arXiv:1904.12182v2 [math.CT] 8 Sep 2020 uco,i smrhct Ext to isomorphic is , ae n .Cra n .Elneg[] sn ehd fhomologic of Hom the methods of using functor [7], derived Eilenberg first S. the and that Cartan showed H. on, Later esnuulyExt usually reason i osrcincnb aiyetne oaeinctgre,wher categories, abelian Ext to class extended the easily that show be can construction His a rtmd yR ari 94 nhswr 1 ] ardfie nad an defined Baer 2], [1, of work kind his This On 1934. ends. Ext in the class Baer at the R. morphisms by identity made the with first if equivalent was another are to sequences one exact two from where equivalence, to up nti ril ewl ou netnin na bla abelian an in extensions on focus will we article this In central multiplic of [10]. from representations topology developed to from has 13] ranging that [4, applications theory with a 15], is [21, extensions of study The netnino nobject an of extension an 1991 e od n phrases. and words Key h uhrtak h rjc AITUiesddNacional PAPIIT-Universidad Project the thanks author The H OEAETADABTAYCPOUT IN COPRODUCTS ARBITRARY AND EXT YONEDA THE ahmtc ujc Classification. Subject Mathematics Abstract. Q aeois ulrsl sas stated. also is to result used dual are a A constructed Colpi categories. R. by the necessar made Furthermore, not question [8]. a with answering categories injectives, construc abelian nor is tives Ab4 in desired colimits, The using contexts. general more oeaEt hc safntreuvln oExt to equivalent functor a is which Ext, Yoneda jects o.Tega fti ae st hwasmlrioopimfo isomorphism similar a show to is paper this of goal The tor. rdcs n rdcsi b bla aeoiswt enoug with category categories such abelian every Ab4 for in Namely, products and products, i ∈ I { Ext 1 A ( i ,A C, } A n i ∈ ( C 1 I hr r elkoniette novn h x bifunctor Ext the involving identities known well are There A h olwn smrhs a ebitExt built be can isomorphism following the , fetnin fa bla group abelian an of extensions of ) ( i X , ,A C, ,weeExt where ), C 1 xesos,Ynd x ordcs,pout b categor Ab4 , products , coproducts , ext Yoneda , Extensions LJNR ARGUD ALEJANDRO ( scle h ru fetnin of extensions of group the called is ) BLA CATEGORIES ABELIAN A ,A C, ya object an by C 1 0 a aua tutr faeingop o this For group. abelian of structure natural a has ) ( → 1. C, n A − Introduction sthe is ,o epcieyt Ext to respectively or ), → A rmr 89 81,18A30. 18G15, , 18E99 Primary ie nobject an given , M 1 n C t eie uco fteHmfunc- Hom the of functor derived -th → sasoteatsequence exact short a is ´ NMONROY IN C → C ( 0 C, n htcnb endin defined be can that − u´ nm eM´exicoAut´onoma IN103317. de A X uco,o Hom or functor, ) A n l nuhprojec- enough ily ya bla group abelian an by n e fob- of set a and dKR ulrin Fuller R. K nd hrceieAb4 characterize e xlctyby explicitly ted L C 1 projectives. h A ( i − ∈ C I by the r A , nti context, this In . ei morphism a is re A ipealgebras simple i C .Ti result This ). ti sdto used is it e X , . tv groups ative co- , n  -th lalgebra, al = ∼ approach iinon dition C ( − ies. A , C ) . 2 A. ARGUD´IN MONROY marked the beginning of a series of research works looking for ways of constructing the derived of the without using projective or injective ob- jects, with the spirit that resolutions should be only a calculation tool for derived functors.

One of this attempts, registered in the work of D. Buchsbaum, B. Mitchell, S. Schanuel, S. Mac Lane, M.C.R. Butler, and G. Horrocks [16, 5, 6, 17], was based in the ideas of N. Yoneda [23, 24], defining what is known today as the theory of n-extensions and the functor called as the Yoneda Ext. An n-extension of an object A by an object C is an of length n

0 → A → M1 →···→ Mn → C → 0 up to equivalence, where the equivalence of exact sequences of length n > 1 is defined in a similar way as was defined for length 1. In this theory, the Baer sum n can be extended to n-extensions, proving that the class ExtC (C, A) of n-extensions of A by C is an .

Recently, the generalization of homological techniques such as Gorenstein or tilting objects to abstract contexts [19, 3, 8, 9], such as abelian categories that do not nec- essarily have projectives or injectives, claim for the introduction of an Ext functor that can be used without restraints. The only problem is that it is not clear if the rich properties of the homological Ext are also valid for the Yoneda Ext. The goal of this work is to make a next step by exploring some properties that the Yoneda Ext shares with the homological Ext.

Namely, we will explore the following property that is well known for cat- egories:

Theorem 1.1. [20, Proposition 7.21] Let R be a , M ∈ Mod R, and {Ni}i∈I be a set of R-modules. Then, there exist an isomorphism

n ∼ n ExtR Ni,M = ExtR (Ni,M) . i∈I ! i∈I M Y

The proof of such theorem can be extended to Ab4 abelian categories with enough projectives. Our goal will be to prove an analogue result for the Yoneda Ext without assuming the existence of enough projectives.

Let us now describe the contents of this paper. Section 2 is devoted to review the basic results of the theory of extensions by following the steps of B. Mitchell in [17]. In section 3 we prove the desired theorem. More precisely, we show that in an Ab4 we can build the desired bijections explicitly by using colimits. Finally, in section 4 we use the bijections constructed in section 3 to characterize Ab4 categories. YONEDA EXT 3

2. Extensions

In this section we will remember the basic theory of extensions. As was mentioned before, the theory of n-extensions was created by Nobuo Yoneda in [23]. In such paper he worked in a category of modules and most of the results are related with the homological tools built by projective and injective modules. Since our goal is to work in an abelian category without depending on the existence of projective or injective objects, we refer the reader to the work of Barry Mitchell [17] for an approach in abelian categories without further assumptions. Throughout this paper, C will denote an abelian category. Definition 2.1. [17, Section 1] Let C ∈ C, and α : A → B, α′ : A′ → B′ be morphisms in C. We set the following notation:

(a) ∇C := ( 1C 1C ): C ⊕ C → C; 1C (b) ∆C := 1C : C → C ⊕ C; ′ α 0 ′ ′ (c) α ⊕ α := ′ : A ⊕ A → B ⊕ B . 0 α  2.1. 1-Extensions. Let us begin by recalling some basic facts and notation about 1-extensions. Definition 2.2. [17, Section 1] Let α : N → N ′, β : M → M ′, and γ : K → K′ be morphisms in C, and consider the following short exact sequences in C

f g f ′ g′ η : 0 → N → M → K → 0 and η′ : 0 → N ′ → M ′ → K′ → 0.

