A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS

HENNING KRAUSE

Abstract. We discuss the Brown Representability Theorem for triangulated categories having arbitrary coproducts.

In this paper we discuss the Brown Representability Theorem for triangulated categories having arbitrary coproducts. This theorem is an extremely useful tool and various versions appear in the literature. All of them require a set of objects which generate the category in some appropriate sense. Depending on the proof, there are essentially two types: the first type is based on the analogue of iterated attaching of cells which 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 Theorem for covariant functors (Theorem B). The final Theorem C establishes a filtration 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 S_{0} perfectly generates T if the following holds: (G1) an object X ∈ T is zero provided that (S, X) = 0 for all S ∈ S_{0}; (G2) for every countable set of maps X_{i} → Y_{i} in T the induced map

(S, ∐_{i} X_{i}) ⟶ (S, ∐_{i} Y_{i})

is surjective for all S ∈ S_{0} provided that (S, X_{i}) → (S, Y_{i}) is surjective for all i and S ∈ S_{0}.

Here, (X, Y) denotes the maps from X to Y. For example, (G2) holds if every S ∈ S_{0} is small, that is, the functor (S, −) preserves arbitrary coproducts. Therefore the following result generalizes the classical Brown Representability Theorem for triangulated 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} → Ab is representable if and only if F is cohomological and sends coproducts in T to products.

In [7], Neeman proves a Brown Representability Theorem for triangulated categories 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 triangulated categories having a set of small generators. Take for instance the stable homotopy category of CW-spectra or the unbounded derived category of modules over an associative ring. Let T_{0} be a set of objects and let T be the smallest full triangulated subcategory which is closed under coproducts and contains T_{0}. 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 groups 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 coherent functors T^{op} → Ab form an additive category with cokernels which we denote by T̂ (see also [3] for this concept). A basic tool is the Yoneda functor

T ⟶ T̂, X ↦ (−, X).

Given an additive functor f: S → T , we denote by f^{∗}: Ŝ → T̂ the right exact functor which sends (−, X) to (−, fX).

Lemma 1. Let T be an additive category.

(1) If T has weak kernels, then T̂ is an abelian category.

(2) If T has arbitrary coproducts, then T̂ 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 family of functors F_{i} having a presentation

(−, X_{i}) ⟶^{(−,ϕ_{i})} (−, Y_{i}) ⟶ F_{i} ⟶ 0

the coproduct F = ∐_{i} F_{i} has a presentation

(−, ∐_{i} X_{i}) \underrightarrow{(−,∐ ϕ_{i})} (−, ∐_{i} Y_{i}) ⟶ F ⟶ 0.

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 kernels. Let S_{0} be a set of objects in T and denote by f: S → T the inclusion for S = Add S_{0}.

(1) S has weak kernels and Ŝ is an abelian category.

(2) The assignment F ↦ F |_{S} induces an exact functor f_{∗}: T̂ ⟶ Ŝ.

(3) The functor f^{∗}: Ŝ ⟶ T̂ is a left adjoint for f_{∗}.

(4) f_{∗} ^{∘} f^{∗} ≅ id and f_{∗} induces an equivalence T̂/Ker f_{∗} → Ŝ.

Proof. First observe that for every X ∈ T , there exists an approximation X′ → X such that X′ ∈ S and (S, X′) → (S, X) is surjective for all S ∈ S. This follows from Yoneda’s lemma if we take X′ = ∐_{S∈S_{0}} X_{S} where X_{S} = ∐_{(S,X)} S.

(1) To prove that Ŝ is abelian it is sufficient to show that every map in S has a weak kernel. To obtain a weak kernel of a map Y → Z in S, take the composite of a weak kernel X → Y in T and an approximation X′ → X.

(2) We need to check that for F ∈ T̂ the restriction F |_{S} belongs to Ŝ. It is sufficient to prove this for F = (−, Y). To obtain a presentation, let X → Y′ be a weak kernel of an approximation Y′ → Y. The composite X′ → Y′ with an approximation X′ → X gives an exact sequence

(−, X′)|_{S} ⟶ (−, Y′)|_{S} ⟶ F |_{S} ⟶ 0.

Clearly, F ↦ F |_{S} is exact.

(3) Let F ∈ Ŝ and G ∈ T̂. 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 exact.

(4) We have (f_{∗} ^{∘} f^{∗})(−, X) = (−, X) for all X ∈ S, and f_{∗} ^{∘} f^{∗} ≅ id follows since f_{∗} ^{∘} 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 S_{0} be a set of objects in T and let S = Add S_{0}. Then the functor

h: T ⟶ Ŝ, X ↦ (−, X)|_{S},

is cohomological. It preserves countable coproducts if and only if (G2) holds for S_{0}.

