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Grothendieck monoids of extriangulated categories

We study the Grothendieck monoid (a monoid version of the Grothendieck group) of an extriangulated category, and give some results which are new even for abelian categories. First, we classify Serre subcategories and dense 2-out-of-3 subcategories using the Grothendieck monoid. Second, in good situations, we show that the Grothendieck monoid of the localization of an extriangulated category is isomorphic to the natural quotient monoid of the original Grothendieck monoid. This includes the cases of the Serre quotient of an abelian category and the Verdier quotient of a triangulated category. As a concrete example, we introduce an intermediate subcategory of the derived category of an abelian category, which lies between the abelian category and its one shift. We show that intermediate subcategories bijectively correspond to torsionfree classes in the abelian category, and then compute the Grothendieck monoid of an intermediate subcategory.


Introduction
In the representation theory of algebras, we often consider extension-closed subcategories of a triangulated category which are neither abelian nor triangulated. An extriangulated category, introduced by Nakaoka-Palu [NP], is an appropriate framework to consider such a class of subcategories. Extriangulated categories unify both exact categories and triangulated categories, and we have the notion of conflations in an extriangulated category, which generalize conflations (short exact sequences) in an exact category and triangles in a triangulated category.
The Grothendieck group is the classical and basic invariant for both a triangulated category and an exact category. As a natural generalization, one can define the Grothendieck group K 0 (C) of an extriangulated category C as in [Hua, ZZ]. On the other hand, the Grothendieck monoid, a natural monoid version of the Grothendieck group, has been studied for an exact category as a more sophisticated invariant ( [BG, Eno, Sai]) than the Grothendieck group. The aim of this paper is to study the Grothendieck monoid of an extriangulated category.
Let C be an extriangulated category. The Grothendieck monoid M(C) can be naturally defined as the universal commutative monoid equipped with a function [−] : C → M(C) which is additive on each conflation. For a triangulated category, the Grothendieck monoid coincides with the Grothendieck group (Example 2.10). As a first use of the Grothendieck monoid, we show that Serre subcategories of an extriangulated category C can be classified purely combinatorially. Here a subcategory S of C is Serre if for any conflation X → Y → Z in C, we have Y ∈ S if and only if both X ∈ S and Z ∈ S.
Theorem A (= Proposition 3.5). Let C be a skeletally small extriangulated category. Then there is a bijection between the following two sets: (1) The set of Serre subcategories of C.
(2) The set of faces of M(C) Here a face F of a monoid M is a submonoid of M such that a + b ∈ F if and only if both a ∈ F and b ∈ F for any a, b ∈ M .
This generalizes [Sai] where C is assumed to be an exact category. We also establish a classification of dense 2-out-of-3 subcategories of C, that is, a subcategory D of C satisfying add D = C and the 2-out-of-3 property for conflations.
Theorem B (= Theorem 3.14, Corollary 3.17). Let C be a skeletally small extriangulated category. Then there is a bijection between the following sets: (1) The set of dense 2-out-of-3 subcategories of C.
(3) The set of subgroups of K 0 (C) such that ρ −1 (H) is cofinal in M(C), where ρ : M(C) → K 0 (C) is the natural map.
Applying Theorem B to a triangulated category, we immediately obtain Thomason's classification [Tho] of dense triangulated subcategories via subgroups of the Grothendieck group (Corollary 3.15). In [Mat], Matsui classified dense resolving subcategories of an exact category containing a generator via certain subgroups of the Grothendieck group (see also [ZZ]). Applying Theorem B to an exact category, we obtain a classification of all dense resolving subcategories, which generalizes [Mat, ZZ] (Corollary 3.20).
Next, we consider the behavior of the Grothendieck monoid with respect to an exact localization of Nakaoka-Ogawa-Sakai [NOS]. An exact localization C/N of an extriangulated category C by an extension-closed subcategory N of C is (if exists) the universal extriangulated category which sends N to 0 (Proposition 4.3). Typical examples are Serre quotients of abelian categories and Verdier quotients of triangulated categories. Under certain assumptions, where an exact localization exists, we show that this localization "commutes with the monoid quotients" as follows. where ι : N ֒→ C and Q : C → C/N are the inclusion and the localization functor respectively. In particular, M(C/N ) is isomorphic to the quotient monoid M(C)/ Im M(ι) (see Definition A.2). This can be applied to the following cases.
(i) C is a triangulated category and N is a thick subcategory of C. In this case, C/N is the usual Verdier quotient of a triangulated category.
(ii) C is a Frobenius category and N is the subcategory of all projective objects in C. In this case, C/N is the usual (triangulated) stable category. (iii) C is an abelian category and N is a Serre subcategory of C. In this case, C/N is the usual Serre quotient of an abelian category. Moreover, if N is a Serre subcategory (e.g. (iii)), then M(ι) : M(N ) → M(C) is an injection.
The above coequalizer diagram immediately yields the right exact sequence of the Grothendieck group (Corollary 4.32): This unifies the well-known results for the abelian case and the triangulated case. Moreover, the injectivity of M(N ) → M(C) for a Serre subcategory N is quite remarkable, since it fails for the Grothendieck groups (see e.g. Example 3.8).
Next, we address a categorification of a monoid localization, which makes certain elements of a monoid invertible (see Definition A.11). For this purpose, we study intermediate subcategories of the derived category in detail, which also gives a concrete example of our study described above for an extriangulated category which is neither abelian nor triangulated. Let A be a skeletally small abelian category and  (Remark 5.6). This result explains why it is natural to study Grothendieck monoids in the generality of extriangulated categories. The Grothendieck monoid of an exact category is reduced, that is, 0 is the only invertible element, while that of a triangulated category is a group, that is, every element is invertible. In particular, most of the monoid localizations of the Grothendieck monoid do not appear as the Grothendieck monoid if we only consider exact categories and triangulated categories. Thus, to realize a localization as the Grothendieck monoid, we have to consider extriangulated categories which are neither abelian nor triangulated.
Finally, we consider Serre subcategories of C := F [1] * A and the behavior of an exact localization of C in detail: Theorem E (= Proposition 5.10, Theorem 5.13). Let A be a skeletally small abelian category and F a torsionfree class in A, and put C := F [1] * A.
(1) The assignment S → F [1] * S gives a bijection between the set of Serre subcategories S of A containing F and the set of Serre subcategories of C.
(2) Let S be a Serre subcategory of A containing F . Then we can apply Theorem C to an exact localization C/(F [1] * S), and we have an exact equivalence A/S ≃ C/(F [1] * S) which makes the following diagram commute: Example F. Let k be a field and Q a quiver 1 ← 2 ← 3, and let A := mod kQ. Then the Auslander-Reiten quiver of D b (A) is as follows: · · · · · · Consider the torsionfree class F := add{1, 2 1 } in A. Then an intermediate subcategory C := F [1] * A is the additive closure of the gray region. By computing the monoid localization, we obtain M(C) ∼ = M(A) MF ∼ = Z ⊕ Z ⊕ N (see Example 5.9 for details). There are exactly two Serre subcategories of A containing F , namely, S := add{1, 2 1 , 2} and A. Therefore, C has exactly two Serre subcategories F [1] * S and C. For example, if we consider S, we have an exact equivalence A/S ≃ C/(F [1] * S), and by taking the Grothendieck monoid of (1.1), we obtain the following commutative diagram of monoids: Organization. In Section 2, we define and construct the Grothendieck monoid of an extriangulated category. In Section 3, we give classifications of some classes of subcategories of an extriangulated category using its Grothendieck monoid, including Theorems A and B. In Section 4, we discuss the relation between the exact localization of an extriangulated category and the monoid quotient, and prove Theorem C. In Section 5, we study intermediate subcategories of the derived category of an abelian category. In Section 6, we give some questions about the Grothendieck monoid. In Appendix A, we collect basic facts about commutative monoids which we shall use in this paper.
Conventions. Throughout this paper, all subcategories are full and closed under isomorphisms. By A ∼ = B for two objects A and B in a category C, we mean that A and B are isomorphic in C.
For a skeletally small category C, we denote by Iso C the set of all isomorphism classes of objects in C. For an object X in C, the isomorphism class of X is denoted by [X] ∈ Iso C.
For a subcategory D of an additive category C, we denote by add D the subcategory of C consisting of direct summands of direct sums of objects in D. This is the smallest additive subcategory of C closed under direct summands containing D.
We denote by Mor C the class of all morphisms in C. For a subclass S ⊆ Mor C and objects X, Y ∈ C, we denote by S(X, Y ) the class of all morphisms in S from X to Y . We simply write C(X, Y ) := (Mor C)(X, Y ).
By a monoid M we mean a commutative monoid with a unit, and we always use an additive notation: the operation is denoted by + with its unit 0. A homomorphism between monoids is a map which preserves both the addition and 0. We denote by N the monoid of non-negative integers. For a subset S of a monoid M , we denote by S N the submonoid of M generated by S. Similarly, S Z denotes the subgroup of M generated by S if M is an abelian group.
We denote by Mon the category of (commutative) monoids and monoid homomorphisms, and by Ab the category of abelian groups and group homomorphisms.
For a noetherian ring Λ, we denote by mod Λ the abelian category of finitely generated right Λ-modules.