(a) We say that (α,β,γ): η → η′ is a morphism of short exact sequences if βf = f ′α and γg = g′β. (b) We denote by η ⊕ η′ to the short exact sequence

f⊕f ′ g⊕g′ 0 → N ⊕ N ′ → M ⊕ M ′ → K ⊕ K′ → 0.

Definition 2.3. [17, Section 1] For N,K ∈ C, let EC(K,N) denote the class of short exact sequences of the form 0 → N → M → K → 0. ′ ′ Remark 2.4. Let A, C ∈C and η, η ∈EC(C, A). Consider the relation η ≡ η given ′ by the existence of a short exact sequence morphism (1A, β, 1C): η → η . By the snake lemma, we know that β is an isomorphism, and hence ≡ is an on EC(C, A). Definition 2.5. [17, Section 1] Consider A, C ∈C.

1 (a) Let ExtC (C, A) := EC(C, A)/ ≡ ; 1 (b) Each object of ExtC (C, A) is refered as an extension from A to C. (c) Every extension from A to C will be denoted with a capital letter E, or by η, in case η is a representative of the class E. 1 ′ 1 ′ ′ (d) Given η ∈ ExtC (C, A) and η ∈ ExtC (C , A ), we will call extension mor- phism from η to η′, to every short exact sequence morphism η → η′. 4 A. ARGUD´IN MONROY

(e) If (α,β,γ): E → E′ and (α′,β′,γ′): E′ → E′′ are extension morphisms, we define the composition morphism as (α′,β′,γ′)(α,β,γ) := (α′α, β′β,γ′γ). Remark 2.6. An essential comment made by B. Mitchell in [17] is that the class 1 ExtC (C, A) may not be a set (see [11, Chapter 6, Exercise A] for an example). Considering this fact, we should be cautious when we talk about correspondences between extensions classes. Nevertheless, by simplicity we will say that a corre- spondence 1 ′ ′ 1 Φ : ExtC (C , A ) → ExtC (C, A) 1 ′ ′ is a function, if it associates to each η ∈ ExtC (C , A ) a single element Φ(η) in 1 ExtC (C, A).

Remember the following result. Proposition 2.7. [16, Lemma 1.2] Consider a morphism α : X → K and an exact sequence g′ f g 0 → N → M → K → 0 in C. If (E, α′,g′) is 0 N E X 0 α′ α the pullback diagram of the morphisms g and α, g then there is an exact short sequence ηα and a 0 N M K 0 ′ morphism (1, α , α): ηα → η.

Of course the construction described above defines a correspondence between the extension classes. ′ Proposition 2.8. [17, Corollary 1.2.] Let η ∈ EC(C, A) and γ ∈ Hom C(C , C). 1 1 ′ Then, the correspondence Φγ : ExtC (C, A) → ExtC (C , A), η 7→ ηγ , is a function.

By duality, given a morphism α : N → X and an exact sequence f η : 0 → N → M → K → 0, the pushout of the morphisms f and α, gives us an exact sequence ηα together with a morphism (α, α′, 1) : η → ηα. Moreover, we also have that the correspondence α 1 1 α Φ : ExtC (K,N) → ExtC (K,X), η 7→ η , is a function. Definition 2.9. [17, Section 1] For α : A → A′ and γ : C′ → C morphisms in C, 1 α and E ∈ ExtC (C, A), we set Eγ := Φγ(E), and αE := Φ (E).

As we have described, there exists a natural action of the morphisms on the exten- sion classes. These actions are associative and respect identities. 1 ′ ′ ′ ′′ Lemma 2.10. [17, Lemma 1.3] Let E ∈ ExtC (C, A), α : A → A , α : A → A , γ : C′ → C, and γ′ : C′′ → C′ be morphisms in C. Then,

(a) 1AE = E and E1C = E; (b) (α′α)E = α′(αE) and E(γγ′) = (Eγ)γ′; (c) (αE)γ = α(Eγ).

Next, we recall the definition of the Baer sum. YONEDA EXT 5

′ 1 Definition 2.11. [17, Section 1] For E, E ∈ ExtC (C, A), the sum extension of E ′ ′ ′ and E is E + E := ∇A (E ⊕ E ) ∆C .

This sum operation is well behaved with the actions before described and gives a structure of abelian group to the extension classes.

Theorem 2.12. [17, Lemma 1.4 and Theorem 1.5.] For any A, C ∈ C, we have 1 that the pair ExtC (C, A) , + is an abelian group, where the identity element is the extension given by the class of exact sequences that split. Furthermore, let 1 ′ 1  ′ ′ ′ ′ ′ E ∈ ExtC (C, A), E ∈ ExtC (C , A ), α ∈ Hom C(A, X), α ∈ Hom C(A ,X ), γ ∈ ′ ′ ′ Hom C(Y, C) and γ ∈ Hom C(Y , C ). Then, the following equalities hold true:

(a) (α ⊕ α′) (E ⊕ E′)= αE ⊕ α′E′; (b) (α + α′) E = αE + α′E; (c) α (E + E′)= αE + αE′; (d) (E ⊕ E′) (γ ⊕ γ′)= Eγ ⊕ E′γ′; (e) E (γ + γ′)= Eγ + Eγ′; (f) (E + E′) γ = Eγ + E′γ; 1 (g) 0E = E0= E0 for every E ∈ ExtC (C, A).

