From triangulated categories to module categories via localisation II: Calculus of fractions

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.


Introduction
Let k be a field and C a skeletally small, triangulated Hom-finite k-category which is Krull-Schmidt and has Serre duality. A standard example of such a category is the bounded derived category of finite dimensional modules over a finite dimensional algebra of finite global dimension (see [6]). In this case, the triangulated category is obtained from the abelian category of modules by Gabriel-Zisman (or Verdier) localisation of the quasi-isomorphisms in the bounded homotopy category of complexes of modules.
Here, our approach is the other way around. Given a triangulated category C as above, we are interested in gaining information about related abelian categories. We are particularly interested in the module categories over (the opposites of) endomorphism algebras of objects in C. An object T in C satisfying Ext 1 (T, T ) = 0 is known as a rigid object. In this case it is known [2] that the category of finite dimensional modules over End(T ) op can be obtained as a Gabriel-Zisman localisation of C, formally inverting the class S of maps which are inverted by the functor Hom C (T, −). However, the class S does not admit a calculus of left or right fractions in the sense of [4, Sect. I.2] (see also [12,Sect. 3]).
If T is a cluster-tilting object then, by a result of Koenig-Zhu [11,Cor. 4.4], the additive quotient C/ΣT , where Σ denotes the suspension functor of C, is equivalent to mod End C (T ) op (see also [8,Prop. 6.2] and [10,Sect. 5.1]; the case where C is 2-Calabi-Yau was proved in [10, Prop. 2.1], generalising [3,Thm. 2.2]). However, when T is rigid, this is no longer the case in general. It is natural to consider instead the quotient C/X T where X T is the class of objects in C sent to zero by the functor Hom C (T, −), since, in the cluster-tilting case, X T = add ΣT . However, one does not obtain the module category this way, since in general C/X T is not abelian.
Our approach here is to show first that C/X T is preabelian, using some arguments generalising those of Koenig-Zhu [11]. This means that, in addition to C/X T being an additive category, every morphism in C/X T has a kernel and a cokernel. This category in general possesses regular morphisms which are not isomorphisms (i.e. morphisms which are both monomorphisms and epimorphisms but which do not have inverses), so it cannot be abelian in general.
However, we show that C/X T does have a nice property. It is integral, i.e. the pullback of any epimorphism (respectively, monomorphism), is again an epimorphism (respectively, monomorphism). This allows us to apply a result of Rump [17, p173] which implies that (C/X T ) R , the localisation of the category C/X T at the class R of regular morphisms, is abelian. We assume that C is skeletally small to ensure that the localisation exists. Furthermore, by the same reference, the class R admits a calculus of left and right fractions.
We go on to show that the projective objects in (C/X T ) R are, up to isomorphism, exactly the objects induced by the objects in the additive subcategory of C generated by T . This implies our main result: Theorem.
Let C be a skeletally small, Hom-finite, Krull-Schmidt triangulated category with Serre duality, containing a rigid object T . Let X T denote the class of objects X in C such that Hom C (T, X) = 0. Let R denote the class of regular morphisms in C/X T . Then R admits a calculus of left and right fractions. Let (C/X T ) R denote the localisation of C/X T at R. Then Let S denote the class of maps in C which are inverted by Hom C (T, −), and let S denote the image of this class in C/X T . Then R = S, and the localisation functor L S : C → C S factors through C/X T . The main result of [2] was the construction of an equivalence G from C S to mod End C (T ) op , such that Hom C (T, −) = GL S . Our theorem above can be seen a refinement of this. It was noted in [2] that S does not admit a calculus of left or right fractions, thus we can observe that the advantage of passing first to the quotient C/X T is that the subsequent localisation does then admit such a calculus.
We note that [14] contains results obtaining abelian categories as subquotients of triangulated categories; we give an explanation of the relationship between the results obtained here and those in [14] in Section 6. We also remark that A. Beligiannis has recently informed us that, in subsequent work using a different approach, he has been able to generalise our main result to the case of a functorially finite rigid subcategory.
There are interesting parallels between our approach here and the construction of the derived category of an abelian category A. We follow [9], [5,III.2,III.4].
The derived category of A can be defined (following Grothendieck) as the Gabriel-Zisman localisation of the category C(A) of complexes over A at the class of quasiisomorphisms. This class does not in general admit a calculus of left and right fractions. However, the more commonly used construction (due to Verdier) of the derived category involves passing first to the homotopy category K(A). Then C(A) is a Frobenius category and K(A) is the corresponding stable category, hence a quotient of C(A). Then the class of quasi-isomorphisms in K(A) admits a calculus of left and right fractions. Localising at this class gives rise to the derived category of A.
In Section 1 we set-up the context in which we work. In Section 2, we recall the definitions of semi-abelian and integral categories and some results of Rump [16,17] which will be useful. In Section 3, we prove that C/X T is integral. In Section 4, we recall the Gabriel-Zisman theory of localisation and calculi of fractions and also how it can be applied (following Rump [17,Sect. 1]) to the case of the regular morphisms in an integral category. In Section 5, we apply this to (C/X T ) R to show that it is abelian. By classifying the projective objects in (C/X T ) R , we deduce the main result. In Section 6, we explain the relationship of the results here to work of Nakaoka [14]. In Section 7, we explain the relationship between our main result and the results in [2].

