On the lattices of exact and weakly exact structures
Rose-Line Baillargeon, Thomas Brüstle, Mikhail Gorsky, Souheila Hassoun
OON THE LATTICE OF WEAKLY EXACT STRUCTURES
ROSE-LINE BAILLARGEON, THOMAS BR ¨USTLE, MIKHAIL GORSKY,AND SOUHEILA HASSOUN
Abstract.
The study of exact structures on an additive category A is closelyrelated to the study of closed additive sub-bifunctors of the maximal extensionbifunctor Ext on A . We initiate in this article the study of “weakly exactstructures”, which are the structures on A corresponding to all additive sub-bifunctors of Ext . We introduce weak counterparts of one-sided exact structuresand show that a left and a right weakly exact structure generate a weakly exactstructure. We further define weakly extriangulated structures on an additivecategory and characterize weakly exact structures among them.We investigate when these structures on A form lattices. We prove thatthe lattice of substructures of a weakly extriangulated structure is isomorphicto the lattice of topologizing subcategories of a certain functor category. Inthe idempotent complete case, this characterises the lattice of all weakly exactstructures. We study in detail the situation when A is additively finite, givinga module-theoretic characterization of closed sub-bifunctors of Ext among alladditive sub-bifunctors. Contents
1. Introduction 22. Acknowledgements 53. Weakly exact and exact structures 63.1. Definitions 63.2. Example 73.3. Weakly exact structures 83.4. The left and right weakly exact structures 103.5. The maximal weakly exact structure 144. Sub-bifunctors and closed sub-bifunctors of Ext A to weakly exact structures 184.3. Example 204.4. Weakly exact structures as bimodules 215. Weakly extriangulated structures 226. Defects and topologizing subcategories 267. Lattice structures 297.1. Definitions 29 a r X i v : . [ m a t h . C T ] O c t N THE LATTICE OF WEAKLY EXACT STRUCTURES 2 A Introduction
Exact structures go back to the work of Yoneda and early versions of exactstructures in [Buch59, BuHo61] originated from studies of relative homologicalalgebra in abelian categories, like studying resolutions with a different set of objectsthan the projectives. In these papers, a mix of structures has been considered,on one hand classes of morphisms satisfying certain properties (“h.f.class”), onthe other hand certain (“closed”) subfunctors of Ext . The authors consideredalso a weaker notion, an f.class , which omits the condition of being closed undercomposition of admissible monics and epics.The “stand alone” concept of an exact structure as a class of short exact se-quences in an additive category A satisfying certain axioms has been laid out byQuillen in [Qu73], however requiring that A be embedded into an abelian category.The independent version of these axioms was formulated by Keller in [Ke90], seealso [GR92]. It allows to develop methods from homological algebra, and definederived categories, see [Ne90, Ke91]. Note that there exist different independantnotions of “exact categories”, like the “Barr-exact categories” or “effective regularcategories”, to not be confused with the one we consider in our work.The comparison to subfunctors of Ext has been re-considered in [AS] and thenin [DRSS], with applications to exact structures originating from one-point ex-tensions, a special case of exact structures associated with bimodule problems in[BrHi]. However, the lack of a unique maximum extension-functor for arbitraryadditive categories was a limiting factor in these studies. If A has kernels andcokernels, the existence of the maximal exact structure was first proved by Siegand Wegner [SW11]. Crivei [Cr11] extended the result to additive categories forwhich every split epimorphism has a kernel, but finally Rump [Ru11] showed thatany additive category admits a unique maximal exact structure E max .In [BHLR] a study of the family of all exact structures Ex ( A ) on an additivecategory A was initiated. The existence of a unique maximum exact structureallows to turn Ex ( A ) into a complete bounded lattice. On the side of bifunctors,this amounts to studying all closed sub-bifunctors of a unique maximum bifunctor E max which corresponds to the exact structure E max . It is natural, on the bifunctor N THE LATTICE OF WEAKLY EXACT STRUCTURES 3 side, to extend the study to all additive sub-bifunctors, which in turn raises thequestion to which structure of exact sequences they correspond. We study in thispaper the corresponding systems of short exact sequences, which we call weaklyexact structures , reminiscent of the f.classes studied in [Buch59]. We show that allthe weakly exact structures on A form a lattice. When the underlying category A is additively finite, this lattice is a finite length modular lattice, a class oflattices studied recently by Haiden, Katzarkov, Kontsevich and Pandit in [HKKP]in connection with weight filtrations and the notion of semi-stability.In order to find a general way to give proofs of various statements concerningexact and triangulated categories that would work for both of these classes of cat-egories (or, rather, classes of structures on additive categories), Nakaoka and Palu[NP19] studied additive bifunctors E : A op × A → Ab equipped with certain extradata called a realization . They found a set of axioms on triples consisting of anadditive category, a bifunctor and a realization that unifies the axioms of exactand of triangulated categories. They called such structures extriangulated . Exten-sions in exact categories are realized by “admissible” kernel-cokernel pairs. In anextriangulated category this role is played by pairs of composable morphisms f, g where f is a weak kernel of g and g is a weak cokernel of f . Moreover, Nakaokaand Palu characterized all triples that define exact structures, in other words,closed additive sub-bifunctors of E max . Hershend, Liu and Nakaoka [HLN] intro-duced n − exangulated structures and proved that the choice of a 1 − exangulatedstructure on an additive category is equivalent to the choice of an extriangulatedstructure. The set of axioms of 1 − exangulated structures is slightly different fromthat of extriangulated categories. In Section 5, we consider 1 − exangulated cate-gories with one of the axioms removed. We prove that such weakly − exangulated ,or weakly extriangulated structures naturally generalize weakly exact structures wedefined earlier.For a finite-dimensional algebra Λ , Buan [Bu01] studied closed sub-bifunctorsof the bifunctor Ext on the category mod Λ. He proved that they correspond tocertain Serre subcategories of the category of finitely presented additive functors( mod Λ) op → Ab (i.e. of finitely presented modules over mod
Λ), defined ascategories of contravariant defects in works of Auslander [A66, A78]. This resultwas later extended to exact structures on additive categories in [En18], see also[FG20].We note that the definition of contravariant defects naturally extends to thesetting of weakly exact structures. Ogawa [Og19] defined contravariant defects inthe setting of extriangulated categories, and we further extend this notion to theframework of weakly extriangulated categories. By adapting arguments of Ogawaand Enomoto [En20], we prove that the category of defects of a weakly extriangu-lated structure on an additive category A is topologizing (in the sense of Rosenberg[Ros]) in the category coh ( A ) of coherent right A− modules. That means that itis closed under subquotients and finite coproducts. For coherent A− modules, we N THE LATTICE OF WEAKLY EXACT STRUCTURES 4 have a natural notion of subobjects and of quotients: these are defined object-wise(for objects in A ). This allows us to define topologizing subcategories of an arbi-trary (not necessarily abelian) full subcategory C of coh ( A ) as full subcategoriesof C which are topologizing in coh ( A ) . Given a weakly extriangulated structure, all its substructures are uniquely char-acterized by their categories of defects, and each topologizing subcategory of agiven category of defects defines a weakly extriangulated substructure. Weaklyextriangulated substructures of a weakly exact structure are necessarily weaklyexact. Thus, whenever we know that an additive category A admits a uniquemaximal weakly exact structure, we can classify all weakly exact structures on A in terms of topologizing structures in a certain abelian category.Topologizing subcategories of an abelian category form a lattice. Topologizingsubcategories of the (not necessarily abelian) category of defects of a weakly ex-triangulated structure on A also form a lattice, which is an interval in the latticeof all topologizing subcategories of coh ( A ). Note that Serre subcategories forma subposet, but not a sublattice of this lattice. Weakly extriangulated substruc-trures of a weakly extriangulated structure also form a natural lattice, extendingthe lattice of weakly exact structues. We establish lattice isomorphisms betweenthese several lattices, summarized in the following figure:The latticeof weaklyexactstructures W on A The latticeof additivesub-bifunctorsof E max The latticeof sub-bimodulesover theAuslanderalgebraThe latticeof topolo-gizingsubcate-gories of def E max Figure 1: Isomorphisms of latticesWe introduce Section 3 the class
Wex ( A ) of all weakly exact structures on anadditive category A . It turns out that, despite the fact that weakly exact structuresare not closed under compositions, some of the properties of exact structures are N THE LATTICE OF WEAKLY EXACT STRUCTURES 5 still valid, in particular, every weakly exact structure satisfies Quillen’s obscureaxiom, see Proposition 3.8. Similar to exact structures, it is sometimes beneficialto dissect the set of axioms into two parts, leading to the notion of left/right weaklyexact structure. We show that any pair of left and right weakly exact structuregives rise to a weakly exact structure, and that all such structures arise in thatway.The existence of a unique maximal weakly exact structure for any additive cat-egory is an open question. Since it is not known in general if there exist weaklyexact structures larger than the maximal exact structure ( E max ), we study in thispaper mainly the weakly exact structures that are included in E max , so we con-sider the interval Wex ( E max ) := [ E min , E max ] ⊆ Wex ( A ). Given a weakly exactstructure W on A , constructing the the group W of W− extensions yields a mapΦ to category of bifunctors from A to abelian groups:Φ : Wex ( A ) −→ BiFun ( A ) W (cid:55)−→ W = Ext W ( − , − ) . This function Φ induces lattice isomorphisms
Wex ( E max ) ←→ BiFun ( E max ) ∪ ∪ Ex ( A ) ←→ CBiFun ( A )where CBiFun ( A ) denotes the subclass of closed sub-bifunctors of E max . Notethat Ex ( A ) is not a sublattice of Wex ( E max ), even if it is a subposet: the joinoperation we consider on both sets is different, as we illustrate by an example inSection 4.3.When the underlying category A is additively finite and Krull-Schmidt, it isknown that the lattice Ex ( A ) is boolean, with each object E ( S ) determined bythe choice of a set S of Auslander-Reiten sequences. The larger lattice Wex ( A )however is not boolean, and it is interesting to characterise the members of Ex ( A )in module-theoretic terms, that is, describe the closed sub-bimodules of E max . Weshow that, when viewed as bimodules over the Auslander algebra of A , elementsin Ex ( A ) can be characterized as follows: For every set S of Auslander-Reitensequences, the closed bimodule E ( S ) of E max introduced above is the maximalsubmodule of E max whose socle is S .2. Acknowledgements
The authors would like to thank Shiping Liu and Hiroyuki Nakaoka for helpfuldiscussions contributing to this version of the work.Most of this work was done while the first author was supported by anNSERC USRA grant. The second author was supported by Bishop’s Universityand NSERC of Canada, and the fourth author acknowledges support from the
N THE LATTICE OF WEAKLY EXACT STRUCTURES 6 ”th´esards ´etoiles” scholarship of ISM for outstanding PhD candidates. This workwas completed during the third author’s participation at the Junior TrimesterProgram ”New Trends in Representation Theory” at the Hausdorff Institute forMathematics in Bonn. He is very grateful to the Institute for the perfect workingconditions. 3.
Weakly exact and exact structures
We introduce in this section the central topic of this paper, exact structureson additive categories. Then, we introduce weakly exact structures, which are ageneralization of exact structures, and study some of their properties.3.1.
Definitions.