2.2. n-Extensions. We are ready for recalling the definition of n-extensions. It is a well known fact that short exact sequences can be sticked together in order to contruct a long exact sequence. Following this thought, the spirit of n-extensions is to define a well behaved 1-extensions composition that constructs long extensions.

Definition 2.13. [17, Section 3] We will make use of the following considerations.

(a) For an exact sequence η : 0 → A → Bn−1 →···→ B0 → C → 0 in C we say that η is an exact sequence of length n, and A and C are the left and right ends of η, respectively. n (b) Let EC (L,N) denote the class of exact sequences of length n with L and N as right and left ends. (c) Consider the following exact sequences in C

µ fn−1 f1 π η : 0 → N → Bn−1 → · · · → B0 → K → 0, ′ f ′ ′ ′ ′ ′ µ ′ n−1 f1 ′ π ′ η : 0 → N → Bn−1 → · · · → B0 → K → 0.

′ A morphism η →η is a collection of n + 2 morphisms (α, βn−1, ··· ,β0,γ) ′ ′ ′ in C, where α : N → N , γ : K → K , and βi : Bi → Bi ∀i ∈ [0,n − 1] are such that

′ ′ ′ βn−1µ = µ α, γπ = π β0 and βi−1fi = fi βi ∀i ∈ [0,n − 1].

Equivalently, we can say that a morphism of exact sequences of length n is a 6 A. ARGUD´IN MONROY

0 N Bn−1 ··· B0 K 0

′ ′ ··· ′ ′ 0 N Bn−1 B0 K 0

In the following lines, we define an equivalence relation for studying the classes of exact sequences of length n. As we did for the case with n = 1, we start by ′ n ′ saying that two exact sequences η, η ∈EC (C, A) are related, denoted by η  η , if ′ there is a morphism (1A,βn−1 ··· ,β0, 1C): η → η . In this case, we say also that this morphism has fixed ends. Observe that, in contrast with the case n = 1, this relation needs not to be symmetric. Thus, for achieving our goal, we most consider the equivalence relation ≡ induced by . Namely, we write η ≡ η′ if there are exact sequences η1, ··· , ηk such that ′ η = η1, ηi  ηi+1 or ηi+1  ηi, and η = ηk. Definition 2.14. [14, Section 9] For n ≥ 1 and A, C ∈ C, we consider the class n n ExtC (C, A) := EC (C, A)/ ≡ , whose elements will be called extensions of length n with C and A as right and left ends. Let η denote the equivalence class of n ′ η ∈ EC (C, A). An extension morphism from η to η is just a morphism from η to η′. Remark 2.15. The definition of the equivalence relation above might seem naive. But actually the relation is built with the purpose of making the composition of extensions associate properly when there is a morphism acting in the involved extensions [17, Section 3][16, Section 5]. In the following lines, we will discuss briefly such matter.

1 ′ 1 ′ ′ Observe how in general, for η ∈EC (C, A), η ∈EC (D, C ) and β : C → C in C, it is false that (ηβ) η′ = η (βη′). The only affirmation that can be made is that there is an extension morphism (ηβ) η′ → η (βη′). To show such morphism, we remember that β induces morphisms ηβ → η and η′ → βη′. Hence, we can build the morphisms (ηβ) η′ → ηη′ and ηη′ → η (βη′), whose composition gives the wanted morphism. Therefore, even if we have the inequality (ηβ) η′ 6= η (βη′) we can conclude that (ηβ) η′ = η (βη′). Definition 2.16. [17, Section 3] Consider the following exact sequences of length n and m, respectively µ π η : 0 → N → Bn →···→ B1 → K → 0,

′ ′ ′ µ ′ ′ π η : 0 → K → Bm →···→ B1 → L → 0.

The composition sequence ηη′, of η with η′, is the exact sequence ′ ′ µ µ π ′ ′ π 0 → N → Bn →···→ B1 → Bm →···→ B1 → L → 0. Remark 2.17. Note that each exact sequence in C

κ : 0 → A → Bn →···→ B1 → C → 0 YONEDA EXT 7 can be written as a composition of n short exact sequences κ = ηn ··· η1, where

ηi := 0 → Ki+1 → Bi → Ki → 0, with Kn+1 := A, K1 := C and Ki = Im(Bi → Bi−1) ∀i ∈ [2,n − 1]. We will refer to such factorization of κ as its natural decomposition.

Of course, the composition of exact sequences induces a composition of extensions. Lemma 2.18. [17, Proposition 3.1] Let m,n > 0, and A,C,D ∈ C. Then, the n m n+m ′ ′ correspondence Φ : ExtC (C, A) × ExtC (D, C) → ExtC (D, A), (η, η ) 7→ ηη , is a function.

We can now define without ambiguity the composition of extensions. n ′ m Definition 2.19. Let E ∈ ExtC (C, A) and E ∈ ExtC (D, C). For E = η and E′ = η′, we define the composition extension EE′ of E with E′, as the extension ′ ′ EE := ηη . If η = ηn ··· η1 is the natural decomposition of η, the induced extension factorization E = ηn ··· η1 is known as a natural decomposition of E.

In the same way, an n-extension can be factored into simpler extensions; a morphism of n-extensions can be factored into a composition of n simpler morphisms. The next lemma shows the basic fact in this matter. Lemma 2.20. [17, Lemma 1.1] Consider a mor- ′ phism of exact sequences (α,β,γ): η → η, with f ′ g′ 0 ′ ′ ′ 0 f g A B C η : 0 → A → B → C → 0 and α β′′ f ′ g′ η′ : 0 → A′ → B′ → C′ → 0. 0 A E C′ 0 ′ ′ β γ Then, ηγ = αη and (α,β,γ) factors through ηγ f g as 0 ABC 0 (α,β,γ)=(1,β′,γ)(α, β′′, 1).

In general, we can make the following affirmation.

′ n Corollary 2.21. Let η, η ∈ EC (C, A) be exact sequences with natural decomposi- ′ ′ ′ tions η = ηn ··· η1 and η = ηn ··· η1. Then, the following statements hold true.