The Grothendieck monoid of an extriangulated category
In this section, we first give basic definitions and properties about extriangulated categories. Next, we define the Grothendieck monoid M(C) of an extriangulated category C by the universal property, and then give a construction of M(C).
2.1. Preliminaries on extriangulated categories. In this paper, we omit the precise definitions and axioms of extriangulated categories. We refer the reader to [NP, LN] for the basics of extriangulated categories.
An extriangulated category (C, E, s) consists of the following data satisfying certain axioms: • C is an additive category.
• E is an additive bifunctor E : C op × C → Ab.
• s is a correspondence which associates to each δ ∈ E(Z, X) an equivalence class of com- For an extriangulated category (C, E, s), we call a complex X f − → Y g − → Z a conflation if its equivalence class is equal to s(δ) for some δ ∈ E(Z, X). In this case, we call f an inflation and g a deflation, and say that this complex realizes δ. We write this situation as follows: and we also call this diagram a conflation in C. In what follows, we often write C = (C, E, s) for an extriangulated category (C, E, s) for simplicity. Triangulated categories and Quillen's exact categories can be naturally considered as extriangulated categories as follows: (1) Let T be a triangulated category with shift functor Σ. Then by setting E(Z, X) := T (Z, ΣX), we may regard T as an extriangulated category. In this case, for each h ∈ (2) Let E be a skeletally small exact category. We have a functor Ext 1 is the set of equivalence classes of conflations (in the sense of exact categories) in E of the form 0 → X → Y → Z → 0. Then by setting E := Ext 1 E and s := id, we may regard E as an extriangulated category. Throughout this paper, we always regard triangulated categories and exact categories (in particular, abelian categories) as extriangulated categories in this way.
For later use, we fix some terminologies for extriangulated categories. Let C = (C, E, s) be an extriangulated category. For any δ ∈ E(Z, X) and any morphisms x ∈ C(X, X ′ ) and z ∈ C(Z ′ , Z), we denote E(Z, x)(δ) and E(z, X)(δ) briefly by x * δ and z * δ respectively.
We often write a morphism of conflations (x, y, z) as follows: For two subcategories C 1 and C 2 of an extriangulated category C, we denote by C 1 * C 2 the subcategory of C consisting of X ∈ C such that there is a conflation in C with C 1 ∈ C 1 and C 2 ∈ C 2 . We recall some properties of a subcategory D of an extriangulated category C.
• D is closed under finite direct sums if it contains a zero object of C and for any objects X, Y ∈ D, their direct sum X ⊕ Y in C belongs to D. We also say that D is an additive subcategory in this case.
We also say that D is an extension-closed subcategory in this case.
An extension-closed subcategory D of an extriangulated category C also has the structure of an extriangulated category induced from that of C, see [NP,Remark 2.18]. In this structure, a conflation X → Y → Z in D is precisely the conflation in C with X, Y , and Z in D. When we consider extension-closed subcategories of an extriangulated category, we always regard D as an extriangulated category in this way.
One of the main topics of this paper is the notion of Serre subcategories, which is a natural generalization of that in abelian categories.
Definition 2.1. Let C be an extriangulated category and D an additive subcategory of C. Then D is called a Serre subcategory if for any conflation X Y Z in C, we have that both X and Z are in D if and only if Y ∈ D.
In particular, a Serre subcategory is extension-closed, so it can be regarded as an extriangulated category in itself.
We also need the natural notion of functors between extriangulated categories, namely, an exact functor : Definition 2.2 ([BS, Definition 2.32]). Let (C i , E i , s i ) be an extriangulated category for i = 1, 2, 3.
(2) For two exact functors F 1 = (F 1 , φ 1 ) : C 1 → C 2 and F 2 = (F 2 , φ 2 ) : C 2 → C 3 , their composition is defined by (3) Let F = (F, φ) and G = (G, ψ) be two exact functors C 1 → C 2 . A natural transformation η : F ⇒ G of exact functors is a natural transformation of additive functors such that, for in C, the following is a morphism of conflations in D: (4) ETCat is the category of skeletally small extriangulated categories and exact functors.
Horizontal compositions and vertical compositions of natural transformations of exact functors are defined by those for natural transformations of additive functors. The compositions are also natural transformations of exact functors. In particular, we can define an exact equivalence of extriangulated categories in the same way as that of additive categories.
Remark 2.3. Let F = (F, φ) : C → D be an exact functor of extriangulated categories, and let (a, b, c) be a morphism of conflations in D by the naturality of φ.
Later, we need the following observation about exact functors to exact categories, which can be easily proved by considering the extriangulated structure on exact categories.
Lemma 2.4. Let C = (C, E, s) be an extriangulated category, E an exact category, and F : C → E an additive functor. Then the following are equivalent.
(2) For each conflation X Due to this fact, we call an additive functor F : C → E satisfying (2) above just an exact functor.

2.2.
Definition and construction of Grothendieck monoid. For a triangulated category or an exact category C, its Grothendieck group K 0 (C) is a basic invariant and has been studied and used in various areas. Recently, the Grothendieck monoid of an exact category, a natural monoid version of the Grothendieck group, has been introduced by Berenstein-Greenstein in [BG] to study Hall algebras, and its relation to the categorical property of an exact category was studied in [Eno]. Let C be an extriangulated category. We can naturally generalize these constructions to an extriangulated category C. The Grothendieck group K 0 (C) was formulated by Zhu-Zhuang [ZZ] and Haugland [Hua] and was used to classify certain subcategories of an extriangulated category. The aim of this paper is to define and investigate the Grothendieck monoid M(C) of C.
First, we define the Grothendieck monoid by the universal property. Recall that all monoids are commutative.
Definition 2.5. Let C be a skeletally small extriangulated category.
(1) We say that a map f : Iso C → M for a monoid M respects conflations if it satisfies the following conditions: which satisfies the following universal property: (a) π respects conflations in C. Since the Grothendieck monoid of C is defined by the universal property, we must show that it actually exists. For later use, we give an explicit construction of M(C).
Definition 2.6. Let C be an extriangulated category and A and B objects in C.
(1) A and B are conflation-related, abbreviated by c-related, if there is a conflation such that either of the following holds in C: In this case, we write A ∼ c B.
(2) A and B in C are conflation-equivalent, abbreviated by c-equivalent, if there are objects A 0 , A 1 , . . . , A n in C for some n ≥ 0 satisfying A 0 ∼ = A, A n ∼ = B, and A i ∼ c A i+1 for each i. In this case, we write A ≈ c B.
It is immediate that ∼ c and ≈ c induce binary relations ∼ c and ≈ c on the set Iso C by putting . Then clearly ≈ c is an equivalence relation on Iso C generated by ∼ c . On the other hand, Iso C can be regarded as a monoid with the addition given by [A] + [B] = [A ⊕ B]. Then we can state the following explicit construction of the Grothendieck monoid.
Proposition 2.7. Let C = (C, E, s) be a skeletally small extriangulated category. Then the following hold.
(1) ≈ c is a monoid congruence on Iso C (see Definition A.1 for a monoid congruence).
(2) The quotient monoid Iso C/≈ c together with the natural projection π : Iso C ։ Iso C/≈ c gives a Grothendieck monoid of C.
On the other hand, we have the following conflation: since s is an additive realization [NP,Definition 2.10]. Moreover, since conflations are closed under finite direct sums (because s is an additive realization), we obtain the following conflation: in Iso C. Therefore, in either case, we obtain (2) By (1), we obtain the quotient monoid Iso C/≈ c and the natural monoid homomorphism π : Iso C ։ Iso C/≈ c . We first show that π : Iso C ։ Iso C/≈ c respects conflations. Since π preserves the additive unit, we have Next, suppose that we have a map f : Iso C → M for a monoid M which respects conflations, and we will show that there is a unique monoid homomorphism f : Iso C/≈ c → M satisfying f = f π. The uniqueness is clear since π is surjective, so we only show the existence of such f . To obtain a well-defined map f : . Moreover, we have the following split conflation In what follows, we often identify M(C) with Iso C/≈ c and write [X] ∈ Iso C/≈ c to represent an element for X ∈ C.
The assignment C → M(C) actually gives a functor: Since the proof is straightforward by using the defining universal property of the Grothendieck monoid, we omit it.
Remark 2.9. The Grothendieck group K 0 (C) of an extriangulated category can be defined similarly by the universal property (see also [Hua, ZZ]). The relation between M(C) and K 0 (C) is as follows. To any commutative monoid M , we can associate its group completion gp M , which is the universal abelian group together with the monoid homomorphism M → gp M (see Definition A.11 and Proposition A.15). Then the defining properties of M(C) and K 0 (C) immediately show that K 0 (C) is isomorphic to gp M(C). Moreover, the group completion defines a functor gp : Mon → Ab, and we have a natural isomorphism of functors (1) Let E be an exact category. Then our M(E) (as an extriangulated category) coincides with the Grothendieck monoid of an exact category which is studied in [BG, Eno]. For example, [Eno,Theorem 4.12] implies that M(E) is a free monoid if and only if E satisfies the Jordan-Hölder-like property. In particular, if E is an abelian length category with n simple objects, then M(E) ∼ = N n holds.
(2) Let T be a triangulated category. Then M(T ) becomes a group, and hence M(T ) and K 0 (T ) coincide. Indeed, for every element [X] ∈ M(T ), there is a triangle This means that we have a conflation X → 0 → ΣX , which implies [X] + [ΣX] = 0 in M(T ). Therefore, every element in M(T ) is invertible, that is, M(T ) is a group. In general, the converse do not hold: there is an extriangulated category which is not triangulated but whose Grothendieck monoid is a group (see Corollary 5.5 for example).