We recall the definition of an exact structure on an additivecategory given by Quillen in [Qu73], using the terminology of [Ke90], see also[GR92]. We refer to [B¨u10] for an exhaustive introduction to exact categories.We fix an additive category A throughout this section. Many of the earlyversions of exact structures were formulated in the context of an abelian category A , using all the short exact sequences on it. For a general additive category A , the notion of short exact sequence is specified to be a kernel-cokernel pair( i, d ), that is, a pair of composable morphims such that i is kernel of d and d iscokernel of i . An exact structure on A is then given by a class E of kernel-cokernelpairs on A satisfying certain axioms which we recall below. We call admissiblemonic a morphism i for which there exists a morphism d such that ( i, d ) ∈ E .An admissible epic is defined dually. Note that admissible monics and admissibleepics are referred to as inflation and deflation in [GR92], respectively. We depictan admissible monic by (cid:47) (cid:47) (cid:47) (cid:47) and an admissible epic by (cid:47) (cid:47) (cid:47) (cid:47) . The pair( i, d ) ∈ E is referred to as admissible short exact sequence , or short exact sequencein E . Definition 3.1. An exact structure E on A is a class of kernel-cokernel pairs ( i, d )in A which is closed under isomorphisms and satisfies the following axioms:(E0) For all objects A in A the identity 1 A is an admissible monic;(E0) op For all objects A in A the identity 1 A is an admissible epic;(E1) The class of admissible monics is closed under composition(E1) op The class of admissible epics is closed under composition;(E2) The push-out of an admissible monic i : A (cid:47) (cid:47) (cid:47) (cid:47) B along an arbitrarymorphism t : A → C exists and yields an admissible monic s C : A t (cid:15) (cid:15) (cid:47) (cid:47) i (cid:47) (cid:47) PO B s B (cid:15) (cid:15) C (cid:47) (cid:47) s C (cid:47) (cid:47) S. N THE LATTICE OF WEAKLY EXACT STRUCTURES 7 (E2) op The pull-back of an admissible epic h along an arbitrary morphism t existsand yields an admissible epic p B P P A (cid:15) (cid:15) p B (cid:47) (cid:47) (cid:47) (cid:47) PB B t (cid:15) (cid:15) A h (cid:47) (cid:47) (cid:47) (cid:47) C. An exact category is a pair ( A , E ) consisting of an additive category A and anexact structure E on A . Note that E is an exact structure on A if and only if E op is an exact structure on A op .We denote by ( Ex ( A ) , ⊆ ) the poset of exact structures E on A , where the partialorder is given by containment E (cid:48) ⊆ E . Note that Ex ( A ) need not actually forma set, but by abuse of language, we still use the term poset when Ex ( A ) is aclass. The poset ( Ex ( A ) , ⊆ ) always contains a unique minimal element, the splitexact structure E min which is formed by all split exact sequences, that is, sequencesisomorphic to A (cid:47) (cid:47) (cid:34) (cid:35) (cid:47) (cid:47) A ⊕ B [0 1] (cid:47) (cid:47) (cid:47) (cid:47) B (see [B¨u10, Lemma 2.7]).Moreover, every additive category admits a unique maximal exact structure E max , see [Ru11, Corollary 2]. When the category A is abelian, then E max isformed by all short exact sequences in A . The construction is more subtle forother classes of additive categories, we refer to [BHLR, Section 2.4] for a moredetailed discussion.3.2. Example.
Consider the category A = rep Q of representations of the quiver Q : 1 (cid:47) (cid:47) (cid:111) (cid:111) Then the Hasse diagram of the poset of exact structures Ex ( A ) has the shape ofa cube (see [BHLR, Example 4.2] for detailed description of the different exactstructures on A ): E min E E , , E E E , E max E , , N THE LATTICE OF WEAKLY EXACT STRUCTURES 8
Let us mention that by taking other forms of the quiver of type A such as Q : 1 2 (cid:111) (cid:111) (cid:47) (cid:47) Q : 1 (cid:47) (cid:47) (cid:47) (cid:47) Ex ( A ) is a Boolean lattice in these cases,with n Auslander-Reiten sequences in A giving rise to exactly 2 n exact structuresand poset structure isomorphic to the power set of the set of Auslander-Reitensequences in A , see [En18].3.3. Weakly exact structures.Definition 3.2.
Let A be an additive category. We define a weakly exact structure W on A as a class of kernel-cokernel pairs ( i, d ) in A which is closed under isomor-phisms and direct sums, and satisfies the axioms ( E E op ,( E
2) and ( E op ofDefinition 3.1.We denote by ( Wex ( A ) , ⊆ ) the poset of all weakly exact structures on A , or-dered by containment. Lemma 3.3. Ex ( A ) is a subclass of Wex ( A ). Proof.
Only the direct sum condition needs to be verified. But this is alwayssatisfied for exact structures, by [B¨u10, Proposition 2.9]. (cid:3)
Remark 3.4.
The proof of [B¨u10, Proposition 2.9] makes heavy use of axioms( E
1) and ( E op , this makes us think that the property of being closed under directsums does not follow from the remaining axioms for weakly exact structures.We now state some of the properties for exact structures that also hold forweakly exact structures: Lemma 3.5.
Let W be a weakly exact structure and let i and i (cid:48) be admissiblemonics of W forming the rows of a commutative square: A (cid:47) (cid:47) i (cid:47) (cid:47) f (cid:15) (cid:15) B f (cid:48) (cid:15) (cid:15) A (cid:48) (cid:47) (cid:47) i (cid:48) (cid:47) (cid:47) B (cid:48) Then the following statements are equivalent:(i) The square is a push-out.(ii) A (cid:47) (cid:47) (cid:34) i − f (cid:35) (cid:47) (cid:47) B ⊕ A (cid:48) [ f (cid:48) i (cid:48) ] (cid:47) (cid:47) (cid:47) (cid:47) B (cid:48) is a short exact sequence belonging to W .(iii) The square is both a push-out and a pull-back. N THE LATTICE OF WEAKLY EXACT STRUCTURES 9 (iv) There exists a commutative commutative diagram with rows being conflationsin W : A (cid:47) (cid:47) i (cid:47) (cid:47) f (cid:15) (cid:15) B f (cid:48) (cid:15) (cid:15) p (cid:47) (cid:47) (cid:47) (cid:47) C C (cid:15) (cid:15) A (cid:48) (cid:47) (cid:47) i (cid:48) (cid:47) (cid:47) B (cid:48) p (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) C Proof.
One can easily verify that the proof of the statement for exact categoriesin [B¨u10, Proposition 2.12] does not use axioms ( E
1) or ( E op when it is done inthe order ( i ) ⇒ ( iv ) ⇒ ( ii ) ⇒ ( iii ) ⇒ ( i ). (cid:3) Remark 3.6.
The dual of Lemma 3.5 is also true. For example, the dual of (i)implies (iv) would be: If d and d’ are admissible epics of W and (g, d) is the push-out of (d’, g’) then the following diagram exists, is commutative and has rows in W : A (cid:47) (cid:47) j (cid:47) (cid:47) A (cid:15) (cid:15) B g (cid:15) (cid:15) d (cid:47) (cid:47) (cid:47) (cid:47) C g (cid:48) (cid:15) (cid:15) A (cid:47) (cid:47) j (cid:48) (cid:47) (cid:47) B (cid:48) d (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) C (cid:48) Commutative squares that are both a pushout and a pullback are called bicarte-sian squares.
Lemma 3.7.
Let W be a weakly exact structure on A . For any morphism ofadmissible short exact sequences A (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) B (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) C (cid:15) (cid:15) A (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) B (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) C (cid:48) in W , there exists a commutative diagram A (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) B (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) CA (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) E (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) C (cid:15) (cid:15) A (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) B (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) C (cid:48) , where the middle row is also an admissible short exact sequence in W and the topleft and bottom right squares are bicartesian. N THE LATTICE OF WEAKLY EXACT STRUCTURES 10
Proof.
The same proof as in [B¨u10, Lemma 3.1] applies here. (cid:3)
Proposition 3.8. (Quillen’s obscure axiom for weakly exact structures)
Let W be a weakly exact structure on an additive category A .(1) Consider morphisms A i (cid:47) (cid:47) B j (cid:47) (cid:47) C in A , where i has a cokernel and ji is an admissible monic of W . Then i is also an admissible monic of W .(2) Consider morphisms X f (cid:47) (cid:47) Y g (cid:47) (cid:47) Z in A , where g has a kernel and gf is an admissible epic of W . Then g is also an admissible epic of W . Proof. (1) The proof given in [B¨u10, Proposition 2.16] also holds for weakly exactcategories: Lemma 3.5 is the equivalent of [B¨u10, Proposition 2.12]. One stepin the proof of [B¨u10, Proposition 2.16] is using axiom ( E W is closed under isomorphisms.(2) The proof is done dually. (cid:3) Lemma 3.9.
The split exact structure E min forms the unique minimal element ofthe poset ( Wex ( A ) , ⊆ ). Proof.
The proof of [B¨u10, Lemma 2.7] does not use axioms ( E
1) and ( E op , sothe statement of [BHLR, Prop 2.12] applies to weakly exact structures as well. (cid:3) The left and right weakly exact structures.
In this subsection, we de-fine left weakly exact structures and right weakly exact structures . We show thattheir combination gives a weakly exact structure and also that every weakly exactstructure can be obtained in this way.These definitions generalise the left and right exact structures introduced in [BC12,Definition 3.1] and studied in [HR20]. Rump is using left and right exact struc-tures in [Ru11] to prove the existence of a unique maximal exact structure on anyadditive category. Unfortunately, his method does not apply to the case of weaklyexact structures, and it is an open question if a unique maximal weakly exactstructure exists for any additive category.
Definition 3.10.
A right weakly exact structure on A is a class of kernels I which is closed under isomorphisms and satisfies the following properties:(Id) For all objects X in A the identity 1 X and the zero monomorphism0 −→ X are in I . N THE LATTICE OF WEAKLY EXACT STRUCTURES 11 (P) The push-out of f : X −→ Y ∈ I along an arbitrary morphism h : X −→ X (cid:48) exists and yields a morphism f (cid:48) ∈ I : X h (cid:15) (cid:15) f (cid:47) (cid:47) PO Y h (cid:48) (cid:15) (cid:15) X (cid:48) f (cid:48) (cid:47) (cid:47) Y (cid:48) (Q) Given A a (cid:47) (cid:47) B b (cid:47) (cid:47) C with ba ∈ I and a has a cokernel, then a is in I .(S) I is closed under direct sums of morphisms. Definition 3.11.
A left weakly exact structure on A is a class of cokernels D which is closed under isomorphisms and satisfies the following properties:(Id op ) For all objects X in A the identity 1 X and the zero epimorphism X −→ D .(P op ) The pullback of f : C −→ F ∈ D along an arbitrary morphism h : E −→ F exists and yields a morphism e ∈ D : B e (cid:15) (cid:15) b (cid:47) (cid:47) PB C f (cid:15) (cid:15) E h (cid:47) (cid:47) F (Q op ) Given A a (cid:47) (cid:47) B b (cid:47) (cid:47) C with ba ∈ D and b has a kernel, then b is in D .(S op ) D is closed under direct sums of morphisms. Remark 3.12.
Note that, contrary to exact structures (see [B¨u10, Proposition2.9]) the properties (S) and (S) op above are not implied by the rest of the propertiesand we need to add them. These properties are necessary to ensure that we get astructure which is equivalent to an additive sub-bifunctor of Ext as we show inSection 4. The reason behind this is that the Baer sum uses the direct sum of twoshort exact sequences in its construction. N THE LATTICE OF WEAKLY EXACT STRUCTURES 12
Theorem 3.13.
Let A be an additive category. A left weakly exact structure D on A can be combined with a right weakly exact structure I to form a weakly exactstructure W given by the short exact sequences A i (cid:47) (cid:47) B d (cid:47) (cid:47) C with i ∈ I and d ∈ D . Proof.