′ (a) There is an exact sequence morphism (α, βn−1, ··· ,β0,γ): η → η if, and only if, there is a collection of extension morphisms ′ (αi,βi−1, αi−1): ηi → ηi ∀i ∈ [1,n]

where αn = α and α0 = γ. ′ (b) If there is an exact sequence morphism (α, βn−1, ··· ,β0,γ): η → η , then there is a collection of morphisms αn−1, ··· , α1 in C satisfying the following equalities: ′ ′ (b1) ηn ··· η1γ = αηn ··· η1, ′ ′ (b2) ηi ··· η1γ = αiηi ··· η1 ∀i ∈ [1,n − 1], and ′ ′ (b3) ηn ··· ηi+1αi = αηn ··· ηi+1 ∀i ∈ [1,n − 1]. 8 A. ARGUD´IN MONROY

Proof. It follows from 2.20. 

By Lemma 2.18, the following actions are well defined. ′ n n Definition 2.22. [17, Section 3] Consider η, η ∈EC (C, A), E := η ∈ ExtC (C, A), ′ ′ n ′ ′ ′ E := η ∈ ExtC (C, A), and let η = ηn ··· η1 and η = ηn ··· η1 be the natural decompositions of η and η′.

′ (a) Given α ∈ Hom C (A, A ), we define αE := αηn ··· η1. ′ (b) Given γ ∈ Hom C(C , C), we define Eγ := ηn ··· η1γ. (c) We define the sum of extensions of length n in the following way ′ ′ E + E := ∇A (E ⊕ E ) ∆C .

Most of the properties, proved earlier for extensions of length 1, can be naturally extended, as can be seen in the following lines. Corollary 2.23. [17, Lemma 3.2 an Theorem 3.3] Let n> 0.

n ′ m ′ ′ ′ ′′ ′ (a) Let E ∈ ExtC (C, A), E ∈ ExtC (D, C ), β ∈ Hom C(C , C), β ∈ Hom C(C , C ), ′ ′ ′ ′′ α ∈ Hom C(A, A ), and α ∈ Hom C(A , A ). Then the following equalities hold true: (a1) (Eβ) E′ = E (βE′); (a2) 1AE = E = E1C; (a3) E (ββ′) = (Eβ) β′; (a4) (α′α) E = α′ (αE). n ′ n ′ ′ m ′ m ′ ′ (b) Let E ∈ ExtC (C, A), E ∈ ExtC (C , A ), F ∈ ExtC (D, C), F ∈ ExtC (D , C ), ′ ′ ′ ′ ′ ′ α ∈ Hom C(A, X), α ∈ Hom C(A ,X ), γ ∈ Hom C (Y, C), and γ ∈ Hom C (Y , C ). Then the following equalities hold true: (b1) (α ⊕ α′) (E ⊕ E′)= αE ⊕ α′E′ and (E ⊕ E′) (γ ⊕ γ′)= Eγ ⊕ E′γ′; (b2) (E ⊕ E′) (F ⊕ F ′)= EF ⊕ E′F ′; (b3) (E + E′) F = EF + E′F and E (F + F ′)= EF + EF ′; (b4) (α + α′)E = αE + α′E and E (γ + γ′)= Eγ + Eγ′; and (b5) α (E + E′)= αE + αE′ and (E + E′) γ = Eγ + E′γ. n (c) The pair (ExtC (C, A) , +) is an abelian group, where the identity element is the extension E0 given by the exact sequence, in case n ≥ 2, 0 → A →1 A →0 · · · →0 C →1 C → 0 .

We conclude this section with the following theorem that focus on characterizing the trivial extensions. n Theorem 2.24. [17, Theorem 4.2] Let n > 1 and η ∈ EC (C, A) with a natural decomposition η = ηn ··· η1. Then , the following statements hold true:

(a) η =0; n (b) there is an exact sequence κ∈EC (C, A) and a pair of morphisms with fixed ends 0 ← κ → η. ′ n (c) there is an exact sequence κ ∈EC (C, A) and a pair of morphisms with fixed ends 0 → κ′ ← η. YONEDA EXT 9

3. Additional structure in Abelian Categories

In this section we will approach our problem dealing with arbitrary products and coproducts. Of course, an abelian category does not necessarily have arbitrary products and coproducts. Hence, we will review briefly the theory of abelian cate- gories with additional structure introduced by A. Grothendieck in [12]. For further reading we suggest [18, Section 2.8].

3.1. Limits and colimits.

Definition 3.1. [18, Section 1.4.] Let C and I be categories, where I is small (that is the class of objects of I is a set). Let F : I →C be a functor and X ∈C. A family of morphisms {αi : F (i) → X}i∈I in C is co-compatible with F , if αi = αj F (λ) for every λ : i → j in I. The colimit (or inductive limit) of F is an object colim F in C with a co-compatible family of morphisms F (λ) F (i) F (j) µi µj {µi : F (i) → colim F }i∈I , such that for every co-compatible family of mor- colim F γ phisms {γi : F (i) → X}i∈I , there is a unique morphism γi γj γ : colim F → X such that γi = γµi for every i ∈ I. X

Let I be a small category and λ : i → j be a morphism in I. The following notation will be useful s(λ) := i and t(λ) := j.

Proposition 3.2. [22, Proposition 8.4] Let C be a preadditive category with coprod- ucts and cokernels, I be a small category, F : I →C be a functor, and

uk : F (k) → F (i) ∀k ∈ I, vλ : F (s(λ)) → F (s(γ)) ∀λ ∈ H := HomI i∈I γ∈H M M be the respective canonical inclusions into the coproducts. Then,

 ϕ  ϕ colim F = Coker F (s(γ)) → F (i) , F (s(γ)) F (i) M M Lγ∈H Li∈I γ∈H i∈I  u u vλ ( s(λ) t(λ) ) where ϕ is the morphism induced by the F (s(λ)) F (s(λ)) ⊕ F (t(λ)) universal property of coproducts applied 1 − to the family of morphisms  F (λ) 

ϕλ := us(λ) − ut(λ)F (λ) λ∈H .  The dual notion of colimit is the limit.