Proof. We apply Lemma 2. To this end write h as composite

h: T ⟶ T̂ ⟶^{f_{∗}} Ŝ.

The Yoneda functor is cohomological and f_{∗} is exact. Therefore h is cohomological. It is clear that h preserves coproducts if and only if f_{∗} preserves coproducts. We know that f_{∗}: T̂ → Ŝ induces an equivalence T̂/Ker f_{∗} → Ŝ, and it is not hard to see that f_{∗} preserves coproducts if and only if Ker f_{∗} is closed under taking coproducts. We fix a coproduct F = ∐_{i} F_{i} in T̂ and for each F_{i} a presentation

(−, X_{i}) ⟶^{(−,ϕ_{i})} (−, Y_{i}) ⟶ F_{i} ⟶ 0.

Now suppose that F_{i} ∈ Ker f_{∗} for all i. Thus (S, ϕ_{i}) is surjective for all S ∈ S and all i. We have F ∈ Ker f_{∗} if and only if the induced map

(S, ∐_{i} X_{i}) ⟶ (S, ∐_{i} Y_{i})

is surjective for all S ∈ S. Clearly, it is sufficient to have this for all S ∈ S_{0}, and we conclude that h preserves countable coproducts if and only if (G2) holds for S_{0}.

Proof of the Theorem A. We fix a perfectly generating set S_{0} of objects in T and put S = Add S_{0}. Replacing S_{0} by {Σ^{n}S | n ∈ ℤ, S ∈ S_{0}}, we may assume that Σ(S_{0}) = S_{0}. Let F: T^{op} → Ab be a cohomological functor which sends coproducts in T to products. We construct inductively a sequence

X_{0} ⟶^{ϕ_{0}} X_{1} ⟶^{ϕ_{1}} X_{2} ⟶^{ϕ_{2}} ⋯

of maps in T and a set of maps π_{i}: (−, X_{i}) → F for i ≥ 0 as follows. Let U = ⋃_{S∈S_{0}} F S. Each x ∈ U corresponds to an element in F S_{x} and we put X_{0} = ∐_{x∈U} S_{x}. We get an element in

∏_{x∈U} F S_{x} ≅ F X_{0},

and using Yoneda’s lemma, this gives a map π_{0} : (−, X_{0}) → F. Suppose we have already constructed π_{i}: (−, X_{i}) → F for some i ≥ 0. Let K_{i} = Ker π_{i} and let U_{i} = ⋃_{S∈S_{0}} K_{i}S. We define T_{i} = ∐_{x∈U_{i}} S_{x} and apply again Yoneda’s lemma to obtain a map T_{i} → X_{i}. We complete this to a triangle

T_{i} ⟶^{υ_{i}} X_{i} ⟶^{ϕ_{i}} X_{i+1} ⟶^{χ_{i}} ΣT_{i}

and get an exact sequence

F (ΣT_{i}) ⟶^{F χ_{i}} F X_{i+1} ⟶^{F ϕ_{i}} F X_{i} ⟶^{F υ_{i}} F T_{i}

since F is cohomological. The construction implies (F υ_{i})π_{i} = 0 and this gives an element π_{i+1} ∈ F X_{i+1} such that (F ϕ_{i})π_{i+1} = π_{i}. Thus we have a factorization

π_{i}: (−, X_{i}) ⟶^{(−,ϕ_{i})} (−, X_{i+1}) ⟶^{π_{i+1}} F.

For each i ≥ 0 the map υ_{i} induces an epimorphism (−, T_{i})|_{S} → K_{i}|_{S} and we get therefore an exact sequence

(−, T_{i})|_{S} ⟶^{(−,υ_{i})|_{S}} (−, X_{i})|_{S} ⟶^{π_{i}|_{S}} F |_{S} ⟶ 0.

We obtain in Ŝ for each i ≥ 0 the following commutative diagram with exact rows

0 ⟶ K_{i}|_{S} ⟶ (−, X_{i})|_{S}  ⟶^{π_{i}|_{S}} F |_{S} ⟶ 0
↓0  ↓ψ_{i}  ↓id
0 ⟶ K_{i+1}|_{S} ⟶ (−, X_{i+1})|_{S} ⟶^{π_{i+1}|_{S}} F |_{S} ⟶ 0

where ψ_{i} = (−, ϕ_{i})|_{S}. Each ψ_{i} has a factorization

ψ_{i}: (−, X_{i})|_{S} ⟶^{π_{i}|_{S}} F |_{S} ⟶^{ι_{i}} (−, X_{i+1})|_{S}

and therefore π_{i+1}|_{S} ^{∘} ι_{i} = id. This gives the following commutative diagram