Classifying subcategories via Grothendieck monoid
Throughout this section, C = (C, E, s) is a skeletally small extriangulated category. In this section, we give classifications of several subcategories of C via its Grothendieck monoid M(C). More precisely, we consider the following assignments: • For a subcategory D of C, we define a subset M D of M(C) by • For a subset N of M(C), we define a subcategory D N of C by In what follows, we will show that these maps give bijections between certain subcategories of C and certain subsets of M(C). We freely identify M(C) with Iso C/≈ c by Proposition 2.7.
3.1. Serre subcategories and faces. In this subsection, we establish a bijection between the set of Serre subcategories of C and the set of faces of M(C). Most results in this subsection are direct generalizations of [Sai]. However, we give more sophisticated proofs using c-equivalences and new results such as Proposition 3.4.
We also say that D is a c-closed subcategory for short.
The following lemma follows immediately from Proposition 2.7. We freely use this characterization in what follows.
Lemma 3.2. The following are equivalent for a subcategory D of C.
(2) For every X, Y ∈ C, if X ∼ c Y holds and X belongs to D, then Y also belongs to D.
(3) For every X, Y ∈ C, if [X] = [Y ] holds in M(C) and X belongs to D, then Y also belongs to D.
The relation between the subcategories D N defined above and c-closed subcategories is as follows. Note that a subcategory is not assumed to be additive.  (2) First, we prove that M DN = N holds for each subset N of M(C). Clearly, we have M DN ⊇ N . Take [X] ∈ M DN . Then there is an object Y ∈ D N such that [X] = [Y ] holds in M(C). Since D N is closed under c-equivalences by (1), we have that X is also in D N , that is, [X] ∈ N . Hence, we obtain M DN = N .
Next, we prove that D MD = D holds for a c-closed subcategory D of C. Clearly, we have D MD ⊇ D. Take X ∈ D MD , then we have [X] ∈ M D , so there exists an object Y ∈ D satisfying [X] = [Y ] in M(C). Since D is closed under c-equivalences, we obtain X ∈ D. Therefore, we obtain D MD = D, which completes the proof. Now we will consider Serre subcategories of an extriangulated category. Recall that a subcategory D of C is called a Serre subcategory if it is additive and for any conflation X → Y → Z in C, we have that X, Z ∈ D if and only if Y ∈ D. We denote by Serre C the set of Serre subcategories of C.
The following characterizes Serre subcategories using c-equivalences.  Proof. Let S be a Serre subcategory of C and F a face of M(C). We already know D MS = S and M DF = F by Propositions 3.3 and 3.4. Hence we only need to show that M S is a face and D F is a Serre subcategory.
We first prove that M S is a face. Note that S is closed under finite direct sums, direct summands, and c-equivalences by Proposition 3.4. It is clear that Next, we prove that D F is a Serre subcategory. It is obvious that D F is closed under finite direct sums since F is a submonoid of M(C). We already know that D F is c-closed by Proposition 3.3 (1). Thus it is enough to show that D F is closed under direct summands by Proposition 3.4. Let X and Y be objects and [Y ] belong to F , which implies both X and Y belong to D F . Therefore D F is a Serre subcategory of C. Example 3.6. Let A be a skeletally small abelian category. Then A can be regarded as an extension-closed subcategory of its bounded derived category D b (A). The natural inclusion functor by Example 2.10 (2). In fact, the composition M(A) → K 0 (A) of the above maps coincides with the group completion of M(A) (see Remark 2.9). Consider the polynomial ring k[T ] over a field k. Then the monoid homomorphism T In spite of this example, if we consider Serre subcategories, then the natural monoid homomorphism is injective: Proposition 3.7. Let S be a Serre subcategory of C and ι : S → C the inclusion functor. Then the monoid homomorphism This injectivity is remarkable since it is false for Grothendieck groups and one has to consider the higher K-group K 1 to deal with its failure.

3.2.
Dense 2-out-of-3 subcategories and cofinal subgroups. In this subsection, we give classifications of dense 2-out-of-3 subcategories of C in terms of M(C) and K 0 (C), which generalize Thomason's classification of dense triangulated subcategories of a triangulated category [Tho] in terms of K 0 (C).
Definition 3.9. Let D be an additive subcategory of C.
(1) D is a dense subcategory if add D = C holds, that is, for every C ∈ C, there is some C ′ ∈ C satisfying C ⊕ C ′ ∈ D.
(2) D is a 2-out-of-3 subcategory if it satisfies 2-out-of-3 for conflations, that is, if two of three objects X, Y, Z in a conflation X → Y → Z belong to D, then so does the third.
We note that 2-out-of-3 subcategories closed under direct summands are called thick subcategories, see Definition 4.2.
Example 3.10. Let T be a triangulated subcategory. Then a subcategory of T is a 2-out-of-3 subcategory if and only if it is a triangulated subcategory (see e.g. [Tho, 1.1]).
. We will freely use this property in what follows.
We can relax the 2-out-of-3 condition of dense 2-out-of-3 subcategories by the following observation.
Proposition 3.12 ([ZZ, Lemma 5.5]). Let D be a dense additive subcategory of C. Then the following are equivalent.
(1) For every conflation X → Y → Z in C, if X and Y belong to D, then so does Z.
(2) For every conflation X → Y → Z in C, if Y and Z belong to D, then so does X.
A key observation in this subsection is as follows.
Proposition 3.13. Let D be a dense 2-out-of-3 subcategory of C. Then D is closed under cequivalences.
Proof. Take any conflation 2) in C. It suffices to show that Y belongs to D if and only if so does X ⊕ Z.
First, suppose that Y belongs to D. Since D is dense, there is some W ∈ C satisfying Y ⊕Z⊕W ∈ D. By taking the direct sum of (3.2) and a split conflation Z → Z ⊕ W → W , we obtain the following conflation.
Conversely, suppose that X ⊕ Z belongs to D. By taking the direct sum of (3.2) and a split conflation Z → Z ⊕ X → X , we obtain the following conflation: Now we can state the following classification of dense 2-out-of-3 subcategories.
Theorem 3.14. Let C be a skeletally small extriangulated category. Then the bijection in Proposition 3.3 induces a bijection between the following two sets.
• The set of dense 2-out-of-3 subcategories of C.
• The set of cofinal subtractive submonoids of M(C) (see Definition A.18).
Proof. Due to Proposition 3.3 (2) and Proposition 3.13, we only have to check the following welldefinedness of maps: (1) M D is a cofinal subtractive submonoid for a dense 2-out-of-3 subcategory D.
(2) D N is a dense 2-out-of-3 subcategory of C for a cofinal subtractive submonoid N .
Next, to show that M D is subtractive, suppose that x + y and x belong to M D . Take X, Y ∈ C satisfying [X] = x and [Y ] = y. Then [X ⊕ Y ] and [X] belong to M D . Since D is c-closed by Proposition 3.13, we have D = D MD by Proposition 3.3. Therefore, X ⊕ Y and X belong to D. As a corollary, we can immediately deduce the following classification of dense triangulated subcategories.
Corollary 3.15 ([Tho, Theorem 2.1]). Let T be a skeletally small triangulated category. Then there exists a bijection between the following two sets: • The set of dense triangulated subcategories of T .
• The set of subgroups of K 0 (T ).
Proof. Since M(T ) ∼ = K 0 (T ) holds by Example 2.10 (2) and dense triangulated subcategories of T are precisely dense 2-out-of-3 subcategories, we only have to check that a subset of K 0 (T ) is a cofinal subtractive submonoid if and only if it is a subgroup. This follows from Remark A.19.
Due to Theorem 3.14, the classification of dense 2-out-of-3 subcategories can be reduced to the classification of cofinal subtractive submonoids of the Grothendieck monoid. Since it is easier to deal with subgroups of a group than with submonoids of a monoid, we study the relation between cofinal subtractive submonoids of a monoid and subgroups of its group completion. The main observation can be summarized as follows: Proposition 3.16. Let M be a monoid and ρ : M → gp M its group completion. Consider the following two maps: Then the following hold. Proof. It is clear that Ψ(H) is a subtractive submonoid of M for every subgroup H of G.
(2) Let N be a cofinal subtractive submonoid of M , and put H := ρ(N ) Z . It suffices to show ρ −1 (H) ⊆ N by (1). Suppose x ∈ ρ −1 (H). Then since ρ(x) ∈ H = ρ(N ) Z and N is a submonoid of M , there are elements n 1 and n 2 in N such that ρ(x) = ρ(n 1 ) − ρ(n 2 ), so ρ(x + n 2 ) = ρ(n 1 ). Therefore, there is an element m ∈ M such that x + n 2 + m = n 1 + m in M (see the argument below Definition A.11). Then, since N is cofinal in M , there is m ′ ∈ M satisfying m + m ′ ∈ N . Then we have x + (n 2 + m + m ′ ) = n 1 + m + m ′ . Now x ∈ N follows since N is subtractive and x + (n 2 + m + m ′ ) and n 2 + m + m ′ belong to N . ( Therefore, by (2) and the above argument, we only have to check that Φ and Ψ are well-defined on these two sets. This is immediate from (2) and definitions.
(4) Every subtractive submonoid containing S is cofinal, and every subgroup H of gp M containing ρ(S) satisfies that ρ −1 (H) is cofinal by S ⊆ ρ −1 (H). Therefore, the two sets in (4) are subsets of the corresponding two sets in (3). Thus it suffices to observe that Φ and Ψ are well-defined on the two sets in (4), which is immediate from definitions.
By combining this with Theorem 3.14, we immediately obtain the following result.
Corollary 3.17. Let C be a skeletally small extriangulated category. Then there is a bijection between the following two sets, where ρ : M(C) → K 0 (C) is the group completion.
• The set of dense 2-out-of-3 subcategories of C.
• The set of subgroups H of K 0 (C) such that ρ −1 (H) is cofinal in M(C).
Using this observation, we can obtain all dense 2-out-of-3 subcategories in an abelian length category with finitely many simples. First, recall the following description of the Grothendieck monoid. Let A be an abelian length category, that is, an abelian category such that every object has a composition series. Suppose that {S 1 , . . . S n } is the set of all non-isomorphic simple objects in A. Then for C ∈ A, define dim C := (x 1 , . . . , x n ) ∈ N n , where x i is the multiplicity of S i in the composition series of C (this is well-defined due to the Jordan-Hölder theorem). Then dim respects conflations, and moreover, it induces the following isomorphisms of monoids and groups: where ρ is the group completion and ι is the natural inclusion.
Corollary 3.18. Let A be an abelian length category with n simple objects up to isomorphism. Then there are bijections between the following two sets: • The set of dense 2-out-of-3 subcategories C of A.
• The set of subgroups H of Z n containing a strictly positive element.
Here an element Proof. By Corollary 3.17 and the isomorphisms in (3.5), we only have to observe that N n ∩ H is cofinal in N n if and only if H contains a strictly positive element for a subgroup H of Z n . This is straightforward, so we omit it.
Certain classes of dense 2-out-of-3 subcategories were classified via the Grothendieck group in [Mat] (for the exact case) and [ZZ] (for the extriangulated case). We explain that their results can be immediately deduced from ours. Let us explain some terminology to state them.
Definition 3.19. Let C be an extriangulated category. Then a set G of objects in C is called a generator if for every C ∈ C there is a conflation X → G → C in C with G ∈ G.
Now we can deduce their results as follows.
Corollary 3.20 ([Mat, Theorem 2.7], [ZZ,Theorem 5.7]). Let C be a skeletally small extriangulated category and G a generator of C. Then there is a bijection between the following two sets: (i) The set of dense 2-out-of-3 subcategories of C containing G.
(ii) The set of subgroups of K 0 (C) containing the image of G.
Proof. We will show that these sets are in bijection with the following one: (iii) The set of subtractive submonoids of M(C) containing the image of G.
. Therefore, by Proposition 3.16 (4), we have a bijection between (ii) and (iii), and every submonoid in (iii) is cofinal. Therefore, (i) and (iii) are subsets of the two sets in Theorem 3.14.
Hence it suffices to observe the following well-definedness, which are immediate from definitions: If D is a dense 2-out-of-3 subcategory containing G, then M D contains [G], and if N is a submonoid of M(C) containing [G], then D N contains G.
Remark 3.21. In [Hua,Theorem 5.1], the above classification of dense 2-out-of-3 subcategories via the Grothendieck group was generalized to an n-exangulated category, which is a generalization of extriangulated category. Therefore, it may be possible to establish classifications similar to this paper for an n-exangulated categories using the Grothendieck monoid of an n-exangulated category.