Definition 3.3. [18, Section 1.4.] Let C and I be categories, with I small. Let F : I →C be a functor and X ∈C. A family of morphisms {αi : X → F (i)}i∈I in 10 A. ARGUD´IN MONROY

C is compatible with F , if αj = F (λ)αi for every λ : i → j in I. The limit (or projective limit) of F is an object lim F in C together with a compatible family of morphisms F (λ) F (i) F (j) µi µj {µi : lim F → F (i)}i∈I such that for any compatible family of morphisms lim F γ {γi : X → F (i)}i∈I there is a unique γ ∈ HomC (X, lim F ) γi γj such that γi = µiγ for every i ∈ I. X

Proposition 3.4. [22, Proposition 8.2] Let C be a preadditive category with products and kernels, I be an small category, F : I →C be a functor, and

uk : F (i) → F (k) ∀k ∈ I, vλ : F (t(γ)) → F (t(λ)) ∀λ ∈ H := HomI i∈I γ∈H Y Y be the respective canonical proyections out of the products. Then,

ϕ lim F = Ker F (i) → F (t(γ)) , ϕ ∈ F (i) γ∈H F (t(γ)) i∈I γ∈H ! i I Y Y where ϕ is the morphism induced by usQ(λ) Q ut(λ) vλ the universal property of products ap- plied to the family of morphisms F(s(λ))⊕ F (t(λ)) F (t(λ)) ( −F (λ) 1 ) ϕλ := F (λ)us(λ) − ut(λ) λ∈H . Definition 3.5. Let I be a small category and C be an abelian category. It is said that a family of objects and morphisms (Mi,fα) is a direct system, if i∈I,α∈HomI there is a functor F : I → C such that F (i)= Mi ∀i ∈ I and F (α) = fα for every α ∈ HomI .

3.2. Ab3 and Ab4 Categories. Definition 3.6. [18, Section 2.8.] An Ab3 category is an abelian category satisfying the following condition:

(Ab3): For every set of objects {Ai}i∈I in C, the coproduct i∈I Ai exists. L We remember the following well known fact. Proposition 3.7. [18, Section 2.8.] Let C be an Ab3 category and

′ fi gi ′′ Xi → Xi → Xi → 0 i∈I be a set of exact sequences inn C. Then, o

fi gi ′ Li∈I Li∈I ′′ Xi → Xi → Xi → 0 i∈I i∈I i∈I M M M is an exact sequence in C.

In general, it is not possible to prove that i∈I fi is a monomorphism if each fi is a monomorphism. For this reason, the following Grothendieck’s condition arised. L YONEDA EXT 11

Definition 3.8. [18, Proposition 8.3.] An Ab4 category is an Ab3 category C satisfying the following condition:

(Ab4): for every set of monomorphisms {fi : Xi → Yi}i∈I in C, the morphism i∈I fi is a monomorphism. L We will refer to the dual condition as Ab4*.

Remark 3.9. Let C be an Ab4 category. Then, for every sets of objects {Ai}i∈I and {Bi}i∈I in C, the correspondence

n n C : ExtC (Ai,Bi) → ExtC Ai, Bi , (ηi) 7→ ηi, i∈I i∈I i∈I ! i∈I Y M M M is a well defined morphism of abelian groups.

3.3. Ext groups and arbitrary products and coproducts. We are finally ready to proceed in our goal’s direction. Lemma 3.10. Let C be an Ab4 category, and

fi gi ηi : 0 → B → Ai → Ci → 0 i∈I be a set of short exact sequencesn in C. Then, there is ao short exact sequence f g η : 0 → B → colim(fi) → Ci → 0 i∈I M such that ηµi = ηi ∀i ∈ I, where µi : Ci → i∈I Ci i∈I is the family of canonical inyections into the coproduct.  L

Proof. Consider the set {fi : B → Ai}i∈I as a direct system. Observe that the set of morphisms of exact sequences 1 0 {(1B,fi, 0) : β → ηi}i∈I with β := 0 → B → B → 0 → 0, is a direct system of exact sequences. We will consider the colimit of such system and prove that, as result, we get a short exact sequence. To this end, we observe that (B, 1i : B → B)i∈I is the colimit of the system {1i : B → B}i∈I and that ( i∈I Ci,µi : Ci → i∈I Ci) is the colimit of the system {0:0 → Ci}i∈I . Hence, by 3.2 we build the L L diagram beside, where the 0 columns are the morphism mentioned in 3.2, the up- 0 B B 0 0 Li∈I i Li∈I i per and central rows are co- products of the sequences β 0 B⊕( B ) B⊕( A ) C 0 Li∈I i Li∈I i Li∈I i and ηi respectively, and the bottom row is the result of B colim(fi ) i∈I Ci 0 the colimits. Thus, by the L snake lemma we get the ex- act sequence 0 0 0 12 A. ARGUD´IN MONROY

η : 0 → B → colim(fi) → Ci → 0. i∈I M Furthermore, the families of morphisms associated to such colimits give us the exact ′  sequence morphisms (1,µi,µi): ηi → η ∀i ∈ I, which proves the statement.

Proposition 3.11. Let C be an Ab4 category and {Ai}i∈I a set of objects in C. Con- sider the coproduct with the canonical inclusions µi : Ai → i∈I Ai i∈I . Then, the correspondence Ψ : Ext1 ( A ,B) → Ext1 (A ,B), defined by E 7→ (Eµ ) , C i i∈I C i L  i i∈I is an isomorphism for every B ∈C. L Q

Proof. We will proceed by proving the following steps:

(a) The correspondence Ψ is an abelian group morphism. (b) Ψ is injective. 1 1 (c) Given (ηi) ∈ i∈I ExtC (Ai,B), there is E ∈ ExtC ( Ai,B) such that Ψ(E) = (ηi). Q L Clearly, proving these statements are enough to conclude the desired proposition.