(−, X_{1})|_{S}  ⟶^{ψ_{1}} (−, X_{2})|_{S}  ⟶^{ψ_{2}} (−, X_{3})|_{S}  ⟶^{ψ_{3}} ⋯
↓≀  ↓≀  ↓≀
F |_{S} ∐ K_{1}|_{S} \underrightarrow{id ∐ 0} F |_{S} ∐ K_{2}|_{S} \underrightarrow{id ∐ 0} F |_{S} ∐ K_{3}|_{S} \underrightarrow{id ∐ 0} ⋯

and taking colimits in Ŝ, we get an exact sequence

0 ⟶ ∐_{i}(−, X_{i})|_{S} ⟶^{id−ψ_{i}} ∐_{i}(−, X_{i})|_{S} ⟶^{(π_{i}|_{S})} F |_{S} ⟶ 0. (∗)

Now consider the triangle

∐_{i} X_{i} ⟶^{id−ϕ_{i}} ∐_{i} X_{i} ⟶ X ⟶ Σ(∐_{i} X_{i})

and observe that

(π_{i}) ∈ ∏_{i} F X_{i} ≅ F (∐_{i} X_{i})

induces a map π: (−, X) → F. We apply the functor

T ⟶ Ŝ, X ↦ (−, X)|_{S},

which is cohomological and preserves countable coproducts by Lemma 3. This gives an exact sequence

∐_{i}(−, X_{i})|_{S} ⟶^{id−ψ_{i}} ∐_{i}(−, X_{i})|_{S} ⟶ (−, X)|_{S} ⟶ ∐_{i}(−, ΣX_{i})|_{S} ⟶^{id−Σψ_{i}} ∐_{i}(−, ΣX_{i})|_{S}.

We compare this sequence with (∗). The map id − Σψ_{i} is a monomorphism, since Σ(S) = S, and it follows that π|_{S}: (−, X)|_{S} → F |_{S} is an isomorphism. Moreover, the subcategory of all Y ∈ T such that π_{Y} is an isomorphism is triangulated, contains S_{0}, and is closed under arbitrary coproducts.

Now let T′ be the localizing subcategory of T which is generated by S_{0}. Thus T′ is the smallest triangulated subcategory of T which contains S_{0} and is closed under coproducts. We claim that T′ = T . To see this let Y ∈ T and apply the construction in the first part of this proof to F = (−, Y). The corresponding map π: (−, X) → (−, Y) is induced by a map X → Y since X ∈ T′. We complete this map to a triangle

W ⟶ X ⟶ Y ⟶ ΣW

and use (G1) to obtain W = 0 = ΣW since (S, X) → (S, Y) is an isomorphism for all S ∈ S_{0}, and Σ(S_{0}) = S_{0} by our assumption. Thus T′ = T and we obtain (−, X) ≅ F in 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 S_{0}. Suppose that T′ is a full triangulated subcategory which is closed under countable coproducts and contains all coproducts of objects in S_{0}. Then T′ = T .

Thus perfect generators are strong generators in the sense of [2]. Note that for every cardinal β > ℵ_{0} a β-perfect generating set in the sense of [7] is automatically perfect as in Definition 1.

Remark. If T has a set S_{0} of perfect generators, then the construction of each X ∈ T implies that the functor X ↦ (−, X)|_{S} preserves arbitrary coproducts. Therefore the following stronger condition holds for S_{0}: (G2′) for every set of maps X_{i} → Y_{i} in T the induced map

(S, ∐_{i} X_{i}) ⟶ (S, ∐_{i} Y_{i})

is surjective for all S ∈ S_{0} provided that (S, X_{i}) → (S, Y_{i}) is surjective for all i and S ∈ S_{0}.

Brown representability for the dual

Let T be a triangulated category with arbitrary coproducts. In this section we prove a Brown Representability Theorem for T^{op}. The first result of this type is due to Neeman [6] and requires T to be generated by a set of small objects. This has been generalized in [7]. The concept which is used here stresses the symmetry between T and T^{op}.

Definition 2. A set S_{0} of objects is a set of symmetric generators for T if the following holds: (G1) an object X ∈ T is zero provided that (S, X) = 0 for all S ∈ S_{0}; (G3) there exists a set T_{0} 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 ∈ S_{0} if and only if (Y, T) → (X, T) is injective for all T ∈ T_{0}.

It is clear that (G3) implies (G2), and that T has a set of symmetric generators if and only if T^{op} has a set of symmetric generators. Therefore the following Brown Representability Theorem for T^{op} is an immediate consequence of Theorem A.

Theorem B. Let T be a triangulated category with arbitrary coproducts, and suppose 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 products.