We adapt the proof of [Ru11, Theorem 1] to the case of weakly exactstructures. Denote by D I the class of morphisms in D that have a kernel in I , andlet I D be the class of morphims in I that have a cokernel in D . Thus W is givenby the short exact sequences A i (cid:47) (cid:47) B d (cid:47) (cid:47) C with i ∈ I D and d ∈ D I .(1) Suppose we have the following commutative diagram where the top row is in W , the bottom row is a short exact sequence and u, v, w are isomorphisms: E : A i (cid:47) (cid:47) u (cid:15) (cid:15) B v (cid:15) (cid:15) d (cid:47) (cid:47) C w (cid:15) (cid:15) E : A (cid:48) i (cid:48) (cid:47) (cid:47) B (cid:48) d (cid:48) (cid:47) (cid:47) C (cid:48) Since E ∈ W we have i ∈ I and d ∈ D . Since I is closed under isomor-phisms, we obtain i (cid:48) ∈ I , and dually, d (cid:48) ∈ D . Therefore, E ∈ W and W isclosed under isomorphisms.(2) Suppose that E and E are in W . It is well-known that the direct sum E ⊕ E : A ⊕ A (cid:48) i ⊕ i (cid:48) (cid:47) (cid:47) B ⊕ B (cid:48) d ⊕ d (cid:48) (cid:47) (cid:47) C ⊕ C (cid:48) is a short exact sequence.With E , E ∈ W we have i, i (cid:48) ∈ I and d, d (cid:48) ∈ D . Axioms (S) and (S) op imply i ⊕ i (cid:48) ∈ I and d ⊕ d (cid:48) ∈ D , so W is closed under direct sums.(3) For any object X in A , X (cid:47) (cid:47) X (cid:47) (cid:47) (cid:47) (cid:47) X (cid:47) (cid:47) X are shortexact sequences. By axiom (Id), X (cid:47) (cid:47) X ∈ I and 0 (cid:47) (cid:47) X ∈ I . Byaxiom (Id) op , X (cid:47) (cid:47) ∈ D and X (cid:47) (cid:47) X ∈ D . Therefore, the twosequences are in W which means that W satisfies (E0) and ( E op .(4) Let us show that W satisfies (E2). To this end, consider a short exactsequence A (cid:47) (cid:47) i (cid:47) (cid:47) B d (cid:47) (cid:47) (cid:47) (cid:47) C ∈ W and f : A −→ A (cid:48) ∈ A . By (P), thepushout of i and f exists and i (cid:48) belongs to I . A f (cid:15) (cid:15) (cid:47) (cid:47) i (cid:47) (cid:47) PO B g (cid:15) (cid:15) d (cid:47) (cid:47) (cid:47) (cid:47) CA (cid:48) i (cid:48) (cid:47) (cid:47) B (cid:48) Since di = 0 = 0 ◦ f , the push-out property gives us that there exists aunique d (cid:48) : B (cid:48) −→ C such that d (cid:48) i (cid:48) = 0 and d (cid:48) g = d . Therefore, we have N THE LATTICE OF WEAKLY EXACT STRUCTURES 13 the following commutative diagram. A (cid:47) (cid:47) i (cid:47) (cid:47) f (cid:15) (cid:15) B g (cid:15) (cid:15) d (cid:47) (cid:47) C C (cid:15) (cid:15) A (cid:48) i (cid:48) (cid:47) (cid:47) B (cid:48) d (cid:48) (cid:47) (cid:47) C We know i (cid:48) ∈ I and we want to show that i (cid:48) ∈ I D . First, we show that d (cid:48) isthe cokernel of i (cid:48) : Let x : B (cid:48) −→ X be such that xi (cid:48) = 0, so xgi = xi (cid:48) f = 0.By the cokernel property of d , there exists a unique h : C −→ X such that hd = xg , so ( xg ) i = hdi = 0 = 0 ◦ f . By the pushout property, there existsa unique w : B (cid:48) −→ X such that wi (cid:48) = 0 and wg = xg . Both x and hd (cid:48) qualify for the defining properties of w , so by uniqueness, we get x = hd (cid:48) . Therefore, d (cid:48) = coker( i (cid:48) ).Since ( i (cid:48) , d (cid:48) ) is a kernel-cokernel pair, we get i (cid:48) = ker( d (cid:48) ).As d (cid:48) g = d ∈ D and d (cid:48) has a kernel, we infer from property (Q) op for D that d (cid:48) ∈ D , hence i (cid:48) ∈ I D . Therefore, A (cid:48) (cid:47) (cid:47) i (cid:48) (cid:47) (cid:47) B (cid:48) d (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) C (cid:48) ∈ W .(5) Dually, W satisfies ( E op . (cid:3) Proposition 3.14.
Every weakly exact structure W on A can be constructed froma right weakly exact structure and a left weakly exact structure as in Theorem 3.13.More precisely, if I is the class of admissible monics of a weakly exact structure W and D is the class of admissible epics of W , then I is a right weakly exactstructure and D is a left weakly exact structure. Proof.
Let W be a weakly exact structure on an additive category A . Let I bethe class of admissible monics of W and D the class of admissible epics of W .(1) First, we show that I and D are closed under isomorphisms. Supposethat the following diagram is commutative, i ∈ I and that u and v areisomorphisms. A (cid:47) (cid:47) i (cid:47) (cid:47) u (cid:15) (cid:15) B v (cid:15) (cid:15) A (cid:48) i (cid:48) (cid:47) (cid:47) B (cid:48) Since i is an admissible monic of W , there exists d : B (cid:47) (cid:47) (cid:47) (cid:47) C ∈ D suchthat ( i, d ) ∈ W . We can show that i (cid:48) = ker ( dv − ) and dv − = cokernel ( i (cid:48) ).So we have the following isomorphism of short exact sequences. A (cid:47) (cid:47) i (cid:47) (cid:47) u (cid:15) (cid:15) B v (cid:15) (cid:15) d (cid:47) (cid:47) (cid:47) (cid:47) C C (cid:15) (cid:15) A (cid:48) i (cid:48) (cid:47) (cid:47) B (cid:48) dv − (cid:47) (cid:47) C N THE LATTICE OF WEAKLY EXACT STRUCTURES 14
Since, ( i, d ) ∈ W and W is closed under isomorphisms, then ( i (cid:48) , d (cid:48) ) ∈ W so i (cid:48) ∈ I . Therefore, I is closed under isomorphisms. Dually, D is closedunder isomorphisms.(2) Since W satisfies ( E
0) and ( E op , it is clear that I satisfies (Id) and D satisfies (Id) op .(3) By Proposition 3.8, I satisfies (Q) and D satisfies (Q) op .(4) Since W is closed under direct sums it is clear that I satisfies (S) and D satisfies (S) op . (cid:3) Remark 3.15.
In view of Proposition 3.14, we write a weakly exact structure W as W = ( I , D ) where I is the right weakly exact structure of kernels and D is theleft weakly exact structure of cokernels in W .3.5. The maximal weakly exact structure.
Rump’s construction of a maxi-mal exact structure in [Ru11] does not generalise to the weakly exact structures.In fact, it is an open question if a unique maximal weakly exact structure existsfor any additive category. However, under certain conditions on A we show thatthe maximal weakly exact structure coincides with the maximal exact structure:this is true under the same conditions as in Crivei’s characterisation of stable shortexact sequences forming the maximal exact structure. We recall the necessarydefinitions first: Definition 3.16. [RW77] A kernel (
A, f ) is in an additive category A is called semi-stable if for every push-out square A t (cid:15) (cid:15) f (cid:47) (cid:47) PO B s B (cid:15) (cid:15) C s C (cid:47) (cid:47) S the morphism s C is also a kernel. We define dually a semi-stable cokernel. A shortexact sequence A (cid:47) (cid:47) i (cid:47) (cid:47) B d (cid:47) (cid:47) (cid:47) (cid:47) C in A is said to be stable if i is a semi-stablekernel and d is a semi-stable cokernel. We denote by E sta the class of all stable short exact sequences.For any additive category A , the class E sta clearly satisfies conditions (E2) and(E2) op , but in general it need not satisfy (E1) or the direct sum property (S). Rumpprovides in [Ru15] an example showing that the class of semi-stable cokernels doesnot satisfy property (S), thus the semi-stable cokernels do not form a left weaklyexact structure. This does not imply that E sta is not always closed for direct sumof conflations. However, it is known by [Cr11, Theorem 3.5] that E sta forms anexact structure when A is weakly idempotent complete , that is, every section hasa cokernel, or equivalently, every retraction has a kernel. Moreover, in this case, N THE LATTICE OF WEAKLY EXACT STRUCTURES 15 the class of stable short exact sequences clearly forms the maximal weakly exactstructure since no larger class will satisfy (E2) and (E2) op . So we get that a uniquemaximum weakly exact structure exists and coincides with E max when A is weaklyidempotent complete: W max = E max = E sta Crivei goes further in [Cr12, Theorem 3.4], characterising maximum exact struc-tures using the idempotent (or Karoubian) completion H : A → ˆ A of the category A . Considering the maximal exact structure ˆ E max in ˆ A , he defines the notion of A being closed under pushouts and pullbacks for ( ˆ A , ˆ E max ) (see [Cr12, Theorem3.4] for details), and obtains the following result: Theorem 3.17. [Cr12, Theorem 3.4] Let A be an additive category, and let H : A → ˆ A be its idempotent completion. Then the class E sta of stable shortexact sequences of A defines an exact structure on A if and only if A is closedunder pushouts and pullbacks for ( ˆ A , ˆ E max ). In this case, E sta is the maximal exactstructure on A .Again, since the class of stable short exact sequences clearly forms the maximalclass satisfying (E2) and (E2) op , we get: Corollary 3.18.
Assume that A is closed under pushouts and pullbacks for( ˆ A , ˆ E max ). Then the class of stable short exact sequences forms the unique maximalweakly exact structure on A : W max = E max = E sta We refer to [Cr12, Corollary 3.5] for an example of an additive category A whichis not weakly idempotent complete, but satisfies that A is closed under pushoutsand pullbacks for ( ˆ A , ˆ E max ).4. Sub-bifunctors and closed sub-bifunctors of Ext We explore in this section the correspondence between exact structures andcertain subfunctors of Ext A .4.1. From weakly exact structures to bifunctors.
Let W be a weakly exactstructure on A . The aim of this section is to associate with W an additive functorto the category of abelian groups W = Ext W ( − , − ) : A op × A → Ab . In the following definition we review the classical construction for abelian cate-gories, stated in [M65], chapter VII, and formulate it in our context.
Definition 4.1.
Define for objects
A, C ∈ A the set W ( C, A ) = Ext W ( C, A ) = (cid:26) ( i, d ) | A i (cid:47) (cid:47) B d (cid:47) (cid:47) C ∈ W (cid:27) , N THE LATTICE OF WEAKLY EXACT STRUCTURES 16 where we denote by ( i, d ) the usual equivalence class of the short exact sequence( i, d ). To define the action of the functor W on morphisms, let E = ( i, d ) ∈ W be a short exact sequence from A to C , and a : A −→ A (cid:48) a morphism. Then wedefine the short exact sequence aE ∈ W to be obtained by taking the pushoutalong i and a (thus defining the image of E under the map W ( C, a ) : E : A i (cid:47) (cid:47) a (cid:15) (cid:15) B (cid:15) (cid:15) d (cid:47) (cid:47) C C (cid:15) (cid:15) aE : A (cid:48) i (cid:48) (cid:47) (cid:47) P O d (cid:48) (cid:47) (cid:47) C Dually, for a morphism c : C (cid:48) −→ C , the pullback Ec along d and c defines theimage of E under the map W ( c, A ) : Ec : A i (cid:48)(cid:48) (cid:47) (cid:47) A (cid:15) (cid:15) P B (cid:15) (cid:15) d (cid:48)(cid:48) (cid:47) (cid:47) C (cid:48) c (cid:15) (cid:15) E : A i (cid:47) (cid:47) B d (cid:47) (cid:47) C Moreover, we define on W ( C, A ) an addition (Baer’s sum) by E + E = ∇ A ( E ⊕ E ) ∆ C where ∇ A and ∆ C are the codiagonal and diagonal maps, and E ⊕ E is the directsum of E and E in W ( C ⊕ C, A ⊕ A ).Note that we use property (E2) (or (P)) from the definition of (right) exactstructure to define the product aE , dually we use (E2) op (or (P) op ) to define theproduct Ec . For the sum E + E , we employ (E2), (E2) op , (S) and (S) op .Given a left weakly exact structure D on A and objects A, C ∈ A , we define D A ( C ) = (cid:26) ( i, d ) | A i (cid:47) (cid:47) B d (cid:47) (cid:47) C is a short exact sequence with d ∈ D (cid:27) We also use the notation Ext D ( C, A ) = D A ( C ). Dually, we define Ext I ( C, A ) = I A ( C ) for a right weakly exact structure I . Lemma 4.2.
Let D be a left weakly exact structure on A . Then for each A ∈ A ,the construction in Definition 4.1 yields a functor D A = Ext D ( − , A ) : A op → Set.Dually, for every object C ∈ A , a right weakly exact structure I defines a functor I C = Ext I ( C, − ) : A →
Set.
Proof.
Adapt [M65, Chapter VII], Lemma 1.3 (i) and (ii) to our context. (cid:3)
Proposition 4.3.
Let W be weakly exact structure on A . Then the constructionin Definition 4.1 yields an additive bifunctor W = Ext W ( − , − ) : A op × A −→ Ab; (
C, A ) (cid:55)−→ W ( C, A ) . N THE LATTICE OF WEAKLY EXACT STRUCTURES 17
Proof.
This is a classical result going back to Yoneda and Baer when A is thecategory of modules over a ring and W = E max is the class of all short exactsequences, see [Mac63, Chapter III]. When A is abelian and W = E is an exactstructure on A , then E is what MacLane calls a proper class, and the result isgiven in Proposition 4.3 of [Mac63, Chapter XII]. For an exact structure W = E on a general additive category A , the result can be obtained using the embeddingof [GR92, Prop. 9.1] and the same techniques as used in [DRSS, Section 1.2].Finally, assume that W is a weakly exact structure on A , and write W = ( I , D )with I a right weakly exact structure and D a left weakly exact structure as inRemark 3.15. The fact that W is a bifunctor then follows from Lemma 4.2 and[M65, Chapter VII, Lemma 1.3 (iii)]. The proof that W is an additive functor tothe category of (big) abelian groups is shown as in [M65, Chapter VII, Theorem1.5], noting that none of the proofs there is using condition (E1) or that A isabelian, all that is needed are the axioms of a weakly exact structure. (cid:3) Lemma 4.4.