Grothendieck monoids and localization of extriangulated categories
The purpose of this section is to prove Theorem 4.17, which describes the Grothendieck monoid of the localization of extriangulated categories as a monoid quotient of the Grothendieck monoid. Throughout this section, we assume that every category, functor, and subcategory is additive.
4.1. Localization of extriangulated categories. We first recall the localization of an extriangulated category following [NOS].
Definition 4.1. Let C be an extriangulated category, and let S ⊆ Mor C be a class of morphisms. A pair (C S , Q) of an extriangulated category C S and an exact functor Q : C → C S is the exact localization of C with respect to S if it satisfies the following conditions: (i) F (s) is an isomorphism in C S for any s ∈ S.
(ii) For any extriangulated category D and any exact functor F : C → D such that F (s) is an isomorphism for any s ∈ S, there exists a unique exact functor F S : If the exact localization exists, it is unique up to exact isomorphisms. Note that the exact localization is closed under exact isomorphisms, but not closed under exact equivalences, see Remark 4.7 below.
Nakaoka-Ogawa-Sakai [NOS] constructs the exact localization of an extriangulated category by a class of morphisms under some assumptions. We only recall the construction of the localization of an extriangulated category by the class of morphisms determined by a thick subcategory, as we shall explain.
From now on, C = (C, E, s) is an extriangulated category.
Definition 4.2. A subcategory N of C is a thick subcategory if it satisfies the following conditions: For a thick subcategory N ⊆ C, we set the following classes of morphisms: We define S N to be the class of all finite compositions of morphisms in L and R. We can easily check L•L ⊆ L and R•R ⊆ R. Thus a morphism s in S N is of the form s = · · · ℓ n−1 r n ℓ n+1 r n+2 · · · for some ℓ i ∈ L and r j ∈ R. The thick subcategory N can be recovered from S N since we have by [NOS,Lemma 4.5]. In the following, we consider the exact localization C/N := C SN of C with respect to S N . This localization satisfies the following natural universal property. Proof. By comparing the claimed properties with Definition 4.1, it suffices to show the following claim: for an extriangulated category D and an exact functor F : C → D, we have that F (N ) ∼ = 0 holds for every N ∈ N if and only if F (s) is an isomorphism for every s ∈ S N . To see the "only if" part, suppose that F (N ) ∼ = 0 holds for every N ∈ N . It suffices to check that F (ℓ) and F (r) are isomorphisms for ℓ ∈ L and r ∈ R respectively. By the definition of L, Since F is an exact functor, we obtain a conflation F (A) is an isomorphism in D by considering the associated long exact sequence (cf. [NP,Corollary 3.12]). The same proof applies to r ∈ R. To see the "if" part, suppose that F (s) is an isomorphism for every s ∈ S N , and let N ∈ N . Then 0 → N clearly belongs to L ⊆ S N , so 0 → F (N ) is an isomorphism in D. Thus the assertion holds.
Let C := C/[N ] be the quotient by the ideal [N ] consisting of morphisms which factor through objects in N , and let p : C → C be the canonical functor. In what follows, we write f := p(f ) for any morphism f in C. Set S N := p(S N ) 1 . We recall a condition in [NOS] under which the exact localization C/N = (C/N , Q) exists.
Condition 4.4. Let N be a thick subcategory of C.
(i) f ∈ S N holds for any split monomorphism f : A → B in C such that f is an isomorphism in C. (This is equivalent to the dual condition by [NOS,Lemma 3.2]: f ∈ S N holds for any split epimorphism f : A → B in C such that f is an isomorphism in C.) (ii) S N satisfies 2-out-of-3 with respect to compositions in C. (iii) S N is a multiplicative system in C.
(iv) The set txs | x is an inflation in C and s, t ∈ S N is closed under compositions. Dually, the set tys | y is a deflation in C and s, t ∈ S N is closed under compositions. (1) C/N is constructed as the category S N −1 C of fractions. In particular, every morphism in C/N can be described as a right or left roof of morphisms in C. Remark 4.7. Some readers may find the definition of exact localizations unsatisfactory since it is not preserved by exact equivalences. In fact, there is a notion of exact 2-localizations, which is preserved by exact equivalences. For two extriangulated categories C and D, we denote by Fun ex (C, D) the category of exact functors C → D and natural transformations of them. Any exact functor F : C → C ′ induces a functor F * : Fun ex (C ′ , D) → Fun ex (C, D) defined by F * (G) := G • F . For a class S ⊆ Mor C of morphisms in an extriangulated category C, the exact 2-localization of C with respect to S is a pair (C S , Q) of an extriangulated category C S and an exact functor Q : C → C S which satisfies the following conditions: (i) F (s) is an isomorphism in C S for any s ∈ S.
(ii) For any extriangulated category D and an exact functor F : C → D such that F (s) is an isomorphism for any s ∈ S, there exist an exact functor F : C S → D and a natural isomorphism F ∼ = F • Q of exact functors. (iii) The functor Q * : Fun ex (C S , D) → Fun ex (C, D) is fully faithful for every extriangulated category D. Actually, the exact localization C/N obtained by Fact 4.5 is also the exact 2-localization by [NOS,Theorem 3.5].
Although Condition 4.4 seems a little technical, there is a sufficient condition which we can easily verify, see Fact 4.12 and 4.13. In this paper, we focus on the situation where Condition 4.4 is satisfied so that we can use the additional properties (1)-(2) of Fact 4.5 freely.
Remark 4.8. We do not assume in Condition 4.4 that S N is a multiplicative system. However, any morphism f : X → Y in C/N is of the form f = Q(s) −1 Q(g) for some morphisms g : X → A in C and s : Y → A in S N . Indeed, let Q : C → S N −1 C = C/N be the canonical functor. For any holds for any morphism φ in C, the claim follows.
We introduce some convenient classes of thick subcategories.
Definition 4.9. Let N be a thick subcategory of C.
(1) N is called biresolving if for any C ∈ C, there exist an inflation C → N and a deflation (2) N is called percolating if for any morphism f : X → Y in C factoring through some object in N , there exist a deflation g : X → N and an inflation h : N → Y satisfying N ∈ N and f = hg 2 .
For the triangulated case, we have the following observation.
Example 4.10. Let T be a triangulated category with shift functor Σ.
(1) A thick subcategory of T in the sense of Definition 4.2 coincides with the usual one, that is, a subcategory of T closed under cones, shifts, and direct summands. We can easily check it by considering the following conflations: (2) Any thick subcategory of T is biresolving because there exist conflations C (3) Similarly, any thick subcategory of T is percolating because every morphism is both an inflation and a deflation in T .
Typical examples of percolating subcategories are Serre subcategories of admissible extriangulated categories, as we shall explain. A morphism f : A → B in C is called admissible if it has a factorization f = i • d such that i is an inflation and d is a deflation. We call this factorization a deflation-inflation factorization. We also say that C is admissible if every morphism in C is admissible. For examples, abelian categories and triangulated categories are admissible.
Example 4.11. Let C be an admissible extriangulated category. Then every Serre subcategory N of C is percolating. Indeed, let f : X → Y be a morphism in C having a factorization X It is obvious that Serre subcategories are thick, and thus N is percolating.
The following two facts are useful conditions where Condition 4.4 is satisfied for biresolving and percolating subcategories.  (EL2) For any split monomorphism f : A → B in C such that f is an isomorphism in C, there exist N ∈ N and j : in C, both of the following hold: (EL4) C is admissible, namely, every morphism f : A → B in C has a factorization f = i • d such that i is an inflation and d is a deflation. Then the following statements hold.
The situation can be summarized as follows.
(1) Consider the following condition: (WIC) Let h = gf be a morphism in C. If h is an inflation, then so is f . Dually, if h is a deflation, then so is g. Then (WIC) implies (EL2) by [NOS,Remark 4.31 (2)]. A triangulated category satisfies (WIC) since every morphism is both an inflation and a deflation. More generally, an extension-closed subcategory of a triangulated category which is closed under direct summands satisfies (WIC), see Lemma 4.16 below.
(3) Under the condition (EL1), the condition (EL3) is equivalent to that f (resp. g) is a monomorphism (resp. an epimorphism) in C for every conflation A (1) Let T be a triangulated category. A thick subcategory N of T is biresolving as mentioned in Example 4.10, so the exact localization T /N exists and becomes a triangulated category by Fact 4.12. This coincides with the Verdier quotient of T by N .
(2) Let F be a Frobenius exact category, and let N be the subcategory of projective objects in F . Then N is a biresolving subcategory of F . In this case, the exact localization F /N coincides with the usual stable (triangulated) category F = F /[N ]. (3) Let A be an abelian category. A Serre subcategory S of A is percolating as mentioned in Example 4.11. Moreover, it satisfies (EL1)-(EL4) in Fact 4.13. Indeed, (EL2) is satisfied by Remark 4.14 (1) since A satisfies (WIC), and (EL3) is satisfied since every inflation and deflation in A is a monomorphism and an epimorphism respectively. Therefore, the exact localization A/S exists and becomes an abelian category by Fact 4.13. This coincides with the Serre quotient of A by S.
We have the following convenient criterion for the condition (WIC).
Lemma 4.16. Let C be an extension-closed subcategory of a triangulated category T with shift functor Σ. If C is closed under direct summands in T , then it satisfies (WIC).
Proof. Let f : A → B and g : B → C be morphisms in C. Suppose that h := gf is an inflation,

