (X; T ) is injective for all T 2 T0.
6 HENNING KRAUSE
It is clear that (G3) implies (G2), and that T has a set of symmetric genera*
*tors
if and only if T ophas a set of symmetric generators. Therefore the following *
*Brown
Representability Theorem for T opis an immediate consequence of Theorem A.
Theorem B. Let T be a triangulated category with arbitrary coproducts, and s*
*uppose
that T has a set of symmetric generators. Then T has arbitrary products, and a *
*functor
F :T ! Ab is representable if and only if F is cohomological and preserves prod*
*ucts.
Proof.We have a set of perfect generators for T and therefore arbitrary product*
*sQin T .
In fact, Theorem A implies that for everyQfamily (Xi)i2Iof objects the functor *
* i(-; Xi)
is represented by an object which is iXi. The set T0 which arises in (G3) is *
*a set of
perfect generators for T op, and it follows from Theorem A that a functor F :T *
*! Ab __
is representable if and only if F is cohomological and preserves products. *
* |__|
An example for a set of symmetric generators is any set S0 of small objects s*
*atisfying
(G1). To see this, take for T0 the set of objects representing the functors
T op-! Ab ; X 7! ((S; X); Q=Z)
where S 2 S0. This shows that the stable homotopy category of CW-spectra or the
unbounded derived category of modules over an associative ring have sets of sym*
*metric
generators.
Remark. Let S0 be a set of perfect generators and let S = Add S0. Each injecti*
*ve object
I 2 bSgives rise to an object in T representing
T op-! Ab ; X 7! ((-; X)|S; I):
Therefore (G3) holds for S0 if and only if bShas an injective cogenerator.
Neeman's Brown Representability Theorem for the dual in [7] involves the exis*
*tence
of an injective cogenerator for a category which is equivalent to some bS; it i*
*s therefore
a consequence of Theorem B.
A filtration
Let T be a triangulatedScategory with arbitrary coproducts. In this section w*
*e study
a filtration T = S which is defined in terms of a set S0 of appropriate gen*
*erators.`
Let ff be a cardinal.`Recall that an object S is ff-small if every map S ! *
*i2IXi
factors through i2JXi for some J I with cardJ < ff.
Theorem C. Let T be a triangulated category with arbitrary coproducts, and s*
*uppose
that T is perfectly generated by a set S0 of ff-small objects. Let be the succ*
*essor of ff
for some cardinal
a
sup{card(S; Si) | S; Si2 S0 and cardI < ff} + cardS0:
i2I
Then the objects X 2 T satisfying card(S; X) < for all S 2 S0 form a subcateg*
*ory S
having the following properties:
(1) S is a triangulated subcategory of T which contains S0;
(2) S is closed under taking coproducts of less than objects;
(3) the isomorphism classes of objects in S form a set;
(4) every subcategory T 0of T which satisfies (1) and (2) contains S ;
(5) every object in S is -small.
A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS 7
A triangulated category which is well generated in the sense of Neeman [7] sa*
*tisfies
the assumption of the preceding theorem. The conclusion of this theorem implies*
* the
condition on a triangulated category which Franke assumes in [2] for his proof *
*of the
Brown Representability Theorem. Note that the proof in [2] is based on a varia*
*nt of
Freyd's Adjoint Functor Theorem. Thus Theorem C provides a link between results
having completely different proofs.
Proof of the Theorem C.(1) is clear. To prove (2), let S 2 S0 and (Xi)i2Ibe a`f*
*amily of
less than objects`in S . Suppose first that Xi2 S0 for all i. Every map`S ! *
*i2IXi
factors through i2JXifor some J I with cardJ < ff. We have card(S; i2JXi) *
* ,
and I has at most (ff)ff= ffsubsets of cardinality less than ff. Therefore
a
card(S; Xi) ff. = ff< :
i
Now let each Xi2 S be arbitrary. We have for each i 2 I a map Ti! Xi such that*
* Ti
is a coproduct of less than objects from S0 and the induced map (S; Ti) ! (S; *
*Xi) is
surjective for all S 2 S0. Using (G20), it follows that the induced map
a a
(S; Ti) -! (S; Xi)
i i
` `
is surjective for all S 2 S0. Thus iXi belongs to S since iTi2 S by the f*
*irst part
of this proof.
(3) and (4) follow from the proof of Theorem A where it is shown that each ob*
*ject in
S can be constructed in countably many steps from objects in S0 by taking copr*
*oducts
of less than factors and cofibers. Note that in each step there is only a set *
*of possible
choices.
(5) follows from (4) since the -small objects form a triangulated subcategory*
* which_
satisfies (1) and (2). *
* |__|
As an example take the stable homotopy category S of CW-spectra. The set S0 =
{nS | n 2 Z} of suspensions of the sphere spectrum S = S0 is a set of perfect g*
*enerators
where ff = @0. For every regular cardinal > @0 the subcategory
S = {X 2 S | cardss*(X) < }
has the properties (1) - (5) of the preceding theorem.
Acknowledgement. I would like to thank Dan Christensen and Nora Ganter for a
number of helpful conversations about the topic of this paper. Thanks in addit*
*ion to
Amnon Neeman for various comments on (t)his work.
References
[1]M. Auslander, Coherent functors, in: Proceedings of the conference on categ*
*orical algebra (La
Jolla, 1965), Springer-Verlag (1966), 189-231.
[2]J. Franke, On the Brown representability theorem for triangulated categorie*
*s, Topology, to appear.
[3]P. Freyd, Stable homotopy, in: Proceedings of the conference on categorical*
* algebra (La Jolla,
1965), Springer-Verlag (1966), 121-172.
[4]P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 3*
*23-448.
[5]A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and *
*Brown representabil-
ity, J. Amer. Math. Soc. 9 (1996), 205-236.
[6]A. Neeman, Brown representability for the dual, Invent. Math. 133 (1998), 9*
*7-105.
[7]A. Neeman, Triangulated categories, Princeton Univ. Press, to appear.
8 HENNING KRAUSE
Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33501 Bielefeld, Germany
E-mail address: henning@mathematik.uni-bielefeld.de