Let V and W be weakly exact structures on A with V ⊆ W . Then V = Ext V ( − , − ) : A op × A → Ab is an additive sub-bifunctor of W . Proof.
Since V is contained in W , the set V ( C, A ) is contained in the abeliangroup W ( C, A ). We show that it is a subgroup: By Lemma 3.9, the set V ( C, A )contains the zero element of W ( C, A ) which is given by the split exact sequence.Moreover, V ( C, A ) is closed under the addition in W ( C, A ). In fact, given shortexact sequences E and E in V ( C, A ), we have that E ⊕ E is in V ( C ⊕ C, A ⊕ A )by definition of weakly exact structure. Axioms ( E
2) and ( E op imply then that E + E = ∇ A ( E ⊕ E ) ∆ C lies in V ( C, A ). Finally, it is well-known that theadditive inverse in V ( C, A ) of the element E = A i (cid:47) (cid:47) B d (cid:47) (cid:47) C is given by theequivalence class of − E = A − i (cid:47) (cid:47) B d (cid:47) (cid:47) C . But − E is the pushout of E along − A , thus is contained in V when E is. This shows that V ( C, A ) is a subgroup of W ( C, A ). Multiplication by morphisms is given by pullback and pushout, and since V is stable under these operations, we have that V is an additive sub-bifunctor of W : A op × A −→ Ab; (
C, A ) (cid:55)−→ W ( C, A ) . (cid:3) Remark 4.5.
The construction in Definition 4.1 associates with Rump’s maximalexact structure E max an additive bifunctor E max = Ext E max ( − , − ). In particular,when A is an abelian category, then E max is the class of all short exact sequences,and we obtain Yoneda’s bifunctor E max = Ext A ( − , − ) . Generalizing this notation, we write Ext A ( − , − ) := Ext E max ( − , − ) for any additivecategory A .Additive functors from a preadditive category C to the category of abelian groupsare usually referred to as C− modules , and they form an abelian category, see e.g.Theorem 4.2 in [Po73, chap. 3]. For C = A op × A and W a weakly exact structure N THE LATTICE OF WEAKLY EXACT STRUCTURES 18 on A , we can thus view W = Ext W ( − , − ) as an object in the abelian category BiFun ( A ) of A − A− bimodules. We consider the partial order on
BiFun ( A )given by F ≤ F (cid:48) ⇐⇒ F ( C, A ) ≤ F (cid:48) ( C, A ) for all
A, C ∈ A that is, F ( C, A ) is a subgroup of F (cid:48) ( C, A ) for every pair of objects in A . The con-struction in Definition 4.1 thus defines a map Φ from the weakly exact structuresincluded in E max on the additive category A to the A − A− bimodules:Φ :
Wex ( A ) −→ BiFun ( A ) W (cid:55)−→ W = Ext W ( − , − ) . Lemma 4.4 shows that Φ is a morphism of posets. The elements in Ex ( A )are sent under the map Φ to subfunctors of Ext A ( − , − ) = E max that enjoy anadditional property, namely they give rise to a long exact sequence of functors: Definition 4.6. ([BuHo61, DRSS]) An additive sub-bifunctor F of Ext A ( − , − ) iscalled closed if for any short exact sequence E : A i (cid:47) (cid:47) B d (cid:47) (cid:47) C whose class lies in F ( C, A ) and any object X in A , the sequences0 → Hom(
X, A ) → Hom(
X, B ) → Hom(
X, C ) → F ( X, A ) → F ( X, B ) → F ( X, C )and0 → Hom(
C, X ) → Hom(
B, X ) → Hom(
A, X ) → F ( C, X ) → F ( B, X ) → F ( A, X )are exact in the category of abelian groups. As noted in [BuHo61], the abovesequences are always exact in all positions except F ( X, B ), respectively F ( B, X ),thus one could equivalently say F is closed if the functors F ( X, − ) and F ( − , X )are middle-exact, or using the terminology of [Rou, 4.1.1], (co-)homological. Proposition 4.7. [DRSS, Prop 1.4] Let E be an exact structure on A . Then thebifunctor Φ( E ) is closed.4.2. From sub-bifunctors of Ext A to weakly exact structures. We definedin the previous section a mapΦ :
Wex ( A ) −→ BiFun ( A ) . The aim of this section is to construct a partial inverse function Ψ. Since we do notknow if there is a maximal weakly exact structure for a general additive category,we need to restrict the construction to the interval of weakly exact structuresincluded in E max , denoted Wex ( E max ) := [ E min , E max ] ⊆ Wex ( A ) . N THE LATTICE OF WEAKLY EXACT STRUCTURES 19
Likewise, we write
BiFun ( E max ) for the class of sub-objects of E max in BiFun ( A ).Formulated in terms of posets, one can say BiFun ( E max ) := [ E min , E max ] ⊆ BiFun ( A )is the interval of all additive bifunctors between the minimum and the maximumexact structure on A . Note that for a weakly idempotent complete category A , ormore generally under the conditions of Corollary 3.18, we have that E max is themaximal weakly exact structure on A , therefore Wex ( E max ) = Wex ( A ) . To define a map Ψ on
BiFun ( E max ), we use the notion of F − exact pairs givenin the following definition: Definition 4.8. [BuHo61, DRSS] Let F : A op × A −→ Ab be an additive sub-bifunctor of Ext A ( − , − ). Define a class W F of short exact sequences by W F := { A i (cid:47) (cid:47) B d (cid:47) (cid:47) C in A | ( i, d ) ∈ F ( C, A ) } . The short exact sequences ( i, d ) in W F are called F -exact pairs. Proposition 4.9. ([BuHo61, DRSS]) The construction in Definition 4.8 yields amap Ψ :
BiFun ( E max ) −→ Wex ( E max ) F (cid:55)−→ W F . Moreover, the functions Φ and Ψ induce mutually inverse poset isomorphisms
Wex ( E max ) ←→ BiFun ( E max ) ∪ ∪ Ex ( A ) ←→ CBiFun ( A )where CBiFun ( A ) denotes the subclass of closed sub-bifunctors of Ext A . Proof.
These results are mostly covered in [DRSS], some going back to [BuHo61]:Let F : A op × A −→ Ab be an additive sub-bifunctor of Ext A ( − , − ). As discussedin Section 1.2 of [DRSS], the collection of F − exact pairs W F is closed underisomorphisms and satisfies condition (E2) and (E2) op from Definition 3.1. Alsoconditions (E0) and (E0) op hold since the identity 1 A splits and thus representsthe zero object which is clearly an F -exact pair. Moreover, Lemma 1.2 in [DRSS]ensures that W F is closed under direct sums of F − exact pairs since F is an additivefunctor. We have thus verified that W F is a weakly exact structure, hence the mapΨ is well-defined. It is also clear that Ψ is a morphism of posets.Proposition 1.4 in [DRSS] shows that the sub-bifunctor F is closed preciselywhen the class W F forms an exact structure, so the restriction of the map Ψ to CBiFun ( A ) maps exactly to Ex ( A ).Finally, it is easy to verify that the maps Φ and Ψ are mutually inverse. (cid:3) N THE LATTICE OF WEAKLY EXACT STRUCTURES 20
Remark 4.10. If A is essentially small then BiFun ( E max ) forms a set: the choiceof a sub-bifunctor of E max is determined by the choice of a subgroup of E max ( C, A )for all objects
C, A in A , which form a set up to isomorphism.Since Wex ( E max ) is in bijection with BiFun ( E max ), we conclude that theclass Wex ( E max ) of weakly exact substructures of E max also forms a set in thiscase, and the same argument applies to the subclasses Ex ( A ) of Wex ( E max ) and CBiFun ( A ) of BiFun ( E max ).4.3. Example.
We reconsider here Example 3.2 in light of the bijection from thelast proposition: Let A = rep Q be the category of representations of the quiver Q : 1 (cid:47) (cid:47) (cid:47) (cid:47) A is as follows: P P S I P S There are (up to equivalence) exactly five non-split exact sequences with inde-composable end terms, where the first three are the Auslander-Reiten sequences:( α ) 0 (cid:47) (cid:47) P a (cid:47) (cid:47) P c (cid:47) (cid:47) S (cid:47) (cid:47) β ) 0 (cid:47) (cid:47) S e (cid:47) (cid:47) I f (cid:47) (cid:47) S (cid:47) (cid:47) γ ) 0 (cid:47) (cid:47) P (cid:47) (cid:47) P ⊕ S (cid:47) (cid:47) I (cid:47) (cid:47) δ ) 0 (cid:47) (cid:47) P (cid:47) (cid:47) P d (cid:47) (cid:47) I (cid:47) (cid:47) (cid:15) ) 0 (cid:47) (cid:47) P b (cid:47) (cid:47) P (cid:47) (cid:47) S (cid:47) (cid:47) A it is thereforesufficient to examine the bimodule structure on the vector space generated bythese five non-split exact sequences with indecomposable end-terms. It is depictedin the following diagram, which indicates the multiplication rules δe = α, aδ = γ, (cid:15)f = γ, c(cid:15) = β (see Definition 3.1) : (cid:15)β γ δ α eafc From there it is easy to see that Ext A admits 13 submodules (including the zerosubmodule and itself), and the submodule lattice is given in Figure 4.3, indicatingeach submodule by a set of generators. The eight closed submodules, correspondingto the eight exact structures on A , are indicated in blue. N THE LATTICE OF WEAKLY EXACT STRUCTURES 21 ∅ α γ βα, γ α, β β, γα, γ, δ α, β, γ β, γ, (cid:15)α, β, γ, δ α, β, γ, (cid:15)α, β, γ, δ, (cid:15) Figure 2: Subbimodules of Ext A ( − , − )4.4. Weakly exact structures as bimodules.
In Section 4.1, we explained howweakly exact structures give rise to bifunctors. In this subsection, we use thesebifunctors to obtain bimodules over the Auslander algebra.
Definition 4.11.
Let A be an additively finite, Hom-finite Krull-Schmidt cat-egory with indecomposables X , . . . , X n and denote by A = End( X ) with X = X ⊕ · · · ⊕ X n its Auslander algebra. The Krull-Schmidt property implies thatthe additive category A is weakly idempotent complete, thus as discussed in Sec-tion 3.5, we know that the maximum weakly exact structure coincides with themaximum weakly exact structure formed by the stable short exact sequences. Thecorresponding bifunctor E max , evaluated at the object X yields a bimoduleB = E max ( X, X )over the Auslander algebra A : We know that B = E max ( X, X ) is an abelian group,and elements of A are morphisms a : X → X, whose action on B is described bythe action of the bifunctor E max . More generally, let W be a weakly exact structure on A , and consider its asso-ciated bifunctor W = Ext W ( − , − ). We showed in Proposition 4.3 that the abeliangroup B W = W ( X, X ) forms a bimodule over the Auslander algebra B , and byProposition 4.9, we obtain that B W is an A − A − subbimodule of B.We denote by Bim (B) the class of all sub-bimodules of A B A ; it forms a poset( Bim (B), ⊆ ) with inclusion as order relation. Example 4.12.
In the example studied in Section 4.3, the Auslander algebra A is the algebra whose quiver is the Auslander-Reiten quiver with mesh relations,and the A − A − bimodule B = E max ( X, X ) is the Ext − bimodule on A , a five-dimensional bimodule with basis given by the elements α, β, γ, δ, (cid:15) . The Figure 4.3describes the bimodule lattice ( Bim (B), ⊆ ) in this example. N THE LATTICE OF WEAKLY EXACT STRUCTURES 22 Weakly extriangulated structures
In Section 4.2, we showed that additive sub-bifunctors of E max give rise toweakly exact sub-structures of the maximal exact structure E max on A . Extrian-gulated structures [NP19] (or, equivalently, 1 − exangulated structures [HLN]) aredefined in terms of additive bifunctors A op × A → Ab equipped with an extradata and satisfying certain axioms. They generalize both exact and triangulatedcategories. In this Section, we define their weak version by removing one of theaxioms and show that this covers weakly exact structures.We first present the definition of 1 − exangulated categories following [HLN]. Definition 5.1.