and thus there is a conflation
in T by taking cone of f . Applying [LN,Proposition 1.20] (see also [Nee,Proposition 1.4.3]) to T , we obtain a morphism x : X → D in T which gives a morphism of conflations Since B, D ∈ C and C is extension-closed, we have X ⊕ C ∈ C, which implies X ∈ C because C is closed under direct summands. Hence f is an inflation in C. We can dually prove that, if h is a deflation, then so is g, so we omit it.

4.2.
The Grothendieck monoid of the localization of an extriangulated category. In this subsection, we investigate the relations between Grothendieck monoids and the exact localization. Throughout this subsection, C is an extriangulated category and N is a thick subcategory of C.
The following is the main theorem of this section. We refer the reader to Definition A.2 for the quotient monoid by a submonoid.
Theorem 4.17. Let C be a skeletally small extriangulated category and N its thick subcategory. Suppose that the following two conditions are satisfied: (i) Condition 4.4 is satisfied. Thus the exact localization Q : C → C/N exists, and we can freely use Fact 4.5. We first prove this theorem, and then discuss the condition (ii) of this theorem.
Lemma 4.18. Assume the same conditions as in Theorem 4.17. Let X and Y be objects in C.
Proof. Let f : X ∼ −→ Y be an isomorphism in C/N . We have morphisms g : X → A in C and s : Y → A in S N such that f = Q(s) −1 Q(g) by Remark 4.8. Then Q(g) is also an isomorphism in C/N , and thus g ∈ S N because S N is saturated. Therefore it is enough to show that [X] ≡ [Y ] mod M N if there is a morphism s : X → Y in S N . Since a morphism in S N is a finite composition of morphisms in L and R, we may assume that s ∈ L or s ∈ R. If s ∈ L, then there is a conflation X in C/N . Thus, by Lemma 4.18, we have This proves the injectivity of φ.
In the following, we discuss the kernel of the localization functor Q : C → C/N and study when S N is saturated. The authors would like to thank Hiroyuki Nakaoka for sharing the results on the kernel of the localization functor. We prepare some lemmas. Proof. The morphism u is a split monomorphism in C since so is u in C. Then the split monomorphism A u − → N ∼ = 0 should be an isomorphism in C. Thus (i) of Condition 4.4 implies that u : A → N belongs to S N . By composing this with N → 0, which is in S N , we obtain that A → 0 is in S N . Dually, 0 → A is also in S N by considering a retraction q : N → A of u. By (4.1), we conclude that A ∈ N .
Lemma 4.20. Suppose (i) of Condition 4.4. Then the following hold.
In each case, all objects in the above conflations belong to N since N is a thick subcategory.
Proof. We only prove (1) since (2) follows dually. The assumption r = 0 implies that it has a Then there is a morphism of conflations The top row implies M ∈ N because N is extension-closed. We can choose b : M → Y so that − −−−− → Z is a conflation by [LN,Proposition 1.20]. This implies that the following diagram is a weak pullback diagram, and hence a section φ : Y → M of b is induced: Since M ∈ N , we obtain Y ∈ N by Lemma 4.19.
Set L := p(L) and R := p(R). We have the following interpretation of (EL3) for the case of percolating subcategories.  Proof. Since (2) is dual of (1), we only prove (1). To show the "only if" part, suppose that f is a Then we have f xℓ = idℓ = 0, and by the assumption, we conclude that xℓ = 0 in C. Since ℓ ∈ L is an epimorphism in C by [NOS,Lemma 4.7 (2)], we have x = 0. This proves that f is a monomorphism. (i) N is a biresolving subcategory.

Then any morphism in S N is both a monomorphism and an epimorphism in C.
Proof. By [NOS,Lemma 4.7], every morphism in L (resp. R) is an epimorphism (resp. a monomorphism) in C. Thus it suffices to show that every morphism in L is a monomorphism in each case, since the proof for R follows dually.
(ii) In this case, N is a percolating subcategory by (EL1). Thus the assertion follows from (EL3) and Lemma 4.21, since every morphism in L is an inflation in C by definition. Now we can show that the kernel of C → C/N coincides with N under some conditions. Recall that a kernel Ker F of an additive functor F : C → D consists of objects C ∈ C satisfying F (C) ∼ = 0 in D. Proof. We have Ker(Q : C → C/N ) ⊇ N by Proposition 4.3. Conversely, suppose that X ∈ C satisfies Q(X) ∼ = 0 in C/N . Then there exists Y ∈ C satisfying 0 ∈ S N (X, Y ) since C/N is constructed as the category of fractions S N −1 C by Fact 4.5. Hence we obtain s ∈ S N (X, Y ) satisfying s = 0 in C. By the construction of S N , we can write either s = tℓ or s = ur for some t, u ∈ S N , ℓ ∈ L, and r ∈ R. We consider the case s = tℓ. Since t is a monomorphism in C by the assumption, s = tℓ = 0 implies ℓ = 0. Thus Lemma 4.20 (2) implies X ∈ N . The case s = ur can be proved similarly using Lemma 4.20 (1).
As a consequence, we can describe the kernel of the localization for the cases we are interested in. Proof. For each case, Condition 4.4 is satisfied by Fact 4.12 and 4.13, and every morphism in S N is a monomorphism by Corollary 4.22. Thus the assertion follows from Proposition 4.23.
Using this result, we next consider whether S N is saturated. Assume Condition 4.4, and let Q : C → C/N be the localization functor. Recall that S N is saturated if f ∈ S N holds for any morphism f in C such that Q(f ) is an isomorphism in C/N . Proposition 4.25. If N is a biresolving subcategory of C, then S N is saturated.
We can choose a morphism g : N → M so that is a conflation by [LN,Proposition 1.20 Thus it is enough to show that [ x f ] belongs to L, that is, M ∈ N . Applying Q to (4.2), we have a morphism of conflation in C/N by Remark 2.3. Since Q(f ) is an isomorphism by the assumption, Q(g) is an isomorphism in C/N by [NP,Corollary 3.6]. Thus we obtain M ∼ = N ∼ = 0 in C/N , which shows M ∈ Ker Q. Since Ker Q = N holds by Corollary 4.24, we obtain M ∈ N . This proves [ x f ] ∈ L by (4.3).
Unfortunately, for the percolating case, the authors do not know whether the condition (EL1)-(EL3) implies that S N is saturated, although we have the following criterion. (1) S N is saturated.
(2) For a morphism f : X → Y in C, if Q(f ) is an isomorphism in C/N , then f is admissible in C, that is, there exists a deflation-inflation factorization of f . Proof.
(1) ⇒ (2): We have S N = L • R by [NOS,Lemma 4.37]. Hence if Q(f ) is an isomorphism in C/N , then f ∈ S N = L • R since S N is saturated, which proves (2) since a morphism in L (resp. R) is an inflation (resp. a deflation).
(2) ⇒ (1): Let f : X → Y be a morphism in C such that Q(f ) is an isomorphism in C/N . There (2). In particular, we have conflations Since Q is an exact functor, we obtain conflations is an epimorphism because C/N is an exact category by Fact 4.13. On the other hand, Q(d) is a split monomorphism since Q(f ) = Q(i) • Q(d) is an isomorphism. Thus Q(d) is an isomorphism in C/N . Dually, Q(i) is also an isomorphism in C/N . Hence both Q(K) and Q(C) are isomorphic to 0 in C/N , which implies K, C ∈ Ker Q = N by Corollary 4.24. We conclude that d ∈ R and i ∈ L, and hence f = i • d ∈ S N . Therefore, if one assumes in addition that (EL4) holds, that is, C is admissible, then we can show the saturatedness of S N . We can summarize our results of the saturatedness as follows. (ii) Since N satisfies (EL1)-(EL3), we can apply Proposition 4.26. In addition, every morphism in C is admissible by (EL4), and hence the condition in Proposition 4.26 (2) automatically holds. Therefore, S N is saturated.
As a consequence, we can deduce the following result for the cases we are interested in. Proof. This follows from Theorem 4.17 and Corollary 4.27.
Remark 4.29. Ogawa [Oga] developed localization theory of triangulated categories with respect to an extension-closed subcategory. We briefly introduce his results and explain the relation to ours. Let T be a triangulated category and N an extension-closed subcategory of T . We consider T as the extriangulated category (T , E, s). Then there is a subfunctor E N ⊆ E satisfying the following: First, by applying this to the abelian case, we obtain the following consequence on the Serre quotient of an abelian category. Proof.
(2) Example 4.15 (3) shows that S satisfies (EL1)-(EL4). Therefore, we can apply Corollary 4.28 to this setting. This may be seen as a "short exact sequence" of monoids: This description of M(A/S) gives the following description of Serre subcategories of A/S, which seems to be a folklore.

