A REFORMULATION OF BROWN REPRESENTABILITY THEOREM

SIMION BREAZ, GEORGE CIPRIAN MODOI

Abstract. A well–known result says: If a triangulated category with small coproducts satisfies Brown Representability Theorem, then every triangulated coproduct preserving functor having as domain the respective category has a right adjoint. We wonder about the converse. In the paper we provide a reformulation of Brown Representability Theorem which has some similarities with that converse.

Introduction

In the present note we outline a reformulation of Brown Representability Theorem. This result is a central piece in the theory of triangulated categories (see [2] or [4]). Consider a triangulated category T which has small coproducts, that is coproducts indexed over small sets. Brown Representability Theorem deals with contravariant functors T → Ab, where Ab is the category of abelian groups, and says that such a functor is representable provided that it is cohomogical and sends coproducts into products. We consider the so called abelianization of T , namely the abelian category mod(T ), which satisfies the following universal property: Every cohomological functor T → A, into an abelian category, extends uniquely to an exact functor mod(T ) → A. Using this property we reformulate Brown Representability Theorem in terms of mod(T ) and exact functors starting from mod(T ). More precisely T satisfies Brown Representability Theorem if and only if every exact coproduct preserving functor mod(T ) → A, where A is abelian AB3 and has enough injectives, has a right adjoint. In this way we obtain also some indications how far we are in order to prove (or most probably to disprove) the converse of the well–known result saying that if T satisfies Brown Representability Theorem then every triangulated coproduct preserving functor starting from T has a right adjoint. The point is the following question: Given an exact functor mod(T ) → A, how can it be lifted, in a natural way, to a triangulated functor starting on T ?

For the undefined notions concerning triangulated categories we refer to [4], and for abelian categories to [1] or [5]. For general theory of categories may be also consulted [3].

2000 Mathematics Subject Classification. 18E15, 18E35, 18A40.

The authors were supported by the grant PN2CD-ID-489.

1. The results

Consider an additive category T . By a right module over T (for shortly T -module) we understand a contravariant functor T → Ab. Dually a left module over T (or T^{op}-module) is a covariant functor T → Ab. The class of all T -module forms an abelian AB5 category, the morphisms being natural transformations, category which is denoted here by Mod(T ). Note that the limits and colimits in Mod(T ) are computed point-wise. Usually, this category has no small Hom-sets (another way to say this, is that Mod(T ) lives in a higher universe that T ), unless T is essentially small (i.e. it has a small skeleton). Sometimes it is useful to restrict us to the full subcategory mod(T ) of Mod(T ), consisting of those T -modules X which are finitely presented, that is, there is an exact sequence

T (−, y) → T (−, z) → X → 0,

with y, z ∈ T . This last category has small Hom-sets, provided that T does, as we may see by Yoneda lemma. The Yoneda embedding

T → Mod(T ), x ↦ T (−, x)

restricts to a well defined fully faithful functor

H = H_{T} : T → mod(T ), H(x) = T (−, x),

called also Yoneda functor, or Yoneda embedding (we will omit the index T if no confusion is possible). Define also mop(T ) = mod(T^{op})^{op}, and denote

H′ = H′_{T} : T → mop(T ), H′(x) = T (x, −).

For the sake of clarity, from now on, we will denote by T (x, −) the respective (projective) object of mod(T^{op}) and by H′(x) the same (injective) object viewed in mop(T ). It is well–known, that mod(T ) (respectively mop(T )) is an additive finitely cocomplete (complete) category with enough projectives (injectives), and any functor f : T → A, into an additive finitely cocomplete (complete) category, extends uniquely, up to a natural isomorphism, to a cokernel (kernel) preserving functor

f̂: mod(T ) → A (f̌: mop(T ) → A),

such that f ≅ f̂∘ H (f ≅ f̌∘ H′). Note that for us, the category A will be always abelian. Obviously, H commutes with products, and H′ commutes with coproducts which exists in T . If, in addition, T has small coproducts (products) then mod(T ) (mop(T )) has also small coproducts (products), and the embedding H (H′) commutes with coproducts (products). If this is the case, the a functor f : T → A preserves coproducts (products) if and only if the induced functor f̂ (respectively f̌) does it. Finally recall that mod(T ) (mop(T )) is abelian, provided that T has weak–kernels (weak– cokernels).

From now on, the category T is triangulated, with small coproducts. Recall that, a functor f : T → A, into an abelian category A, is called homological (respectively cohomological) if it is covariant (contravariant) and sends triangles into long exact sequences. Since T has both weak-kernels and weak-cokernels mod(T ) and mop(T ) are both abelian. Moreover for a functor f : T → A, into an abelian category A, the induced functor f̂: mod(T ) → A (f̌: mop(T ) → A) is exact if and only if f is homological.