Let E : A op × A → Ab be an additive bifunctor. Given a pairof objects
A, C ∈ A , we call an element δ ∈ E ( C, A ) an E − extension. When wewant to emphasize A and C , we also write A δ C .Since E is a bifunctor, each morphism a ∈ Hom(
A, A (cid:48) ) induces the extension a ∗ ( δ ) := E ( C, a )( δ ) ∈ E ( C, A (cid:48) ). Similarly, each morphism c ∈ Hom( C (cid:48) , C ) inducesthe extension c ∗ ( δ ) := E ( c, A )( δ ) ∈ E ( C (cid:48) , A ).Moreover, we have E ( c, a )( δ ) = c ∗ a ∗ ( δ ) = a ∗ c ∗ ( δ ) . By the Yoneda lemma, each extension A δ C induces a pair of natural transfor-mations δ (cid:93) : Hom( − , C ) → E ( − , A ) and δ (cid:93) : Hom( A, − ) → E ( C, − ) . Namely, for each X ∈ A , we have( δ (cid:93) ) X : Hom( X, C ) → E ( X, A ) , c (cid:55)→ c ∗ ( δ );( δ (cid:93) ) X : Hom( A, X ) → E ( C, X ) , a (cid:55)→ a ∗ ( δ ) . Definition 5.2.
A morphism of extensions A δ C → B ρ D is a pair of morphisms( a, c ) ∈ Hom(
A, B ) × Hom(
C, D ) such that a ∗ ( δ ) = c ∗ ( ρ ) . Definition 5.3. A weak cokernel of a morphism f : A → B in A is a morphism g : B → C such that for all X ∈ A , the induced sequence of abelian groupsHom( C, X ) → Hom(
B, X ) → Hom(
A, X )is exact, i.e. the sequence of functorsHom( C, − ) → Hom( B, − ) → Hom( A, − )is exact. Equivalently, g is a weak cokernel of f if g ◦ f = 0 and for each morphism h : B → X such that h ◦ f = 0 , there exists a (not necessarily unique) morphism l : C → X such that h = l ◦ g. Weak kernel is a weak cokernel in A op . Note that weak (co)kernels satisfy the same factorization properties as usual(co)kernels, but without requiring uniqueness. Clearly, a weak (co)kernel g of f isa (co)kernel of f if and only if g is a monomorphsim (resp. an epimorphism). N THE LATTICE OF WEAKLY EXACT STRUCTURES 23
Definition 5.4.
We call a pair of composable morphisms A f → B g → C a weak kernel-cokernel pair if f is a weak kernel of g and g is a weak cokernel of f . By definition, in each weak kernel-cokernel pair as above the composition g ◦ f is 0 , so the pair can be understood as an element of the category C [0 , ( A ) (cid:44) → C ( A )of complexes over A concentrated in the degrees 0 , . Let C w ( A ) be the full subcategory of C [0 , ( A ) with objects being weak kernel-cokernel pairs.Consider morphisms of complexes in C w ( A ) A A f (cid:47) (cid:47) B b (cid:15) (cid:15) g (cid:47) (cid:47) C C A f (cid:48) (cid:47) (cid:47) B (cid:48) g (cid:48) (cid:47) (cid:47) C (1)with leftmost and rightmost vertical morphisms being identities. Lemma 5.5.
For a diagram of the form (1), the following are equivalent: • The morphism f • = (1 A , b, C ) is an isomorphism in C w ( A ); • The morphism b is an isomorphism; • The morphism f • is a homotopy equivalence in C [0 , ( A ).Here by homotopy equivalence in C [0 , ( A ) we mean that there exists a morphism g • in C [0 , ( A ) and morphisms φ : B → A, φ : C → B, ψ : B (cid:48) → A, ψ : C → B (cid:48) such that the pair ( φ , φ ) yields a chain homotopy g • ◦ f • ∼ φ , φ )yields a chain homotopy f • ◦ g • ∼ . Proof.
This is a reformulation of [HLN, Lemma 4.1], see also [HLN, Claim 2.8]. (cid:3)
Morphisms f • = (1 A , b, C ) satisfying either of conditions in Lemma 5.5 definean equivalence relation on objects in C w ( A ) . We denote by [ A f → B g → C ] theequivalence class of the complex A f → B g → C in C w ( A ) under this equivalence. Definition 5.6. (cf. [HLN, Definition 2.22]) Let s be a correspondence whichassociates an equivalence class s ( δ ) = [ A f → B g → C ]in C ( A ) to each extension δ = A δ C . Such s is called a realization of E if it satisfiesthe following condition for any s ( δ ) = [ A f → B g → C ] and any s ( ρ ) = [ A (cid:48) f (cid:48) → B (cid:48) g (cid:48) → C (cid:48) ] : N THE LATTICE OF WEAKLY EXACT STRUCTURES 24 (R0) For any morphism of extensions ( a, c ) : δ → ρ, there exists a morphism b : B → B (cid:48) such that f • = ( a, b, c ) is a morphism in C [0 , ( A ) : A A f (cid:47) (cid:47) B b (cid:15) (cid:15) g (cid:47) (cid:47) C C (cid:8) (cid:8) A f (cid:48) (cid:47) (cid:47) B (cid:48) g (cid:48) (cid:47) (cid:47) C. Such f • is called a lift of ( a, c ) . We say that [ A f → B g → C ] realizes δ whenever we have s ( δ ) = [ A f → B g → C ] . Each weak kernel-cokernel pair A f → B g → C realizing an extension δ induces apair of sequences of functorsHom( C, − ) → Hom( B, − ) → Hom( A, − ) → E ( C, − );(2) Hom( − , A ) → Hom( − , B ) → Hom( − , C ) → E ( − , A ) . (3) Definition 5.7. (cf. [HLN, Definition 2.22])A realization s is called exact if the following two conditions are satisfied:(R1) For each extension δ , for each A f → B g → C realizing δ , both sequences (2)are exact (i.e. exact when applied to each object in A );(R2) For each object A ∈ A , we have s ( A ) = [ A A → A → , s ( A ) = [0 → A A → A ] . Remark 5.8.
Note that since we require realizations to be given by weak kernel-cokernel pairs, sequences (2) are automatically exact at Hom( B, − ) , resp. atHom( − , B ) . In other words, condition (R1) concerns only exactness at Hom( A, − ) , resp. at Hom( − , C ) . Remark 5.9.
By [HLN, Proposition 2.16], condition (R1) does not depend on thechoice of a representative in the equivalence class s ( δ ) . Definition 5.10. ([HLN, Definition 2.23], [NP19, Definition 2.15, Definition2.19]) Let s be an exact realization of E . Pairs δ, s ( δ ) are called ( distinguished ) E − triangles. If a complex A f → B g → C is a representative in s ( δ ) for some δ, it is called a conflation . In this case, themorphism f is called an inflation and the morphism g is called a deflation. Lemma 5.11. ([HLN, Proposition 3.2]) The class of conflations and the class of E − triangles are both closed under direct sums and direct summands. N THE LATTICE OF WEAKLY EXACT STRUCTURES 25
Since we consider weak kernel-cokernel pairs as complexes, we can considermapping cones and cocones of morphisms between them. We use the minor mod-ification of the usual definition that was considered in [HLN] and applies only forcertain morphisms.
Definition 5.12. [HLN, Definition 2.27] Let f • = (1 A , b, c ) be a morphism in C [0 , ( A ) : A A f (cid:47) (cid:47) B b (cid:15) (cid:15) g (cid:47) (cid:47) C c (cid:15) (cid:15) A f (cid:48) (cid:47) (cid:47) B (cid:48) g (cid:48) (cid:47) (cid:47) C (cid:48) . Its mapping cone M • f is defined to be the complex B − gb → C ⊕ B (cid:48) (cid:104) c g (cid:48) (cid:105) → C (cid:48) . In other words, this is the usual mapping cone of the morphism of complexes B b (cid:15) (cid:15) g (cid:47) (cid:47) C c (cid:15) (cid:15) B (cid:48) g (cid:48) (cid:47) (cid:47) C (cid:48) . The mapping cocones of morphisms of the form ( a, b, C ) are defined dually. Definition 5.13. ([HLN, Definition 2.32 for n = 1]) A 1-exangulated category isa triplet ( A , E , s ) of an additive category A , additive bifunctor E : A op × A → Ab,and its exact realization s , satisfying the following conditions.(EA1) The composition of two inflations is an inflation. Dually, the compositionof two deflations is a deflation.(EA2) For each ρ ∈ E ( C (cid:48) , A ) and c ∈ Hom(
C, C (cid:48) ) , for each pair of realizations A f → B g → C of c ∗ ρ and A f (cid:48) → B (cid:48) g (cid:48) → C (cid:48) of ρ, the morphism (1 A , c ) : c ∗ ρ → ρ admits a good lift f • = (1 A , b, c ), in the sense that M • f realizes f ∗ ρ. (EA2) op Dual of (EA2).
Proposition 5.14. ([HLN, Proposition 4.3]) A triplet ( A , E , s ) is a 1-exangulatedcategory if and only if it is an extriangulated category as defined by Nakaoka andPalu [NP19].This result motivates the following definition. N THE LATTICE OF WEAKLY EXACT STRUCTURES 26
Definition 5.15. A weakly extriangulated (= weakly − exangulated ) structure onan additive category A is a pair ( W , s ) of an additive bifunctor W : A op × A → Aband its exact realization s ) satisfying axioms (EA2) and (EA2) op . Lemma 5.16.
A weakly exact strucure W on A defines a weakly extriangulatedstructure ( A , W , s ) . Proof.
Using Lemma 3.7, all the arguments from [NP19, Example 2.13], except forthose concerning (ET4) and (ET4) op , apply here word for word. That means thata weakly exact structure defines a pair of a bifunctor and its exact realization.Axioms (EA2) and (EA2) op follow directly from axioms (E2) and (E2) combinedwith Lemma 3.5 and its dual. (cid:3) We can also characterize weakly exact structures among weakly extriangulatedones.
Lemma 5.17. (cf. [NP19, Corollary 3.18]) Let ( A , W , s ) be a weakly extrian-gulated category, in which each inflation is monomorphic, and each deflation isepimorphic. If we let W be the class of conflations given by the W − triangles, then( A , W ) is a weakly exact category. Proof.
If an inflation in a conflation is monomorphic, it is not just a weak kernelof the deflation, but the actual kernel. Similarly, if a deflation is epimorphic, itis the cokernel of an inflation. Therefore, if each inflation is monomorphic, andeach deflation is epimorphic, all conflations are kernel-cokernel pairs. From theexactness of the realization, it follows that the class of conflations is closed underdirect sums and axioms (E0) and (E0) op are satisfied. Axioms (EA2) and (EA2) op imply the axioms (E2) and (E2) op by Lemma 3.5 and its dual. (cid:3) Defects and topologizing subcategories
In this section, we extend the notion of contravariant defects to the settingof weakly extriangulated categories. These categories were used in [Bu01, En18,En20, FG20] to classify exact structures on an additive category and, more gen-erally, extriangulated substructures of an extriangulated structure. We show thattheir results extend to our framework. First, let us recall some necessary notions.
Definition 6.1.
Let A be an essentially small additive category. Contravariantadditive functors A op → Ab to the category of abelian groups are called right A− modules . They form an abelian category Mod A . Dually, left A− modules arecovariant additive functors to abelian groups, they form an abelian category thatcan be seen as Mod A op . These categories have enough projectives. Those are precisely the directsummands of direct sums of representable functors Hom( − , A ) ∈ Mod A , resp.Hom( A, − ) ∈ Mod A op . N THE LATTICE OF WEAKLY EXACT STRUCTURES 27
We will work with certain full subcategories of categories of A− modules. First,we need to recall several classical definitions: Definition 6.2. An A -module M is called finitely generated if admits an epi-morphism Hom( − , X ) (cid:16) M from a representable functor. It is moreover finitelypresented if it is a cokernel of a morphism of representable functors. A module iscalled coherent if it is finitely presented and each of its finitely generated submod-ule is also finitely presented. Note that every finitely generated submodule of acoherent module is automatically coherent.By definition, we have a chain of embeddings of full additive categories coh ( A ) (cid:44) → fp ( A ) (cid:44) → fg ( A ) (cid:44) → Mod A , where the first three categories are the categories of coherent, finitely presentedand finitely generated right A− modules, respectively.The category of finitely presented modules fp ( A ) is known to be abelian if andonly the category A has weak kernels. The category of coherent modules behavesbetter, as the following standard fact shows: Proposition 6.3. ([He97, Proposition 1.5], see also [Fi16, Appendix B]) Thecategory coh ( A ) is abelian and the canonical embedding coh ( A ) (cid:44) → Mod A isexact. In particular, coh ( A ) is closed under kernels, cokernels and extensions inMod ( A ) .Two more important full subcategories of categories of modules over abeliancategories has been studied thoroughly since 1950s and 1960s: the category ofeffaceable functors, studied already by Grothendieck [Gr57], and the category ofdefects introduced by Auslander [A66, A78, ARS]. These notions have been gen-eralized to the setting of exact categories (see e.g. [Ke90, Fi16, En18]) and, byOgawa [Og19] and Enomoto [En20], to that of extriangulated categories. Ogawa’sdefinition uses only part of the axioms of extriangulated categories, and so we canformulate it in our broader context.Let ( A , W , s ) be a weakly extriangulated category. Definition 6.4.