Corollary 4.31. Let A be a skeletally small abelian category and S a Serre subcategory of A.
Then there is a bijection between the following two sets: (1) Serre(A/S).

Proof. There is a bijection between
It is well-known that the similar sequence to (4.4) for the Grothendieck group is only right exact. Actually, we can deduce it using our result as follows.
Corollary 4.32. Suppose that Condition 4.4 are satisfied and S N is saturated (e.g. the assumption in Corollary 4.27 holds). Then the following sequence is exact: where ι : N ֒→ C is the inclusion functor and Q : C → C/N is the exact localization functor.
Proof. The diagram is a coequalizer diagram in Mon by Theorem 4.17 (see also Proposition A.3). Since the group completion functor gp : Mon → Ab is a left adjoint of the forgetful functor Ab ֒→ Mon by Proposition A.16, it preserves colimits. Hence we have a coequalizer diagram in Ab because gp •M = K 0 by Remark 2.9. This is nothing but the claimed exact sequence.

Intermediate subcategories of the derived category
In this section, we give a concrete example of an extriangulated category and its thick subcategory which satisfies the condition (EL1)-(EL4) in Fact 4.13, intermediate subcategories of the derived category, and compute its Grothendieck monoid and the exact localization.
Throughout this section, A denotes a skeletally small abelian category, and we assume that every category, functor, and subcategory is additive. We denote by D b (A) the bounded derived category of A, which we regard as an extriangulated category. We also denote by H i : D b (A) → A the i-th cohomology functor. We often identify A with the essential image of the canonical embedding The following observation is useful throughout this section, and can be proved easily by using the truncation functor.
Lemma 5.1. Let A be a skeletally small abelian category and B and B ′ subcategories of A.
(1) We have the following equality in D b (A): (2) For every X ∈ A[1] * A, we have the following conflation in D b (A), which is natural on X: Moreover, every conflation A[1] → X → B with A, B ∈ A is isomorphic to the above conflation.
complex concentrated in degree 0 and −1.

Classification of intermediate subcategories. Let us introduce the main topic of this section.
Definition 5.2. Let A be a skeletally small abelian category. An intermediate subcategory of is a subcategory C satisfying the following conditions: (1) A ⊆ C ⊆ A[1] * A holds.
(2) C is closed under extensions in D b (A).
(3) C is closed under direct summands in D b (A).
By (2) First, we will see that intermediate subcategories can be described using torsionfree classes in A. Here a subcategory F of A is called a torsionfree class if F is closed under subobjects and extensions. Note that torsionfree classes do not necessarily correspond to torsion classes and torsion pairs in general.
Theorem 5.3. Let A be a skeletally small abelian category. Then the following hold. ( It clearly suffices to show that H −1 (Y ) ∈ F . Since H is a cohomological functor, we have the following exact sequence in A: Since we have H −1 (X), H −1 (Z) ∈ F and F is closed under subobjects and extensions, we obtain (2) Suppose that C is an intermediate subcategory of Take any C ∈ C and we will see H −1 (C)[1] ∈ C. By Lemma 5.1 (3), we may assume that C = C • is a complex concentrated in degree 0 and −1, that is, C is of the form C = [· · · → 0 → C −1 → C 0 → 0 → · · · ]. Lemma 5.1 (2) gives the following triangle: On the other hand, define the cochain map g : C 0 → C by the following: Then by consider the mapping cocone of [f, g] : H −1 (C)[1] ⊕ C 0 → C, we obtain the following triangle in D b (A): We claim that A belongs to A. To show that, consider the cohomology long exact sequence induced from the above triangle: On the other hand, f is isomorphic to H −1 (f ), which is an isomorphism. Hence H −1 (A) = 0 holds. Moreover, g is isomorphic to H 0 (g) : C 0 → H 0 (C), which is surjective. Hence H 1 (A) = 0 holds. Therefore, we have shown that H i (A) = 0 for i = 0. Thus A ∈ A holds. It follows that H −1 (C)[1] ⊕ C 0 belongs to C, since A ⊆ C and C is closed under extensions. Therefore, we obtain H −1 (C)[1] ∈ C because C is closed under direct summands. Thus we have shown that C = H −1 (C)[1] * A holds.
We claim that H −1 (C) is a torsionfree class in A. Put F := H −1 (C) for simplicity, and then we have C = F [1] * A by the above argument. Suppose that we have a short exact sequence . If X and Z belong to F , then X[1] and Z[1] belongs to F [1] ⊆ C. Therefore, the conflation Similarly, suppose that Y belongs to F . Then we have a conflation in D b (A), and we have Z ∈ A ⊆ C and Y [1] ∈ F [1] ⊆ C. Therefore, X[1] ∈ C holds since C is closed under extensions. Hence we obtain X = H −1 (X[1]) ∈ H −1 (C) = F . Therefore, F is a torsionfree class in A.  We first check the uniqueness. Suppose that there is such a map ϕ. Let C ∈ C be any object.
Since ϕ is a monoid homomorphism, it preserves the inverse. Therefore, we must have Therefore, the uniqueness of ϕ holds. Next, we will construct ϕ. Consider the following map ψ : Iso C → M : Note that ϕ[H −1 (X)] has an inverse in M since H −1 (X) ∈ F . We will show that ψ respects conflations in C. Clearly ψ[0] = 0 holds. Take any conflation Since this is a triangle in D b (A), we obtain the following long exact sequence in A: By decomposing this long exact sequence into short exact sequences in A, we can easily obtain the equality in M(A). Therefore, by applying ϕ, we obtain As an immediate consequence, we obtain the following example of an extriangulated category whose Grothendieck monoid is a group: Next, we describe the Grothendieck monoid of an intermediate subcategory of D b (A) when A is an abelian length category with finitely many simples. Recall that an abelian length category is a skeletally small abelian category such that every object has a finite length. For an abelian length category A, we denote by sim A the set of isomorphism classes of simple objects in A. The following observation on the localization of a monoid can be easily checked.
Lemma 5.7. Let M and N be monoids. Then the localization of M ⊕ N with respect to M is isomorphic to (gp M ) ⊕ N .
Corollary 5.8. Let A be an abelian length category and F a torsionfree class in A. Define sim F A as follows: ∈ sim A | S appears as a composition factor of a some object in F }.