(a) It follows by 2.21. (b) Suppose that E is an extension with representative

f g η : 0 → B → C → Ai → 0 i∈I M such that Eµi = 0 ∀i ∈ I. Suppose that (1,pi,µi): Eµi → E is the mor- phism induced by µi, and that each extension Eµi has as representative the exact fi gi sequence ηi : 0 → B → Ci →

Ai → 0. By definition, there is a hi morphism hi : Ai → Ci such that 0 B Ci Ai 0 fi gi gihi = 1Ai . Thus, by the coproduct universal property, there is a unique pi µi f g morphism h : Ai → C such that i∈I 0 BC Ai 0 Li∈I hµi = pihi ∀i ∈{1, 2}. Therefore, by L h ghµi = gpihi = µigihi = µi ∀i ∈ I, we have that gh =1 by the coproduct universal property; and thus, i∈I Ai E = 0. L (c) It follows by 3.10.



Theorem 3.12. Let C be an Ab4 category, n ≥ 1, and {Ai}i∈I be a set of objects in

C. Consider the coproduct i∈I Ai and the canonic inclusions µi : Ai → i∈I Ai i∈I . Then, the correspondence Ψ : Extn ( A ,B) → Extn (A ,B), E 7→ (Eµ ) , Ln C i i∈I C i L i i∈I is an isomorphism of abelian groups for every B ∈C. L Q YONEDA EXT 13

Proof. We will proceed by proving the following statements:

(a) The correspondence Ψn is a morphism of abelian groups; (b) Ψn is injective; n n (c) For every (ηi) ∈ i∈I ExtC (Ai,B), there is E ∈ ExtC i∈I Ai,B such that Ψn(E) = (ηi). Q L  It is worth to mention that the result was already proved in 3.11 for n = 1. Fur- thermore, in the proof of 3.11(c) it was shown explicitly the inverse function of Ψ1. −1 We will denote such correspondence as Ψ1 .

(a) It follows by 2.21. (b) Let η be an extension with a natural decomposition η = ηn ··· η1 such that ηµi = 0 ∀i ∈ I. By 2.24 this means that for every i ∈ I there is a pair of exact sequences morphisms with fixed ends ηµi ← κi → 0. Suppose that each exact sequence κi has the natural decomposition κi = κ(i)n ··· κ(i)1. It follows from the morphism κi → 0 that

′ fi gi κ(i)n := κi : 0 → B → Yi → Xi → 0 ′ ′ 1 is a splitting exact sequence. Let (κi) := (κi)i∈I ∈ i∈I ExtC (Xi,B). By 3.11(c), we know that Ψ−1(κ′ ) ∈ Ext1 X ,B is an extension 1 i C i∈I i Q such that Ψ−1(κ′ )µ′ = κ′ ∀i ∈ I, where each µ′ : X → X is the 1 i i i L i i  i∈I i −1 ′ canonical inclusion. Let κ := Ψ1 (κi) i∈I κ(i)n−1 ···L i∈I κ(i)1 . We will show that there is a pair of exact sequence morphisms with fixed L  L  ends η ← κ → 0, which will prove (b) by 2.24. Indeed, by the fact that for every i ∈ I there is a morphism with fixed ends ηµi ← κi, it follows that there is a morphism with fixed right end ηn−1 ··· η1µi ← κ(i)n−1 ··· κ(i)1, inducing by the coproduct universal property a morphism with fixed right end

ηn−1 ··· η1 ← κ(i)n−1 ··· κ(i)1 . i∈I ! i∈I ! M M −1 ′ Furthermore, by the proof of 3.11 we know that Ψ1 (κi) has as representa- f g tive the exact sequence 0 → B → colim(fi) → i∈I Xi → 0. Hence, using the colimit universal property, is easy to see that there is a morphism with −1 ′ L fixed left end ηn ← Ψ1 (κi). Therefore, with the last morphisms we can build a morphism with fixed ends η ← κ. For showing the existence of a morphism with fixed ends κ → 0, it is enough to show that f is a splitting monomorphism, which follows straightforward from the colimit universal property together with the fact that every fi is a splitting monomorphism. n (c) Let(ηi) ∈ i∈I ExtC (Ai,B). We observe the following facts for every i ∈ I. i i Suppose ηiQ= κn ··· κ1 is a natural decomposition, where i i i i κk : 0 → Bk+1 → Ck → Bk → 0 ∀k ∈{n, ··· , 1}. i i i Consider the coproduct canonical inclusions uk : Bk → i∈I Bk. Observe i i i i i that u1 = µi ∀i ∈ I. By 2.20 we can see that i∈I κk uLk = uk+1κk for all k ∈{1, ··· ,n +1} . Hence, by 3.11(c), for every i ∈ I the extension defined L  14 A. ARGUD´IN MONROY

−1 i i i as η := Ψ1 (κn)i∈I i∈I κ n−1 ··· i∈I κ 1 , satisfies by recursion the following equalities L  L 

i −1 i i i i ηµi = ηu1 =Ψ1 (κn)i∈I κ n−1 ··· κ 1 u1 i∈I ! i∈I ! M M

−1 i i i i =Ψ1 (κn)i∈I κ n−1 ··· κ 2 u2κ 1 i∈I ! i∈I ! M M

−1 i i i i i =Ψ1 (κn)i∈I κ n−1 ··· κ 3 u3κ2κ 1 i∈I ! i∈I ! M M . .

−1 i i i i =Ψ1 (κn)i∈I κ n−1 un−1κn−2 ··· κ1 i∈I ! M −1 i i i =Ψ1 (κn)i∈I unκn−1 ··· κ1 i i i = κnκn−1 ··· κ1 = ηi.



By duality we have the following result.

Theorem 3.13. Let C be an Ab4* category, n ≥ 1, and {Ai}i∈I be a set of ob- jects in C. Consider the product πi : i∈I Ai → Ai i∈I . Then, the correspondence Φ : Extn B, A → Extn (B, A ), E 7→ (π E) , is an isomorphism of n C i∈I i i∈I C Q i  i i∈I abelian groups for every B ∈C. Q  Q

We will end this section introducing an application related to the developed in recent years. Namely, R. Colpi and K. R. Fuller developed a theory of tilting objects of projective dimension ≤ 1 for abelian categories in [8], and P. Coupekˇ and J. St’ov´ıˇcekˇ developed a theory of cotilting objects of injective dimension ≤ 1 for Grothendieck categories in [9]. A fundamental result needed in these theories is that

1 1 ExtA Ai,X = 0 if and only if ExtA(Ai,X)=0 ∀i ∈ I. i∈I ! M Such result is proved showing that, in any Ab3 abelian category A, there is an 1 1 injective correspondence ExtA i∈I Ai,X → i∈I ExtA(Ai,X) (see [8, Proposi- tion 8.1, Proposition 8.2] and [9, Proposition A.1] or the proof of 3.11). Now, for extending the theory to tiltingL objects of projective Q dimension ≤ n, it is needed a similar result for Extn. But, it is not known in general if there is an injective n n correspondence ExtA i∈I Ai,X → i∈I ExtA(Ai,X). L  Q YONEDA EXT 15

The following result follows from 3.13 and 3.12. It is worth to mention that it extends [8, Corollary 8.3] and the dual of [9, Corollary A.2] when the category is Ab4.