We say that a module F ∈ Mod A is weakly effaceable with respectto ( W , s ) if the following condition is satisfied:For any Z ∈ A and any z ∈ F ( Z ), there exists a deflation g : Y (cid:16) Z such that F ( g )( z ) = 0. Definition 6.5.
Given a conflation X f (cid:26) Y g (cid:16) Z, we define its contravariantdefect to be the cokernel of Hom( − , g ) : Hom( − , Y ) → Hom( − , Z ) in Mod A . Full subcategories of abelian categories, which are closed under kernels, cokernels and exten-sions, are sometimes also called wide subcategories.
N THE LATTICE OF WEAKLY EXACT STRUCTURES 28
We denote by Eff W the category of weakly effaceable functors and by def W the full subcategory of right A− modules isomorphic to defects of conflations.If ( A , W , s ) corresponded to a weakly exact structure W on A , we also write Eff W := Eff W and def W := Eff W . For abelian categories endowed with maximal exact structures, the followingtwo statements are standard, see e.g. [Gr57], resp. [ARS].
Lemma 6.6.
The category Eff W is closed under subquotients and finite directsums in Mod A . Proof.
Let 0 → F µ → G ν → H → A . Assume that G is weakly effaceable withrespect to ( W , s ). Let Z be an object of A . Choose an element z ∈ F ( Z ) and adeflation f : P → Z such that0 = G ( f ) ◦ µ ( Z )( z ) = µ ( P ) ◦ F ( f )( z ) . Since µ is monic, F ( f )( z ) = 0 . Thus, F is weakly effaceable with respect to ( W , s ).So Eff W is closed under subobjects. The rest is proved by similar straightforwarddiagram chasing. (cid:3) Lemma 6.7.
The category def W is closed under kernels and cokernels in Mod A . Proof.
The same argument as in [Og19, Lemma 2.6] applies here. A morphismof defects of two conflations gives rise to a morphism ( a, c ) of these conflations.Then the kernel is given by the defect of the mapping cone of any good lift of themorphism (1 , c ) and the cokernel is given by the defect of the mapping cocone ofany good lift of the morphism ( a, . (cid:3) The following notion was introduced by Rosenberg [Ros] in his works on non-commutative algebraic geometry and reconstruction of schemes.
Definition 6.8.
A full subcategory of an abelian category is called topologizing ifit is closed under subquotients and finite direct sums.
Proposition 6.9.
Let ( A , W , s ) be a weakly extriangulated category. We have def W = Eff W (cid:92) coh ( A )and this category is topologizing. Proof.
The same argument as in the proof of [En20, Proposition 2.9] applies here.The only difference is that in our generality Eff W is not closed under extensionsin Mod A , but only under finite direct sums. (cid:3) For A− modules, we have natural notions of subobjects, quotients and exten-sions: these are defined object-wise (for objects in A ). N THE LATTICE OF WEAKLY EXACT STRUCTURES 29
Definition 6.10.
We say that a subcategory of an arbitrary (not necessarilyabelian) full subcategory C of coh ( A ) is topologizing if it is closed under sub-quotients (considered object-wise) and finite direct sums. Equivalently, it is topol-ogizing if it is a full subcategory of C which is topologizing in coh ( A ) . Similarly, we say that a subcategory of C is Serre if it is topologizing and closedunder extensions; equivalently, if it is a full subcategory of C and a Serre subcate-gory in coh ( A ) . Note that this definition ensures that a Serre subcategory of C is abelian. Corollary 6.11.
Let ( A , W , s ) be a weakly extriangulated category and let( A , W (cid:48) , s | W (cid:48) ) be a weakly extriangulated substructure on A (that is, W (cid:48) is anadditive sub-bifunctor of W ). Then the category def W (cid:48) is a topologizing subcat-egory of def W . Corollary 6.12.
Let W (cid:48) be a weakly exact substructure of a weakly exact struc-ture W . Then the category def W (cid:48) is a topologizing subcategory of def W . Lattice structures
We study in this section lattice structures on the different posets introduced inthe previous parts of this article.7.1.
Definitions.
We recall the following well known notions:
Definition 7.1.
A poset P is called a join-semilattice if for every pair ( p, q ) ofelements of P there exists a supremum p ∨ q (also called join). It is called a meet-semilattice if for every pair ( p, q ) of elements of P there exists an infimum p ∧ q (also called meet). Finally, P is lattice if it is both a join-semilattice anda meet-semilattice. Equivalently, a lattice is a set P equipped with two binaryoperations ∨ and ∧ : P × P → P satisfying the following axioms:(1) ∨ is associative and commutative,(2) ∧ is associative and commutative,(3) ∧ and ∨ satisfy the following property: m ∨ ( m ∧ n ) = m = m ∧ ( m ∨ n ) for all m, n ∈ P. A lattice is called complete if all its subsets have both a join and a meet, similarfor semilattices. A bounded lattice is a lattice that has a greatest element (alsocalled maximum) and a least element (also called minimum).
Remark 7.2.
As a consequence of the axioms above we have the following prop-erty for lattices: m ∨ m = m and m ∧ m = m for all m ∈ P. N THE LATTICE OF WEAKLY EXACT STRUCTURES 30
Definition 7.3.
A lattice ( P, ≤ , ∧ , ∨ ) is modular if the following property is sat-isfied for all r, s, t ∈ P with r ≤ s : s ∧ ( r ∨ t ) = r ∨ ( s ∧ t ) . Definition 7.4. [Da02, 2.16, 2.17] Let P and Q be two lattices, then a function f : P → Q is a morphism of lattices if for all m, n ∈ P one has: f ( m ∨ n ) = f ( m ) ∨ f ( n ) and f ( m ∧ n ) = f ( m ) ∧ f ( n ) . An isomorphism of lattices is a bijective morphism of lattices (in which case itsinverse is also an isomorphism). Definition 7.5.
Let ( P, (cid:54) ) be a partially ordered set with a unique minimalelement 0. An atom is an element a ∈ P with a > (cid:54) x (cid:54) a implies x = 0 or x = a . In other words, atoms are the elements that are directlyabove the minimal element.7.2. Lattices of right and left weakly exact structures.
In this subsectionwe study a lattice structure on the class of all right (or left) weakly exact struc-tures. These results generalise the one obtained in [HR20, Proposition 8.4] on thecomplete lattice structure of the class of (strong) one-sided exact structures.
Definition 7.6.
We denote by LW ( A ) (respectively RW ( A )) the class of all left(right) weakly exact structures on A . Lemma 7.7.
Let {L i } i ∈ ω ( {R i } i ∈ ω ) be a family of left (right) weakly exact struc-tures on A . Then the intersection ∩ i ∈ ω L i ( ∩ i ∈ ω R i ) is also a left (right) weaklyexact structure. Proof.
Same as Lemma 5.2 of [BHLR]. (cid:3)
Proposition 7.8.
Let A be an additive category. Then LW ( A ) and RW ( A )) arecomplete meet-semi lattices. Proof.
Let L and L (cid:48) be two left weakly exact structures on A . The partial orderon LW ( A ) is given by containment. We define the meet given by L ∧ L (cid:48) = L ∩ L (cid:48) .These operations define the structure of complete meet-semilattice on LW ( A ) byLemma 7.7. (cid:3) Remark 7.9.
If there exists a unique maximal left weakly exact structure L max on A , then LW ( A ) is a complete lattice (similarly for RW ( A )). In this case, the join can be defined by the usual construction L ∨ L L (cid:48) = ∩{L (cid:48)(cid:48) ∈ LW ( A ) | L ⊆ L (cid:48)(cid:48) , L (cid:48) ⊆ L (cid:48)(cid:48) } . The intersection in the definition of the join is well defined since the set includes L max by assumption. These operations define a lattice structure on LW ( A ). Sincethe lattice has a minimal element L min , formed by all retractions, and a maximalelement L max , it is a bounded lattice. Likewise, any interval in the poset LW ( A )forms a complete bounded lattice. N THE LATTICE OF WEAKLY EXACT STRUCTURES 31
Remark 7.10.
The constructions in Section 3.4 can be reformulated in terms ofthe lattices studied in this section as follows: As stated in Remark 3.15, there is asplicing function s : Wex ( A ) −→ LW ( A ) × RW ( A ) , W (cid:55)−→ ( L W , R W )where L W := { d | ( i, d ) ∈ W} is the class of all W− cokernels or W− admissible de-flations and R W := { i | ( i, d ) ∈ W} is the class of all W− kernels or W− admissibleinflations.Moreover, Theorem 3.13 shows that there is a gluing function: g : LW ( A ) × RW ( A ) −→ Wex ( A ) , ( L , R ) (cid:55)−→ W ( L , R ) where W ( L , R ) is formed by all short exact sequences ( i, d ) in A with i ∈ R , d ∈ L} . The maps s and g are not bijective, but it seems interesting to study theirproperties.7.3. Lattice of weakly exact structures.
Lattice of exact structures revisited.
We know by [BHLR, Theorem 5.3] thatthe class of exact structures on an additive category Ex ( A ) forms a lattice. Inorder to study the properties of this lattice, we show that it is isomorphic to thelattice of closed additive sub-bifunctors of Ext A ( − , − ) defined in Section 4. Theorem 7.11. [BHLR, 5.1, 5.3, 5.4] Let A be an additive category. The poset Ex ( A ) of exact structures E on A forms a bounded complete lattice( Ex ( A ) , ⊆ , ∧ , ∨ E )under the following operations:(1) The partial order is given by containment E (cid:48) ⊆ E (2) The meet ∧ is defined by E ∧ E (cid:48) = E ∩ E (cid:48) (3) the join ∨ E is defined by E ∨ E E (cid:48) = (cid:92) {F ∈ Ex ( A ) | E ⊆ F , E (cid:48) ⊆ F } . Lattice structure on the class of all weakly exact structures of a given additivecategory.
Lemma 7.12.
Let {W i } i ∈ ω be a family of weakly exact structures on A . Thenthe intersection ∩ i ∈ ω W i is also a weakly exact structure. Proof.
Same as Lemma 5.2 of [BHLR]. (cid:3)
Theorem 7.13.
Let A be an additive category and E max the maximal exact struc-ture on A . Then the weakly exact structures that are included in E max form acomplete bounded lattice: ( Wex ( E max ) , ⊆ , ∧ , ∨ W ) N THE LATTICE OF WEAKLY EXACT STRUCTURES 32
Proof.
It follows from Lemma 7.12 that
Wex ( A ) forms a meet semi-lattice:( Wex ( A ) , ⊆ , ∧ ) with order relation given by inclusion and meet operation givenby inclusion. Moreover, the weakly exact structures that are included in E max forma complete bounded lattice ( Wex ( E max ) , ⊆ , ∧ , ∨ W ) where the join ∨ W is definedby W ∨ W W (cid:48) = ∩{V ∈ Wex ( A ) | W ⊆ V , W (cid:48) ⊆ V} This join is well-defined for
Wex ( E max ) since the set includes E max by assumption.Since the lattice Wex ( E max ) has a minimal element E min and a maximal element E max , it is a bounded lattice. (cid:3) Remark 7.14.
While the partial order and the meet coincide for Ex ( A ) and Wex ( A ), the join ∨ E is different from the join for weakly exact structures sincewe intersect over a smaller set , making the join larger when both are viewed inthe poset Wex ( E max ): E ∨ W E (cid:48) ≤ E ∨ E E (cid:48) for all E , E (cid:48) ∈ Ex ( A ). In fact, in the example from Section 4.3, if we consider theexact structures E = (cid:104) α (cid:105) and E (cid:48) = (cid:104) γ (cid:105) , then E ∨ W E (cid:48) = (cid:104) α, γ (cid:105) which is striclysmaller than E ∨ E E (cid:48) = (cid:104) α, γ, δ (cid:105) . This shows that Ex ( A ) is a meet-subsemilatticeof Wex ( E max ), but it is not a sublattice in general.We now describe the join of two weakly exact structures in a more constructiveway, motivated by the sum of bifunctors: Definition 7.15.