Proof. Since
Then Lemma 5.7 immediately deduces the assertion. .
On Proposition 5.10. Let A be a skeletally small abelian category and F a torsionfree class in A. Then there is a bijection between the following two sets.
The maps are given as follows: for D in (1), we consider A∩D in (2), and for S in (2)  Im δ H 0 (X) Coker δ 0.
Since Im δ is a quotient of H −1 (Z) ∈ F ⊆ S, we have Im δ ∈ S. Thus H 0 (X) ∈ S holds since S is closed under extensions. Therefore, we obtain X, Z Remark 5.11. The bijection in Proposition 5.10 can also be constructed purely combinatorially as follows: Proposition 3.5, Theorem 5.4, and Proposition A.14 give the following diagram consisting of bijections.
Therefore, we can think of Proposition 5.10 as a categorification of Proposition A.14 about faces of the localization of a monoid.
Next, we discuss the localization of an intermediate subcategory by its Serre subcategory. It is always possible and yields an abelian category:  [NOS,Remark 4.31 (2)].
(EL3) First, we will check the first condition in (EL3). Let f : X → Y be an inflation in F [1] * A and ϕ : W → X in F [1] * A a morphism satisfying f ϕ = 0. We have the following triangle in Here bϕ = 0 holds since both vanish. Therefore, we obtain the above ψ satisfying ϕ = aψ. Thus ϕ = hϕ = haψ holds, so ϕ factors through the object F ∈ F . On the other hand, by the proof of Proposition 5.10, we have F ⊆ D. Hence ϕ factors through an object in D.
Next, we will prove the second condition. Let g : Y → Z be a deflation in F [1] * A and ϕ : Z → W in F [1] * A a morphism satisfying ϕg = 0. We have the following triangle in D b (A) with X ∈ F [1] * A: Since ϕg = 0 and h is a weak cokernel of g in D b (A), it follows that ϕ factors through h, so we obtain the above ϕ : X[1] → W satisfying ϕ = ϕh. Then, since W ∈ F [1] * A, we have the following triangle in D b (A) with F ∈ F and A ∈ A. Let f : X → Y be any morphism in F [1] * A. By taking the mapping cocone K of f and the truncation of K and using the octahedral axiom, we obtain the following commutative diagram consisting of triangles in D b (A): Theorem 5.13. Let A be a skeletally small abelian category, F a torsionfree class in F , and S a Serre subcategory of A with F ⊆ S. We have the following commutative diagram consisting of exact functors of extriangulated categories, where Q 1 and Q 2 are localization functors: We claim that the composition Q 1 H 0 is an exact functor. To show this, take any conflation Then since it is a triangle in D b (A), we obtain the following long exact sequence in A: Then Lemma 5.1 (1) implies H −1 (Z) ∈ F . Now since F is contained in S, we obtain the following short exact sequence in A/S by applying an exact functor Q 1 to the above exact sequence: Therefore, Lemma 2.4 shows that Q 1 H 0 is an exact functor between extriangulated categories. Hence an exact functor Ψ in (5.3) is induced. Now we obtain the following strict commutative diagram consisting of additive functors between additive categories.
Thus the universal property of Q 1 as an exact 2-localization (see Remark 4.7) implies that ΨΦ is naturally isomorphic to the identity functor. Similarly, we have the following strict commutative diagram.
We claim that Q 2 ιH 0 is naturally isomorphic to Q 2 . In fact, for any morphism f : X → Y in F [1] * A, we obtain the following morphism of conflations in F [1] * A: Since Q 2 is an exact functor which sends every object in F [1] to 0, we obtain the following exact commutative diagram in an abelian category (F [1] * A) (F [1] * S): Therefore, Q 2 ιH 0 is naturally isomorphic to Q 2 . Thus the universal property of an exact 2localization Q 1 (Remark 4.7) implies that ΦΨ is naturally isomorphic to the identity functor. Therefore, Φ and Ψ are mutually quasi-inverse.

Questions
In this section, we give some questions on the Grothendieck monoid of an extriangulated category. Throughout this section, C denotes a skeletally small extriangulated category.
6.1. Invertible elements in the Grothendieck monoid. We begin with considering M(C) × , which consists of invertible elements in the Grothendieck monoid (Remark A.5). We can interpret this as the smallest Serre subcategory: Proposition 6.1. In the bijection of Serre C and Face M(C) in Proposition 3.5, the face M(C) × corresponds to the unique smallest Serre subcategory of C.
Proof. This follows from Proposition 3.5 and the fact that M(C) × is the unique smallest face of M(C) by Remark A.5.
We are interested in the extreme case, that is, M(C) × = 0 and M(C) × = M(C). The former one holds when C is an exact category. Actually, we have the following implications.
Proposition 6.2. Let C be a skeletally small extriangulated category. Consider the following conditions.
(1) C is an exact category (with the usual extriangulated structure).
Proof. The conditions in (2) are easily seen to be equivalent by considering the long exact sequence associated to the conflation.
In the first version of this paper, we posed the question whether all of the above conditions are equivalent. However, there is a counterexample for (5) ⇒ (1) and (3) ⇒ (1) (see [BHST,Example 5.2,Remark 5.3]). Thus we modify this question to the following form.
Question 6.3. Let C be a skeletally small extriangulated category. Are the following conditions equivalent?
(1) 0 is a Serre subcategory of C.
Next, we consider the condition M(C) × = M(C). Clearly this is equivalent to that M(C) is a group. If C is triangulated, then this is the case, but there are examples where C is not triangulated and M(C) is a group (see Example 2.10 and Corollary 5.5). Thus we have the following natural question.

Question 6.4. When does M(C) become a group?
We remark that this is equivalent to Serre C = {C} by Proposition 6.1. Finally, we discuss the "decomposition" of a monoid into a group and a reduced monoid. We have the following monoid-theoretic decomposition of an arbitrary monoid.
Proposition 6.5. Let M be a monoid. Then the following holds.
(2) Let L be a submonoid of M . Suppose that L is a group and that M/L is a reduced monoid. Then L = M × holds. Proof.
(1) This follows from the fact that M × is a face of M and Lemma A.13 (1).
(2) Since L is a group, clearly L ⊆ M × holds. Conversely, let x ∈ M × , so there is an element y ∈ M satisfying x + y = 0. Then for the natural quotient map π : M ։ M/L, we have π(x) + π(y) = 0. This implies π(x) = 0 since M/L is reduced. Therefore, there is some l 1 , l 2 ∈ L satisfying x + l 1 = l 2 . Since L is a group, l 1 is invertible, so we obtain x = l 2 − l 1 ∈ L.
Roughly speaking, this claims the existence and the uniqueness of an "short exact sequence" of monoids 0 M × M M/M × 0 such that M × is a group and M/M × is a reduced monoid. We are interested whether the corresponding exact localization exists for M := M(C), which we shall explain in detail. Proposition 6.1 shows that M(C) × corresponds to the smallest Serre subcategory of C, which we denote by S 0 . If S 0 satisfies some good properties like Corollary 4.28, then we get the exact localization C/S 0 such that M(C/S 0 ) ∼ = M(C)/M S0 = M(C)/M(C) × . Moreover, if Question 6.3 is true, then C/S 0 should be an exact category. This can be considered as a decomposition of an extriangulated category C into the "exact (reduced) part" C/S 0 and the "triangulated-like (group) part" S 0 . Thus our question can be summarized as follows.
Question 6.6. Let S 0 be the smallest Serre subcategory of C.
(1) When does there exist the exact localization C → C/S 0 ? Example 6.7. Question 6.6 has a positive answer for the following classes of extriangulated categories C.
• C is an exact category. In this case, we have S 0 = 0, the exact localization is the identity functor, and M(C) is already reduced. • C is a triangulated category. In this case, we have S 0 = C, the exact localization is C → 0, and M(C) is already a group.
• C is an intermediate subcategory of D b (A) for an abelian category A. Let F be a torsionfree class in A with C = F [1] * A and S the smallest Serre subcategory of A containing F . Then we have S 0 = F [1] * S by Proposition 5.10. In this case, the exact localization exists and commutes with the monoid quotient by Proposition 5.12. Note that C/S 0 is actually not only an exact category but an abelian category.
We end this section with the following natural (but maybe difficult) inverse problem.
Question 6.8. For a given commutative monoid M , does there exist an extriangulated category C satisfying M(C) ∼ = M ?
Related to this, we remark that for any reduced monoid M , there is a split exact category C satisfying M(C) ∼ = M by the argument similar to [Fac, Section 2].
6.2. Quasi-split extriangulated categories and c-closed subcategories. In Section 3, we have classified certain subcategories of C via its Grothendieck monoid M(C). In particular, we have introduced c-closed subcategories, the largest class of subcategories which can be classified via M(C) (Proposition 3.3). In this subsection, we will discuss the conditions under which all subcategories are classified via M(C), that is, all subcategories are c-closed.
Let us begin with a simple observation. Note that a subcategory is assumed to be closed under isomorphisms, but not assumed to be additive here.
Proposition 6.9. Let C be a skeletally small extriangulated category. Then the following conditions are equivalent.
(1) All subcategories of C are c-closed.
(1) ⇒ (2): Let A → B → C be a conflation in C. Consider the subcategory X consisting of objects isomorphic to B. Since X is c-closed, it contains A ⊕ C. This means A ⊕ C ∼ = B.
(2) ⇒ (1): Let X be a subcategory of C. For any conflation A → B → C , we have B ∼ = A ⊕ C by the assumption (2). Thus B ∈ X if and only if A ⊕ C ∈ X . This proves that X is c-closed.