Recall that a functor f : T → T′ between two triangulated categories is called triangulated or exact if it sends triangles into triangles (this implies it should commute with ”shifts”). If T and T′ are triangulated categories, and f : T → T′ is an exact functor, then H_{T′} ∘ f is homological, so it induces a unique, up to isomorphism, exact functor f^{∗} : mod(T ) → mod(T′), such that H_{T′} ∘ f ≅ f^{∗} ∘ H_{T}. The duality functor T → T^{op} is (contravariant) exact, so it induces as before a unique (contravariant) functor mod(T ) → mod(T^{op}), which is not hard to see that is a duality. Therefore we obtain an equivalence of categories E : mod(T ) → mop(T ), such that E ∘ H ≅ H′.

We call an abelian category A admissible if it is AB3 and has enough injectives. It is well–known that such a category must be also AB4.

Theorem 1.1. The following are equivalent, for a triangulated category with arbitrary coproducts T :

(i) T satisfies the Brown representability theorem.

(ii) For every homological, coproducts preserving functor f : T → A, into an admissible abelian category A, the induced functor

f̂: mod(T ) → A

has a right adjoint. (iii) Every exact, coproducts preserving functor F : mod(T ) → A, into

an admissible, abelian category A, has a right adjoint. (iv) Every exact, coproducts preserving functor F : mod(T ) → Ab^{op} has

a right adjoint.

Proof. (i)⇒(ii). Let f : T → A be a homological functor into an abelian, admissible category A. Let I ∈ A be an injective object. Then the functor

A(f(−), I) = A(−, I) ∘ f : T → Ab

is cohomological, and sends coproducts into products. Then it is representable, by Brown representability theorem; so there is a unique, up to a natural isomorphism, x_{I} ∈ T , such that A(f(−), I) ≅ T (−, x_{I}). Since A has enough injectives, the assignment I ↦ H(x_{I}) defines a unique, up to isomorphism, left exact functor G : A → mod(T ). It is directly verified that G is the right adjoint of f̂.

(ii)⇒(iii) is obvious, since, under the assumptions of (iii), we have F ≅

{ \atop F ∘ H}.

(iii)⇒(iv) follows immediately since Ab^{op} is admissible.

(vi)⇒(i). Let f : T → Ab be a cohomological functor, sending coproducts into products. Then the functor

F : mod(T ) → Ab^{op}, F (X) = mod(T )(X, f)

is exact, coproducts preserving (actually F is the composition of f̂ with the duality functor of Ab). By hypothesis, F has a right adjoint G : Ab^{op} → mod(T ). We deduce F (X) ≅ mod(T )(X, G(ℤ)). Further f ≅ G(ℤ), showing that f ∈ mod(T ) and f has to be injective, since it represents the exact functor F. Therefore, f ≅ T (−, x), for some x ∈ T . □

We record also the dual of the preceding results (for this, we will say that the abelian category A is co-admissible if A^{op} is admissible):

Theorem 1.2. The following are equivalent, for a triangulated category with arbitrary products T :

(i) T^{op} satisfies the Brown representability theorem.

(ii) For every homological, products preserving functor f : T → A, into a co-admissible, abelian category A, the induced functor

f̌: mop(T ) → A

has a left adjoint. (iii) Every exact, products preserving functor F : mop(T ) → A, into a

co-admissible, abelian category A, has a left adjoint. (iv) Every exact, products preserving functor F : mop(T ) → Ab has a

left adjoint.

Remark 1.3. Since, for a triangulated category T , we have an equivalence of categories E : mod(T ) → mop(T ), such that E ∘ H = H′, we may freely interchange mod(T ) and mop(T ) in Theorems 1.1 and 1.2.

Remark 1.4. Consider a triangulated category, with arbitrary coproducts (products) T . Theorems 1.1 and 1.2 provide reformulations of the Brown representability theorem for T respectively T^{op} in terms of abelian category mod(T ). Note also, Brown Representability Theorem for T implies every triangulated coproduct preserving functor f : T → T′, into another triangulated category T′ has a right adjoint. Therefore the above reformulation has some similarities with the converse of that implication, namely T satisfies Brown Representability Theorem if and only if every exact coproduct preserving functor mod(T ) → A, into an admissible abelian category A has a right adjoint.

References

[1] P. Gabriel, De catégories abeliennes, Bull. Soc. Math. France, 90(1962) 323-448.

[2] H. Krause, A Brown representability theorem via coherent functors, Topology, 41(2002), 853-561.

[3] B. Mitchell, Theory of Categories, Academic Press, New York and London, 1965.

[4] A. Neeman, Triangulated Categories, Annals of Mathematics Studies, 148, Princeton University Press, Princeton, NJ, 2001.

[5] N. Popescu, L. Popescu, Theory of Categories, Editura Academiei, Bucureşti and Sijthoff & Noordhoff International Publishers, 1979.

”Babeş-Bolyai” University, RO-400084, Cluj-Napoca, Romania E-mail address, Simion Breaz: bodo@math.ubbcluj.ro E-mail address, George Ciprian Modoi: cmodoi@math.ubbcluj.ro