Let W , W ∈ Wex ( E max ) be two weakly exact structures con-tained in E max . Then, W = W + W is defined as W := (cid:83) A,C ∈A W ( C, A ) where W ( C, A ) := { η + η | η ∈ W ( C, A ) , η ∈ W ( C, A ) } with W k ( C, A ) := { η : A i (cid:47) (cid:47) B d (cid:47) (cid:47) C | η ∈ W k } for k = 1 ,
2. Here, for η ∈ W ( C, A ) and η ∈ W ( C, A ), the sum η + η := ∇ A ( η ⊕ η )∆ C is the Baersum for short exact sequences. Since W and W are included in E max and theBaer sum in well defined in E max , we have W ⊆ E max . Proposition 7.16.
Let W , W be two weakly exact structures contained in E max . Then(a) W + W is weakly exact(b) W + W is the join W ∨ W W in the lattice Wex ( E max ). Proof.
For part (a), W has to satisfy properties ( E E op , ( E E op andneeds to be closed under direct sums.To show ( E A . Since W and W are weakly exactstructures, by ( E
0) for W and W the short exact sequence E : X (cid:47) (cid:47) X (cid:47) (cid:47) W and in W . Since E + E = E with the first E in W and the second in W ,we obtain that E is in W by definition. The proof for ( E op is dual. N THE LATTICE OF WEAKLY EXACT STRUCTURES 33
For ( E η : A i (cid:47) (cid:47) B d (cid:47) (cid:47) C ∈ W and a : A −→ A (cid:48) ∈ A .We show that the push-out aη of η by a exists and is in W . Since η ∈ W ( C, A ),there exist η ∈ W ( C, A ) and η ∈ W ( C, A ) such that η = η + η . Using [M65,Lemma 1.4(iii)] we have aη = a ( η + η ) = aη + aη . Since W and W satisfy( E
2) we have aη ∈ W ( C, A (cid:48) ) and aη ∈ W ( C, A (cid:48) ). Therefore, aη = aη + aη ∈W ( C, A (cid:48) ) ⊆ W . The proof for ( E op is dual.For the direct sums, let α and β be in W . We want to show that α ⊕ β ∈ W .Suppose that α : A i (cid:47) (cid:47) B d (cid:47) (cid:47) C ∈ W ( C, A ) and β : D j (cid:47) (cid:47) E e (cid:47) (cid:47) F ∈W ( F, D ). Then there exist α ∈ W ( C, A ), α ∈ W ( C, A ), β ∈ W ( F, D ) and β ∈ W ( F, D ) such that α = α + α and β = β + β , hence α ⊕ β = ( α + α ) ⊕ ( β + β ) = ( ∇ A ( α + α )∆ C ) ⊕ ( ∇ D ( β + β )∆ F ) . Since W and W are closed under direct sums, we get α ⊕ β ∈ W ( C ⊕ F, A ⊕ D )and α ⊕ β ∈ W ( C ⊕ F, A ⊕ D ), so ( α ⊕ β ) + ( α ⊕ β ) ∈ W ( C ⊕ F, A ⊕ D ). Wehave ( α ⊕ β ) + ( α ⊕ β ) = ∇ A ⊕ D (( α ⊕ β ) ⊕ ( α ⊕ β ))∆ C ⊕ F = ∇ A ⊕ D (( α + α ) ⊕ ( β + β ))∆ C ⊕ F . Note that the direct sum of the diagrams for ( ∇ A ( α + α )∆ C )and ( ∇ D ( β + β )∆ F ) is the diagram for ∇ A ⊕ D (( α + α ) ⊕ ( β + β ))∆ C ⊕ F . Thismeans that α ⊕ β = ( α ⊕ β ) + ( α ⊕ β ) ∈ W ( C ⊕ F, A ⊕ D ) ⊆ W . Therefore W is closed under direct sums and it is a weakly exact structure.To show part (b), recall that the join W ∨ W W is the smallest (by inclusion)weakly exact structure on A containing both W and W . We have that W ⊂W + W since η = η +0 ∈ W + W for any η ∈ W . Likewise for W , so W + W contains both W and W , hence by definition of the join, W ∨ W W ⊆ W + W . To show the converse inclusion, let W be any weakly exact structure containingboth W and W . Since W satisfies the direct sum property (S), we have η ⊕ η ∈W for all η ∈ W , η ∈ W . By definition of Baer sum and property (E2) and(E2) op for W we have η + η ∈ W . This shows W + W ⊂ W for all W containingboth W and W , so this also holds for the smallest one (their intersection) : W + W ⊆ W ∨ W W . (cid:3) Proposition 7.17.
Let α be an Auslander-Reiten sequence in A , and denote by E α = { X ⊕ Y | X ∈ E min , Y ∈ add ( α ) } the (weakly) exact structure generated by α . Then E α is an atom of both lattices ( Ex ( A ) , ⊆ , ∧ , ∨ E ) and ( Wex ( A ) , ⊆ , ∧ , ∨ W ). Proof.
This property amounts to showing that the Auslander-Reiten sequence liesin the socle of the bifunctor Ext A ( − , − ), that is, multiplication with morphismsdoes not generate any new non-split sequences. This is a well-known property ofalmost split sequences. (cid:3) N THE LATTICE OF WEAKLY EXACT STRUCTURES 34
Lattice of additive sub-bifunctors of Ext A . In Section 4, we dis-cussed additive sub-bifunctors of Ext A := E max = Ext E max and closed additivesub-bifunctors, and we denote these classes respectively by BiFun ( E max ) and CBiFun ( A ). In this section, we construct lattice structures of both classes. Theorem 7.18.
The additive sub-bifunctors of E max form a lattice( BiFun ( E max ) , ≤ , ∧ , ∨ bf ) . Proof.
For
F, F (cid:48) ∈ BiFun ( E max ), we write F ≤ F (cid:48) if F is a sub-bifunctor of F (cid:48) .The meet of F and F (cid:48) is given by the sub-bifunctor F ∧ F (cid:48) of E max satisfying( F ∧ F (cid:48) )( C, A ) = F ( C, A ) ∩ F (cid:48) ( C, A ) for all
A, C ∈ A . The join is given by the sub-bifunctor F + F (cid:48) = F ∨ bf F (cid:48) of E max satisfying( F ∨ bf F (cid:48) )( C, A ) = F ( C, A ) + F (cid:48) ( C, A ) for all
A, C ∈ A , where the sum is the sum of abelian subgroups of E max ( C, A ). Since
BiFun ( E max )has a maximal element E max , one can show similarly to the proof of Proposition7.16 that the join can also be expressed by F ∨ bf F (cid:48) = ∧{ G ∈ BiFun ( E max ) | F ≤ G, F (cid:48) ≤ G } . (cid:3) Lattice of closed additive sub-bifunctors.
As discussed in Proposition4.9, for any additive category A there is a bijection between exact structuresand closed additive sub-bifunctors of E max . We already know that the ex-act structures form a lattice [BHLR, Theorem 5.3]. In this section we define alattice structure on the class CBiFun ( A ) of closed additive sub-bifunctors of E max . Lemma 7.19. [DRSS, corollary 1.5] Consider a family { F i } i ∈ I of closed sub-bifunctors of E max . Then the intersection ∩ i ∈ I F i is a closed sub-bifunctor of E max bifunctor, given by {∩ F i } ( C, A ) = ∩{ F i ( C, A ) } on objects. Remark 7.20. If F and F (cid:48) are closed bifunctors in CBiFun ( A ) then their sum F + F (cid:48) is the sub-bifunctor of E max given by { F + F (cid:48) } ( C, A ) = F ( C, A ) + F (cid:48) ( C, A )on objects. Note that the sum of closed sub-bifunctors is not always closed.
Theorem 7.21.
For an additive category A , the closed additive sub-bifunctors of E max form a complete bounded lattice ( CBiFun ( A ) , ≤ , ∧ , ∨ cbf ). Proof.
The lattice structure is given as follows: the meet is defined by F ∧ F (cid:48) = F ∩ F (cid:48) while the join is defined by F ∨ cbf F (cid:48) = ∩{ F (cid:48)(cid:48) ∈ CBiFun ( A ) | F ≤ F (cid:48)(cid:48) , F (cid:48) ≤ F (cid:48)(cid:48) } , N THE LATTICE OF WEAKLY EXACT STRUCTURES 35 which is well defined since the intersection is always a non empty, containing E max .Lemma 7.19 ensures that CBiFun ( A ) forms a closed meet-semilattice, and thedefinition of join turns it into a closed lattice, which is bounded by E min belowand E max above. (cid:3) Remark 7.22.
The closed sub-bifunctors (
CBiFun ( A ), ≤ ) form a subposet of( BiFun ( E max ) , ≤ ). However, ( CBiFun ( A , ≤ , ∧ , ∨ cbf ) is not a sublattice of BiFun ( E max ) , ≤ , ∧ , ∨ bf ) because their joins are different. In fact, for F, F (cid:48) ∈ CBiFun ( A ) , the join F ∨ bf F (cid:48) = F + F (cid:48) is not necessarily closed. As discussedin Remark 7.14, the join of < α > with < γ > in BiFun ( E max ) is < α, γ > whichis not closed. The join of < α > with < γ > , in CBiFun ( A ) is < α, γ, δ > . Ingeneral, for F, F (cid:48) ∈ CBiFun ( A ) we have that F ∨ bf F (cid:48) ≤ F ∨ cbf F (cid:48) .7.5. Lattice of bimodules over the Auslander algebra.
We return now to thestudy of the bimodule B over the Auslander algebra A defined in Section 4.4. As isthe case for any module over a ring, recall that the set Bim (B) of sub-bimodulesof B forms a complete bounded modular lattice(
Bim (B) , ≤ , ∧ Bim , ∨ Bim ) , where the meet is given by intersection and the join is given by the sum N + N (cid:48) of sub-bimodules. Definition 7.23.
An element N ∈ Bim (B) is said to be a closed bimodule if thereexists a closed sub-bifunctor F of Ext E max such that Ev X ( F ) = N where Ev X : CBiFun ( A ) −→ Bim (B) F (cid:55)→ F ( X, X )is the evaluation at the object X ∈ A . Lemma 7.24.
The intersection of two closed sub-bimodules of B is again closed.
Proof.
Let N and P be two closed sub-bimodules of B such that Φ( F ) = N andΦ( G ) = P . We consider the sub-bifunctor H of Ext E max given by the meet of F ∧ G = H . By Lemma 7.19, H is closed. Since N ∩ P = F ( X, X ) ∩ G ( X, X ) = H ( X, X ) , the intersection is a closed sub-bimodule of B. (cid:3) Theorem 7.25.
The subset
Cbim (B) of closed sub-bimodules of B forms a com-plete bounded lattice (
Cbim (B) , ⊆ , ∩ , ∨ Cbim ) . Proof.
First this class is a poset ordered by inclusion. Second it is a meet-semi-lattice using the associative, commutative intersection of modules. Third, it is ajoin-semi-lattice using the following operation ∨ Cbim : Cbim (B) × Cbim (B) −→ Cbim (B)
N THE LATTICE OF WEAKLY EXACT STRUCTURES 36 ( N, P ) (cid:55)→ N ∨ P = ∩{ R ∈ Cbim ( B ) | N ⊂ R, P ⊂ R } which is associative commutative and satisfies the following property: P ∨ ( P ∧ N ) = N = N ∧ ( N ∨ P ) for all N, P ∈ Cbim (B) . The intersection in this definition of the join is well defined since the set includesB by assumption. These operations define a lattice structure on
Cbim (B). Sincethe lattice has a minimal element 0 and a maximal element B, it is a boundedlattice. Let { N λ } λ ∈ Λ by a family of weakly exact structures in Cbim ( B ). Theirmeet is given by ∩ λ ∈ Λ N λ and the join is given by ∩{ N (cid:48)(cid:48) ∈ Cbim (B) | N λ ⊆ N (cid:48)(cid:48) , ∀ λ ∈ Λ } . Therefore, the lattice is complete. (cid:3)
In the setting of this subsection, the bimodule B = E max ( X, X ) is finite-dimensional, thus B and all of its submodules have a non-zero socle. We knowfrom Proposition 7.17 that the Auslander-Reiten sequences lie in the socle of thebimodule B , and since all non-projective objects admit an Auslander-Reiten se-quence in A ending there, one can derive that the socle is precisely formed byall Auslander-Reiten sequences in A . Based on Auslander’s concept of defects,Enomoto shows in [En18] that the lattice Cbim (B) is an atomic lattice, in fact itis a boolean lattice determined by its atoms, the Auslander-Reiten sequences in A (see also [FG20, Theorem 2.26]).Reformulated in module-theoretic terms, that means that the closed sub-bimodules of B = E max ( X, X ) are uniquely determined by their socle, and forevery choice of elements in the socle, there is a unique closed sub-bimodule of B having precisely these elements as its socle. If the socle is formed by a set S ofAuslander-Reiten sequences, we can thus denote by E ( S ) the subbimodule of B determined by S . For all elements σ ∈ S , denote by E σ the bimodule correspond-ing to the exact structure E σ introduced in Proposition 7.17. Since the lattice Cbim (B) is atomic, we conclude that E ( S ) = (cid:95) σ ∈ S E σ . There may be several submodules of B with the same socle S , but only one ofthem is closed. As explained in the proof of [FG20, Theorem 2.26], this closedsubmodule with socle S corresponds to a Serre subcategory S generated by thesimple objects contained in the set S . All other submodules of B with socle S correspond to certain subcategories of S , but only the closed one is given by theabelian length category formed by all extensions of its simple objects. In otherwords, E ( S ) is maximal, so we derive the following result: Proposition 7.26.