Let us call a conflation
There are subtle differences between split and quasi-split conflations.
Example 6.10. There are two short exact sequences of abelian groups: Then the short exact sequence obtained by their direct sum is quasi-split but not split. Indeed, by the natural isomorphism Ext 1 , the element δ ⊕ ǫ corresponds to (δ, 0), and it is nonzero since δ is not a split exact sequence.
From this example, a quasi-split conflation is not a split conflation in general. However, there are the following examples where this holds.
Example 6.11. In the following classes of extriangulated categories C, every quasi-split conflation is a split conflation.
• C is an exact R-category over a commutative ring R such that C(A, B) is an R-module of finite length for any A, B ∈ C. Indeed, for any quasi-split conflation 0 → A → B f − → C → 0, we have an exact sequence of R-modules. Now we have C(C, B) ∼ = C(C, A) ⊕ C(C, C) by the assumption. Considering the lengths of R-modules in the above exact sequence, we conclude that C(C, f ) is surjective, and this implies that the conflation 0 → A → B f − → C → 0 splits. • C is an extension-closed subcategory of mod Λ, where Λ is an algebra over a commutative noetherian ring R with Λ ∈ mod R. This follows from the result in [Miy,Theorem 1], which states that any quasi-split short exact sequence in mod Λ actually splits.
The above examples lead us to the next question.
Question 6.12. When does a quasi-split conflation become a split conflation? More concretely, in the following classes of exact categories C, does this question have a positive answer?
• C is an exact R-category over a commutative noetherian ring R such that C(A, B) ∈ mod R holds for any A, B ∈ C. • C is a noetherian exact category, that is for any X ∈ C, the poset of admissible subobjects of X satisfies the ascending chain condition (see [Eno,Section 2] for this poset).
We now return to considering when all subcategories are classified via M(C). An extriangulated category C is said to be quasi-split if every conflation in C is quasi-split. Recall that this condition is equivalent to the condition under which all subcategories of C can be classified via the Grothendieck monoid by Proposition 6.9. On the other hand, if every conflation of an extriangulated category C is a split conflation, then every inflation is a monomorphism, and every deflation is an epimorphism. Thus C becomes an exact category (see [NP,Corollary 3.18]). Such an exact category C is called a split exact category. It is clear that a split exact category is quasi-split. Thus we have the following natural question.
Question 6.13. When does a quasi-split extriangulated category C become a split exact category?
Example 6.14. Question 6.13 has a positive answer for the following classes of extriangulated categories C.
• C is one of the exact categories in Example 6.11.
• More generally, C is an exact category in which Question 6.12 has a positive answer.
• C is an extriangulated category with enough projectives. See Proposition 6.15 below.
We recall some terminology to prove Proposition 6.15. An object P of C is projective if E(P, X) = 0 for any X ∈ C. We denote by Proj C the category of projective objects of C. Clearly Proj C is closed under extensions and direct summands. An extriangulated category C has enough projectives if for any A ∈ C, there exists a deflation P → A from a projective object P .
Proposition 6.15. Suppose that an extriangulated category C has enough projectives. Then the following conditions are equivalent.
(1) Every subcategory of C is c-closed.
(2) Every additive subcategory of C is c-closed.
(3) Every extension-closed subcategory of C is c-closed.
(4) C is a split exact category.
Proof. The implications (1) ⇒ (2) ⇒ (3) and (4) ⇒ (5) ⇒ (1) are clear, so we only prove (3) ⇒ (4). Let A → B → C be a conflation in C. Since C has enough projectives, there is a conflation X → P → C in C with P ∈ Proj C. The assumption (3) implies that Proj C is c-closed, and hence C ⊕ X ∈ Proj C. Then C is also projective since Proj C is closed under direct summands. Therefore the conflation A → B → C splits. This proves that C is a split exact category.
Appendix A. Basics on commutative monoids In this section, we collect some definitions and properties of commutative monoids.
A monoid is a semigroup with a unit. In this paper, every monoid is assumed to be commutative. Hence we use the additive notation, that is, the binary operation is denoted by +, and the unit is denoted by 0. A homomorphism of monoids is a map f : M → N satisfying f (x + y) = f (x) + f (y) and f (0 M ) = 0 N . We denote Mon by the category of (commutative) monoids and homomorphisms of them. The category Mon has arbitrary small limits and colimits (see [Ogu,Section I.1.1]). We can define the product i∈I M i and the direct sum (= coproduct) i∈I M i of monoids similarly to abelian groups. In particular, finite products and finite direct sums coincide.
A basic example of monoids is the set N of non-negative integers with the arithmetic addition. A monoid M is said to be free if it is isomorphic to N ⊕I for some index set I. In this case, the cardinality of I is called the rank of M . A basis of a free monoid is defined by a similar way to abelian groups.
The notion of quotients of monoids slightly differs from that of abelian groups. We introduce a class of equivalence relations ∼ on a monoid M to guarantee that the quotient set M/∼ becomes a monoid.
Definition A.1. The equivalence relation ∼ on a monoid M is called a congruence if x ∼ y implies a + x ∼ a + y for every a, x, y ∈ M .
We can check easily that the quotient set M/∼ of a monoid M by a congruence ∼ has the unique monoid structure such that the quotient map M → M/∼ is a monoid homomorphism. To each submonoid of a monoid, we can associate the congruence as follows.
Definition A.2. Let M be a monoid and N its submonoid. Define a congruence on M as follows: x ∼ y :⇔ there exist n, n ′ ∈ N such that x + n = y + n ′ .
Then the monoid M/N := M/∼ is called the quotient monoid of M by N . We write x ≡ y mod N if x ∼ y holds. The equivalence class of x ∈ M is denoted by x mod N .
It is easily seen that the quotient monoids have the following universal property. Next, we introduce a class of submonoids which corresponds to Serre subcategories (see Section 3.1).
Definition A.4. Let M be a monoid.
(1) A submonoid F of M is called a face if for all x, y ∈ M , we have that x + y ∈ F if and only if both x ∈ F and y ∈ F . (2) Face M denotes the set of faces of M .
Remark A.5. Let M be a monoid. An element x ∈ M is called a unit if there exists y ∈ M such that x + y = 0. The set M × of units is the smallest face, and M itself is the largest face. In particular, Face M is a singleton if and only if M is a group.
Example A.6. Let M be a free monoid of rank 2 with a basis e 1 and e 2 .
(1) N(e 1 + e 2 ) is a submonoid of M but not a face.
Let us give an explicit description of the face generated by a subset, which is useful to study faces.
(1) The submonoid S N of M generated by S is the smallest submonoid of M containing S. We can describe it as follows: n i x i m, n i ∈ N, x i ∈ S .
(2) The face S face of M generated by S is the smallest face of M containing S. We can describe it as follows: S face := {x ∈ M | there exists y ∈ M such that x + y ∈ S N } .
Unlike the case of abelian groups, submonoids of M/N do not correspond to those of M containing N : (1) If X is a face of M containing N , then X/N := π(X) is also a face of M/N .
(2) If X ′ is a face of M/N , then π −1 (X ′ ) is also a face of M containing N .
(1) It is clear that X/N is a submonoid of M/N . Let a and b elements in M such that π(a) + π(b) ∈ X/N . Then there exist x ∈ X and n, n ′ ∈ N such that x + n = (a + b) + n ′ . Since x + n belongs to the face X of M , both a ∈ X and b ∈ X hold. Thus both π(a) and π(b) belong to X/N , which shows that X/N is a face of M/N .
(2) This is clear from the definitions.
(3) We have that π −1 (X ′ )/N = π(π −1 (X ′ )) = X ′ for any face X ′ of M/N since π is surjective. Let X be a face of M containing N . It is easily seen that π −1 (X/N ) ⊇ X. Take a ∈ π −1 (X/N ). Then we have π(a) ∈ X/N , which implies that there exist x ∈ X and n, n ′ ∈ N such that x + n = a + n ′ . Since a + n ′ = x + n belongs to X the face of M , we have a ∈ X. Therefore we conclude that π −1 (X/N ) = X. Next, we recall the localization of a monoid, which is a monoid analogue of the localization of a commutative ring.
Definition A.11. Let M be a monoid and S a subset of M . The localization of M with respect to S is a monoid M S together with a monoid homomorphism ρ : M → M S which satisfies the following universal property: (i) ρ(s) is invertible in M S for each s ∈ S.
(ii) For any monoid homomorphism φ : M → X such that φ(s) is invertible for each s ∈ S, there is a unique monoid homomorphism φ : M S → X satisfying φ = φρ.
The localization of a monoid M with respect to a subset S ⊆ M actually exists, which is constructed as follows: Define a binary relation on M × S N by (x, s) ∼ (y, t) :⇔ there exist u ∈ S N such that x + t + u = y + s + u in M .
It is a congruence on the monoid M × S N , and hence the quotient set M S := M × S N /∼ becomes a monoid. We denote by [x, s] the equivalence class of (x, s) ∈ M × S N . We can think of [x, s] as "x−s." Then it is straightforward to check that the monoid M S together with a monoid morphism ρ : M → M S defined by ρ(m) = [m, 0] is the localization of M with respect to S. We call ρ : M → M S the localization homomorphism of M with respect to S.
We reveal the relationship between faces of M S and those of M in Proposition A.14 below. Let us prove two lemmas for this purpose. (1) M/S is reduced (see Definition A.17).
(1) Let x, y ∈ M such that x + y ≡ 0 mod S. There are elements s, t ∈ S such that x + y + s = t in M . Since S is a face, both x and y belong to S, which implies both x ≡ 0 mod S and y ≡ 0 mod S. Therefore M/S is reduced.
( Proposition A.14. Let M be a monoid and S a subset of M . Then there is an inclusionpreserving bijection between the following sets: (1) Face(M S ).
Proof. We have the following inclusion-preserving bijections: Proof. This follows from Proposition A.15 and the fact that units are preserved by monoid homomorphisms.
Finally, we list properties of monoids which we use in this paper.
Definition A.17. A monoid M is reduced if a + b = 0 implies a = b = 0 for any a, b ∈ M . This is equivalent to M × = 0.
Definition A.18. Let S be a subset of a monoid M .
(1) S is subtractive if x + y ∈ S and x ∈ S imply y ∈ S for any x, y ∈ M .
(2) S is cofinal if, for any x ∈ M , there exists y ∈ M satisfying x + y ∈ S.
Remark A.19. Let M be a monoid.
(1) If M is a group, then a subtractive submonoid is nothing but a subgroup.
(2) If M is a group, then any submonoid N of M is cofinal since x + (−x) = 0 ∈ N for all x ∈ M . (3) We can define a pre-order ≤ on any monoid M by x ≤ y :⇔ there exists some a ∈ M such that y = x + a.
A cofinal subset of M defined as above is nothing but a cofinal subset of M with respect to this pre-order ≤.