Corollary 3.14. Let C be an abelian category, n ≥ 1, {Ai}i∈I be a set of objects in C, and B ∈C. Then, the following statements hold true:

n n (a) If C is Ab4, then ExtC i∈I Ai,B =0 if and only if ExtC (Ai,B)=0 ∀i ∈ I. Ln  n (b) If C is Ab4*, then ExtC B, i∈I Ai = 0 if and only if ExtC (B, Ai) = 0 ∀i ∈ I. Q 

4. A characterization of Ab4

This section is inspired by the comments made by Sergio Estrada during the Colo- quio Latinoamericano de Algebra´ XXIII. The goal is to prove that if the correspon- 1 1 dence Ψ : ExtC ( Ai,B) → i∈I ExtC (Ai,B) defined above is always biyective for an Ab3 category C, then C is Ab4. L Q Throughout this section for every natural number n > 0 we will consider the cor- n n respondence Ψn : ExtC ( Xi, Y ) → i∈I ExtC (Xi, Y ) defined above.

In 3.10, it was proved that,L if C is Ab4,Q then given a set of exact sequences

fi ηi : 0 → B → Ai → Ci → 0 , i∈I n o f it can be built an exact sequence 0 → B → colim(fi) → i∈I Ci → 0, where f is part of the co-compatible family of morphisms associated to colim fi. In case C is only an Ab3 category, then by doing a similar constructionL we get an exact sequence B → colim fi → i∈I Ci → 0. Indeed, consider the direct system of exact sequences L 1 0 B B 0 0 1 fi fi 0 B Ai Ci 0 f g Then, we have an exact sequence B → colim fi → i∈I Ci → 0, where f and g are induced by the colimit universal property (see [18, page 55]). Such exact sequence L we shall name it Θ(ηi).

As a first step we will show that, even if the category is not Ab4, if the correspon- dence Ψ is biyective, then the inverse correspondence is given by Θ. That is, if 1 1 Ψ : ExtC ( Ai,B) → i∈I ExtC (Ai,B) is biyective, then for every set of exact se- fi quences ηi : 0 → B → Ai → Ci → 0 , then the morphism f in Φ(ηi) is monic, L Q i∈I and ΨΘ(nηi) = 1. o 16 A. ARGUD´IN MONROY

Lemma 4.1. Let C be an AB3 category, {Ai}i∈I be a set of objects in C, and B ∈C.

Consider the coproduct i∈I Ai with the canonical inclusions µi : Ai → i∈I Ai i∈I , fi gi and a set of exact sequencesL ηi : 0 → B → Ei → Ai → 0 . If there isL an exact i∈I f ′ ng′ o sequence η : 0 → B → E → i∈I Ai → 0 such that ηµi = ηi ∀i ∈ I, then the morphism f in the exact sequence L f g Θ(ηi): B → colim fi → Ci → 0 i∈I M is a monomorphism and η = Θ(ηi).

Proof. Consider the direct system {fi : B → Ei}i∈I . We know that ηµi = ηi ∀i ∈ I. Hence, for every i ∈ I there is a morphism of exact sequences

(1,νi,µi): ηi → η. E A 0 B i g i 0 Observe that the set of morphisms fi i νi µi {νi : Ei → E}i∈I , together with the mor- ′ ′ ′ f g phism f : B → E, is a co-compatible 0 B E Ai 0 family of morphisms. Therefore, there is a unique morphism ω : colim fi → E such thatLωσi = νi for ′ every i in I and ωf = f , where {σi : Ei → colim fi}i∈I ∪{f : B → colim fi} is the co-compatible family associated to the colimit. Notice that ωf = f ′ is a 0 B Ei Ai 0 fi gi monomorphism, so f is also a monomor- σi µi phism. f g It remains to prove that η = Θ(ηi). Ob- B colim(fi) Ai 0 serve that, by the cokernel universal prop- L erty, we can build a morphism of exact sequences ′ (1,ω,ω ):Θ(ηi) → η. 0 B colim(fi) Ai 0 It is enough to show that ω′ = 1. With f g ′ that goal, we see that ω L ω f ′ g′ ′ ′ ′ ′ ω µigi = ω gσi = g ωσi = g νi = µigi. 0 B E Ai 0 L ′ Hence, by the fact that gi is an epimorphism, ω µi = µi ∀i ∈ I. Then, by the universal coproduct property we can conclude that ω′ = 1.



Corollary 4.2. Let C be an AB3 abelian category, {Ai}i∈I be a set of objects in

C, and B ∈ C. Consider the coproduct µi : Ai → i∈I Ai i∈I , and the corre- spondence Ψ : Ext1 ( A ,B) → Ext1 (A ,B). If Ψ is biyective, then the 1 C i i∈I  C i L 1 inverse correspondence maps each (η ) ∈ Ext1 (A ,B), with representatives L Qi i∈I C i

fi Q ηi : 0 → B → Ei → Ai → 0 ∀i ∈ I, YONEDA EXT 17 to the extension given by the exact sequence

0 → B → colim(fi) → Ai → 0. i∈I M Theorem 4.3. Let C be an Ab3 category. Then, C is an Ab4 category if, and only 1 1 if, the correspondence Ψ1 : ExtC ( Xi, Y ) → i∈I ExtC (Xi, Y ) is biyective for every Y ∈C and every set of objects {Xi} . L i∈I Q