For every set S of Auslander-Reiten sequences, the closedbimodule E ( S ) of B introduced above is the maximal submodule of B whose socleis S . N THE LATTICE OF WEAKLY EXACT STRUCTURES 37
This fact is illustrated nicely in the example in Section 4.3. It is also shownindependently for Nakayama algebras in [BHT, Theorem 6.9].7.6.
Lattice of topologizing subcategories.
Topologizing subcategories of anabelian category C form a complete lattice. The order is given by the canonicalinclusion of categories and the meet is given by the usual intersection. This is acomplete semi-lattice and, therefore, it has a canonical join operation upgrading itto a complete lattice. It is straightforward to check from the definitions that thejoin is given by the closure of the union by finite direct sums: (cid:95) : T op ( C ) × T op ( C ) → T op ( C )( T, T (cid:48) ) (cid:55)−→ ⊕{ T ∪ T (cid:48) } . Since this lattice has a canonical minimal element, it is moreover bounded.By definition, each Serre subcategory of an abelian category is topologizing.Thus, Serre subcategories form a subposet of the lattice of topologizing subcat-egories. By similar arguments this subposet admits a lattice structure, with thejoin given by the closure of the union by finite extensions. Since the closure ofthe union by finite direct sums is, in general, not extension-closed, the join ofSerre subcategories in the lattice of topologizing subcategories is different fromtheir join in the lattice of Serre subcategories. In other words, the lattice of Serresubcategories is a subposet, but not a sublattice of the lattice of all topologizingsubcategories.Given a topologizing subcategory C of the category coh ( A ), its topologizing sub-categories in the sense of definition 6.10 form a lattice, which is an interval in thelattice of all topologizing subcategories in coh ( A ) . Serre subcategories of C forma lattice, which is an interval in the lattice of all Serre subcategories in coh ( A ) . Itis a subposet, but not a sublattice of the lattice of topologizing subctegories of C . We formulate this observation explicitly in the case of the categories of defectsof weakly extriangulated structures:
Proposition 7.27.
Let A be an essentially small category and ( W , s ) a weaklyextriangulated structure on it, then the topologizing subcategories of def W forma bounded complete lattice ( Top ( W ) , ⊆ , (cid:92) , (cid:95) ) . Serre subcategories of def W also form a lattice, which is a subposet, but not asublattice of (
Top ( W )) . Lattices of extriangulated and weakly extriangulated substructures.
Let A be an essentially small additive category. We consider the class of all weaklyextriangulated structures on A . Lemma 7.28.
Let { W i } i ∈ ω be a family of weakly extriangulated structures on A .Then the intersection ∩ i ∈ ω W i is also a weakly extriangulated structure. N THE LATTICE OF WEAKLY EXACT STRUCTURES 38
Proof.
Similar to Lemma 5.2 of [BHLR]. (cid:3)
Theorem 7.29.
Let ( A , W , s ) be a weakly extriangulated category. Then all itsweakly extriangulated substructures form a bounded complete lattice:( WET ( A ) , ≤ , (cid:94) , (cid:95) ) Proof.
We consider the set
WET ( A ) of all the additive sub-bifunctors of W onthe essentially small category A . They are ordered by W ≤ W (cid:48) ⇐⇒ W ( C, A ) ⊆ Ab W (cid:48) ( C, A ) for all
A, C ∈ A that is, W ( C, A ) is a subgroup of W (cid:48) ( C, A ) for every pair of objects in A . Itfollows from 7.28 that ( WET ( A ) , ≤ , (cid:86) ) is a meet semi-lattice with the meet( W (cid:86) W (cid:48) )( C, A ) = W ( C, A ) ∩ W (cid:48) ( C, A ) , ∀ A, C ∈ A , by using the intersectionof abelian groups.It also forms a join semi-lattice where the join is defined by
W ∨ W W (cid:48) = (cid:94) {V ∈ WET ( A ) | W ⊆ V , W (cid:48) ⊆ V} This join is well-defined for
WET ( A ) since the set includes W by assumption,and so WET ( A ) is a complete meet semi-lattice: W is its unique maximal element.These operations satisfy the axioms of 7.1 and form then a structure of a completelattice. Moreover the lattice structure defined above on WET ( A ) has a minimalelement given by the split weakly extriangulated structure W min , so it is a boundedlattice. (cid:3) Corollary 7.30.
Let ( A , E , s ) be an extriangulated category. Then all the additivesub-bifunctors of E form a bounded complete lattice.7.8. Isomorphims of lattices.
The three large isomorphic lattices.
Theorem 7.31.
Let A be an additive category. The map Φ : W (cid:55)→
Ext W ( − , − )induces a lattice isomorphism ( Wex ( E max ) , ⊆ , ∩ , ∨ W ) ∼ = ( BiFun ( E max ) , ≤ , ∧ , ∨ bf ) . Proof.
We have already shown in Proposition 4.9 that Φ is an isomorphism ofposets. We need to verify that it preserves the meet and the join. Let W and W (cid:48) be two weakly exact structures, then W ∧ W (cid:48) is also an exact structure. Let Aand C be two objects in A .Ext W∧W (cid:48) ( C, A ) = { ( i, d ) | A i (cid:47) (cid:47) B d (cid:47) (cid:47) C ∈ W ∧ W (cid:48) } = { ( i, d ) | ( i, d ) ∈ W} ∩ { ( i, d ) | ( i, d ) ∈ W (cid:48) } = Ext W ( C, A ) ∩ Ext W (cid:48) ( C, A ) N THE LATTICE OF WEAKLY EXACT STRUCTURES 39
Therefore the two sub-bifunctors Ext W∧W (cid:48) ( − , − ) and Ext W ( − , − ) ∧ Ext W (cid:48) ( − , − )coincide, which shows that Φ is a morphism of meet-semilattices. Moreover, thejoin is defined in both lattices in the same way using intersections (meet), henceΦ is a morphism of lattices. (cid:3) Theorem 7.32.
Consider the setting of an additively finite category A as inSection 4.4 and the bimodule B over the Auslander algebra A defined there. Thenthe evaluation map yields an isomorphism of lattices Ev X : BiFun ( E max ) −→ Bim (B) F (cid:55)→ F ( X, X ) Proof. (1) We first show that the map is well defined:As F is an additive sub-bifunctor of E max , we get that F ( X, X ) is a sub-bimoduleof the A − A − bimodule B = E max ( X ), which shows F ( X, X ) ∈ Bim ( B ).(2) Injectivity: Consider two bifunctors F, G ∈ BiFun ( E max ) such that theirimages under Ev X are equal as A − A − bimodules: F ( X, X ) = G ( X, X ). Decom-posing X into indecomposables X ∼ = X ⊕ · · · ⊕ X n , we consider the idempotentelements e i in A given by projection pr i : X → X i onto X i and followed by in-clusion in i : X i → X of X i . Being equal as bimodules implies that also theirimages F ( X i , X i ) under the maps F ( e i , e j ) are equal, thus F ( X i , X j ) = G ( X i , X j )for all i, j = 1 , . . . , n . Since the functors F, G are additive, and every element in A decomposes uniquely as a direct sum of the X i ’s, this shows that F = G assubfunctors of E max . (3) Surjectivity: Let N ∈ Bim (B) be a sub-bimodule of B. As explained in theinjectivity part, this yields subgroups e j N e i of E ( X j , X i ) for all i, j = 1 , . . . , n .Setting F ( X j , X i ) := e j N e i allows then to define an additive sub-bifunctor F ∈ BiFun ( E max ) with Ev X ( F ) = N. (4) Morphism of posets: If F is a sub-bifunctor of F (cid:48) then F ( X, X ) is a sub-bimodule of F (cid:48) ( X, X ).(5) Morphism of lattices: The meet of
F, F (cid:48) in BiFun ( E max ) is given by inter-section ( F ∧ bf F (cid:48) )( C, A ) = F ( C, A ) ∩ Ab F (cid:48) ( C, A ) for any two objects
A, C of A .Applying to A = C = X yields the meet ( F ∧ bf F (cid:48) )( X, X ) in the lattice
Bim (B).We conclude that Ev X induces an isomorphism of lattices. (cid:3) Corollary 7.33. If A is an additively finite, Hom-finite Krull-Schmidt categorythen the three lattice structures we defined on Wex ( A ), BiFun ( A ) and Bim ( B )are isomorphic. Proof.
Combine 7.31 and 7.32. (cid:3)
N THE LATTICE OF WEAKLY EXACT STRUCTURES 40
The three small isomorphic lattices.
Theorem 7.34.
Let A be an additive category. The map Φ : E (cid:55)→
Ext E ( − , − )induces a lattice isomorphism between ( Ex ( A ) , ⊆ , ∩ , ∨ ) and CBiFun ( A ) , ≤ , ∧ , ∨ ). Proof.
Same as for Theorem 7.31. (cid:3)
Theorem 7.35. If A is an additively finite, Hom-finite Krull-Schmidt categorythen the two lattices ( CBiFun ( A ) , ≤ , ∧ , ∨ Cbf ) and (
Cbim ( B ) , ⊆ , ∩ , ∨ Cbim ) areisomorphic.
Proof.
As already verified in Theorem 7.32, the evaluation map Ev X preserves theorder and the meet-semi-lattice structure. But the join for closed sub-bimodulesis given by intersections on both sides, therefore Ev X also preserves the join-semi-lattice structure. (cid:3) Corollary 7.36. If A is an additively finite, Hom-finite Krull-Schmidt cate-gory then the three lattice structures defined above on Ex ( A ), CBiFun ( A ) and Cbim (B) are isomorphic.
Proof.
By 7.34 and 7.35. (cid:3)
General isomorphism of lattices.
Proposition 7.37.
Let ( A , W , s ) be a weakly extriangulated category. Then thereis a lattice isomophism between the lattice of additive sub-bifunctors of W andthe lattice of topologizing subcategories of def W . Proof.
The proof of [En20, Theorem B], with Step 3 removed, applies word forword. (cid:3)
Corollary 7.38.
Let W be a weakly exact structure on A . Then there is alattice isomorphism between the interval [ W add , W ] in the lattice of weakly exactstructures on A and the lattice of topologizing subcategories of def W . Corollary 7.39.
When the category A admits a unique maximal weakly exactstructure W max , the lattice of weakly exact structures on A is isomorphic to thelattice of topologizing subcategories of def W max . In particular we get the following summarising result:
Corollary 7.40.
Let A be an idempotent complete essentially small additivecategory, then the following four lattices are isomorphic: Wex ( A ) ∼ → BiFun ( A ) ∼ → Bim ( B ) ∼ → def E max . Proof.
It follows from 3.18, 7.33 and 7.39. (cid:3)
Note that when A is idempotent complete, we can use arguments from [En18,En19, FG20] instead. In particular, this approach would give another proof of theexistence of W max in this generality. N THE LATTICE OF WEAKLY EXACT STRUCTURES 41
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Bishop’s University, 2600 College Street, Sherbrooke, Qu´ebec J1M 1Z7Universit´e de Sherbrooke, 2500, boul. de l’Universit´e, Sherbrooke, Qu´ebec J1K2R1
N THE LATTICE OF WEAKLY EXACT STRUCTURES 43
Institute of Algebra and Number Theory, University of Stuttgart, Pfaffen-waldring 57, 70569 Stuttgart, Germany
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