Proof. We have a set of perfect generators for T and therefore arbitrary products in T . In fact, Theorem A implies that for every family (X_{i})_{i∈I} of objects the functor ∏_{i}(−, X_{i}) is represented by an object which is ∏_{i} X_{i}. The set T_{0} 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 S_{0} of small objects satisfying (G1). To see this, take for T_{0} the set of objects representing the functors

T^{op} ⟶ Ab, X ↦ ((S, X), ℚ/ℤ)

where S ∈ S_{0}. This shows that the stable homotopy category of CW-spectra or the unbounded derived category of modules over an associative ring have sets of symmetric generators.

Remark. Let S_{0} be a set of perfect generators and let S = Add S_{0}. Each injective object I ∈ Ŝ gives rise to an object in T representing

T^{op} ⟶ Ab, X ↦ ((−, X)|_{S}, I).

Therefore (G3) holds for S_{0} if and only if Ŝ has an injective cogenerator.

Neeman’s Brown Representability Theorem for the dual in [7] involves the existence of an injective cogenerator for a category which is equivalent to some Ŝ; it is therefore a consequence of Theorem B.

A filtration

Let T be a triangulated category with arbitrary coproducts. In this section we study a filtration T = ⋃_{κ} S_{κ} which is defined in terms of a set S_{0} of appropriate generators.

Let α be a cardinal. Recall that an object S is α-small if every map S → ∐_{i∈I} X_{i} factors through ∐_{i∈J} X_{i} for some J ⊆ I with card J < α.

Theorem C. Let T be a triangulated category with arbitrary coproducts, and suppose that T is perfectly generated by a set S_{0} of α-small objects. Let κ be the successor of λ^{α} for some cardinal

λ ≥ sup{card(S, ∐_{i∈I} S_{i}) | S, S_{i} ∈ S_{0} and card I < α} + card S_{0}.

Then the objects X ∈ T satisfying card(S, X) < κ for all S ∈ S_{0} form a subcategory S_{κ} having the following properties:

(1) S_{κ} is a triangulated subcategory of T which contains S_{0};

(2) S_{κ} is closed under taking coproducts of less than κ objects;

(3) the isomorphism classes of objects in S_{κ} form a set;

(4) every subcategory T′ of T which satisfies (1) and (2) contains S_{κ};

(5) every object in S_{κ} is κ-small.

A triangulated category which is well generated in the sense of Neeman [7] satisfies 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 variant 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 ∈ S_{0} and (X_{i})_{i∈I} be a family of less than κ objects in S_{κ}. Suppose first that X_{i} ∈ S_{0} for all i. Every map S → ∐_{i∈I} X_{i} factors through ∐_{i∈J} X_{i} for some J ⊆ I with card J < α. We have card(S, ∐_{i∈J} X_{i}) ≤ λ, and I has at most (λ^{α})^{α} = λ^{α} subsets of cardinality less than α. Therefore

card(S, ∐_{i} X_{i}) ≤ λ^{α} · λ = λ^{α} < κ.

Now let each X_{i} ∈ S_{κ} be arbitrary. We have for each i ∈ I a map T_{i} → X_{i} such that T_{i} is a coproduct of less than κ objects from S_{0} and the induced map (S, T_{i}) → (S, X_{i}) is surjective for all S ∈ S_{0}. Using (G2′), it follows that the induced map

(S, ∐_{i} T_{i}) ⟶ (S, ∐_{i} X_{i})

is surjective for all S ∈ S_{0}. Thus ∐_{i} X_{i} belongs to S_{κ} since ∐_{i} T_{i} ∈ S_{κ} by the first part of this proof.

(3) and (4) follow from the proof of Theorem A where it is shown that each object in S_{κ} can be constructed in countably many steps from objects in S_{0} by taking coproducts of less than κ factors and cofibers. Note that in each step there is only a set of possible choices.

(5) follows from (4) since the κ-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 S_{0} = {Σ^{n}S | n ∈ ℤ} of suspensions of the sphere spectrum S = S^{0} is a set of perfect generators where α = ℵ_{0}. For every regular cardinal κ > ℵ_{0} the subcategory

S_{κ} = {X ∈ S | card π_{∗}(X) < κ}

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 addition to Amnon Neeman for various comments on (t)his work.

References

[1] M. Auslander, Coherent functors, in: Proceedings of the conference on categorical algebra (La Jolla, 1965), Springer-Verlag (1966), 189–231.

[2] J. Franke, On the Brown representability theorem for triangulated categories, 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 catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.

[5] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.

[6] A. Neeman, Brown representability for the dual, Invent. Math. 133 (1998), 97–105.

[7] A. Neeman, Triangulated categories, Princeton Univ. Press, to appear.

8 HENNING KRAUSE

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: henning@mathematik.uni-bielefeld.de