Proof. By 3.11, it is enough to prove that if Ψ is biyective for every Y ∈ C and every set of objects {Xi}i∈I , then C is Ab4. With this purpose, we will consider a αi βi set of exact sequences ηi : 0 → Ai → Bi → Ci → 0 and prove that the mor- i∈I phism i∈I αi : i∈I Ani → i∈I Bi is a monomorphism.o Consider the coproduct i∈I Ai and the canonic inclusions µi : Ai → i∈I Ai. By theL dual resultL of 2.7, forL every i in L L I we have an exact sequence morphism αi βi ′ 0 Ai Bi Ci 0 (µi,µi, 1) : ηi → µiηi, where

f µi ′ i µi µiηi : 0 → Ai → Ei → Ci → 0. fi i∈I M 0 i∈I Ai Ei Ci 0 Consider the correspondence L

1 1 Ψ : ExtC Ci, Ai → ExtC Ci, Ai . i∈I i∈I ! i∈I i∈I ! M M Y M −1 1 By 4.2, we know that Ψ maps (µiηi) ∈ i∈I ExtC Ci, i∈I Ai to the extension given by the exact sequence Q L  f 0 → Ai → colim(fi) → Ci → 0. i∈I i∈I M M

We will show that colim fi = i∈I Bi to conclude αi that i∈I αi is a monomorphism. Indeed, consider a A B L i i family of morphisms {gi : Bi → X} . By the uni- L i∈I ′ versal property of the coproduct Ai, there is µi µi i∈I gi a unique morphism α : Ai → X such that i∈I L Ai Ei giαi = αµi∀i ∈ I. Now, by the universal property fi γi L L α of the pushout on the last equality, for every i ∈ I X there is a unique morphism γi : Ei → X such that ′ gi = γiµi and α = γifi.

Before going further, consider the co-compatible family of morphisms associated to the colimit {uk : Ek → colim fi}k∈I . Observe that, by the universal property of the colimit on the last equalities, there is a unique morphism Λ : colim fi → X such that Λui = γi ∀i ∈ I and Λf = α. 18 A. ARGUD´IN MONROY

In particular, if X = i∈I Bi and g : B → B is the set of canonic α i i i∈I i i∈I L i inclusions, there is a unique morphism Ai Bi  L Λ : colim fi → i∈I Bi such that ′ µi µi Λui = γi ∀i ∈ I and Λf = α. Further- L more, by the universal property of the co- Ai Ei fi product i∈I Bi, there is a unique mor- gi ′ L phism Λ : i∈I Bi → colim fi such that ui ′ L′ f Λ gi = uiµi ∀i ∈ I. colim fi L γi ′ We shall now prove that Λ is an isomorphism α uiµi and Λ′ =Λ−1. Observe that Λ ΛΛ′g =Λu µ′ = γ µ′ = g ∀i ∈ I. i i i i i i X Hence, by the universal property of the ′ coproduct i∈I B, we can conclude that Λ ΛΛ′ = 1. Next, we proveL that Λ′Λ = 1. Observe that colim fi ′ uiµiαi = uifiµi = fµi ∀i ∈ I, and also that ′ ′ ′ ′ uiµiαi =Λ giαi =Λ αµi = (Λ Λf) µi ∀i ∈ I.

Hence, by the last equalities and the universal property of the coproduct i∈I Ai, we can conclude that f =Λ′Λf. Furthermore, observe that L ′ ′ (ui)µi = uiµi and (ui) fi = f ∀i ∈ I, and also that ′ ′ ′ ′ ′ ′ ′ ′ (Λ Λui)µi =Λ γiµi =Λ gi = uiµi and (Λ Λui) fi =Λ Λf = f ∀i ∈ I. ′ Hence, by the last equalities and the universal property of the pushout (Ei,fi,µi), ′ we can conclude that Λ Λui = ui. Now, it follows from the universal property of the colimit that Λ′Λ = 1. Therefore, Λ is an isomorphism and Λ′ =Λ−1.

′ By the last remark, without loss of generality colim fi = i∈I Bi,Λ=1=Λ , and ′ gi = uiµ ∀i ∈ I. Now, observe that i L fµi = giαi ∀i ∈ I.

Hence, by the universal property of the corpoduct i∈I Ai, we can conclude that f = αi. Therefore, αi is a monomorphism.  i∈I i∈I L L L We have the following equivalences. Theorem 4.4. Let C be an Ab3 category. Then, the following statements are equivalent:

(a) C is an Ab4 category. 1 1 (b) The correspondence Ψ : ExtC i∈I Xi, Y → i∈I ExtC (Xi, Y ) is biyec- tive for every Y ∈C and every set of objects {Xi} . L  Q i∈I YONEDA EXT 19

n n (c) The correspondence Ψn : ExtC i∈I Xi, Y → i∈I ExtC (Xi, Y ) is biyec- tive ∀Y ∈C, every set of objects {Xi} , and ∀n> 0. L i∈I  Q

Proof. It follows from 3.12 and 4.3. 

Remark 4.5. For an example of an Ab3 category that is not Ab4, see the dual category of [9, Example A.4].

By duality we have the following result. Theorem 4.6. Let C be an Ab3* category. Then, the following statements are equivalent:

(a) C is an Ab4* category. 1 1 (b) The correspondence Ψ : ExtC Y, i∈I Xi → i∈I ExtC (Y,Xi) is biyective for every Y ∈C and every set of objects {Xi}i∈I . n Q  Q n (c) The correspondence Ψn : ExtC Y, i∈I Xi → i∈I ExtC (Y,Xi) is biyec- tive ∀Y ∈C, every set of objects {Xi} , and ∀n> 0. Q i∈I  Q

Acknowledgements

I wish to thank my advisor Octavio Mendoza for encouraging me to publish this work and for proof reading the article. I am also grateful to Sergio Estrada whose comments greatly improved the quality of this paper.

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Instituto de Matematicas,´ Universidad Nacional Autonoma´ de Mexico,´ Circuito Exte- rior, Ciudad Universitaria, C.P.04510 Mexico City, Mexico.

E-mail address: [email protected]