Notation
We first set up the context in which we work and define some notation. Let k be a field and C be a skeletally small, triangulated, Hom-finite, Krull-Schmidt kcategory with suspension functor Σ. We need the skeletally small assumption to ensure that the localisations we need exist. We assume that C has a Serre duality, i.e. an autoequivalence ν : C → C such that Hom C (X, Y ) ≃ D Hom C (Y, νX) (natural in X and Y ) for all objects X and Y in C, where D denotes the duality Hom k (−, k). Let T be a rigid object in C and set Γ = End C (T ) op .
For a full subcategory X of C, let X ⊥ = {C ∈ C | Ext 1 (X, C) = 0 for each X ∈ X }, and define ⊥ X dually. For an object X in C, let add X denote its additive closure, and let We also recall the triangulated version of Wakamatsu's Lemma; see e.g. [8, Section 2]. Lemma 1.1. Let X be an extension-closed subcategory of a triangulated category C.
(a) Suppose that X → C is a minimal right X -approximation of C and Σ −1 C → Y → X → C a completion to a triangle. Then Y is in X ⊥ , and the map Then Z is in ⊥ X , and the map Z → ΣC is a right ⊥ X -approximation of ΣC.
Using this, we obtain: Let T be a rigid object in C. Then the subcategory X T of C is functorially finite.
Proof. This follows from combining Wakamatsu's Lemma (Lemma 1.1) with the existence of Serre duality.

Preabelian categories
Recall that an additive category A is said to be preabelian if every morphism has a kernel and a cokernel. In this section we shall recall some of the theory of preabelian categories that we need in order to study C/X T . A morphism is said to be regular (or a bimorphism) if it is both an epimorphism and a monomorphism.
According to [17, Sect. 1] a preabelian category is called left semi-abelian (respectively, right semi-abelian) if every morphism f has a factorisation of the form ip where p is a cokernel and i is a monomorphism (respectively, where p is an epimorphism and i is a kernel); see [17,Sect. 1], where it is pointed out that in the left semi-abelian case p is necessarily coim(f ) = coker(ker(f )) and in the right semiabelian case i is necessarily im(f ) = ker(coker(f )). A preabelian category is said to be semi-abelian if it is both left and right semi-abelian.
We remark that pullbacks and pushouts always exist in a preabelian category. For the pullback of maps c : B → D and d : C → D, we can take the kernel of the map B C → D whose components are c and −d, obtaining a pullback diagram: There is a dual construction for the pushout. We recall the following characterisation of semi-abelian categories: A dual characterisation in terms of pushout diagrams holds for right semi-abelian categories.
A preabelian category is said to be left integral provided that, in any pullback diagram as above, a is an epimorphism whenever d is an epimorphism. A dual definition involving pushouts is used to define right integral categories. A preabelian category which is both left and right integral is said to be integral.
The following then follows from Proposition 2. We recall the following two results from [17].

Lemma 2.3. [17, Lemma 1] Let A be a preabelian category. In a pullback diagram (1), whenever d is a monomorphism, a is a monomorphism.
We also recall the following (which also includes a dual statement involving pushouts which we will not need here). Finally, we note the following, which is easy to show using the definitions.
Lemma 2.5. Let h be a map in an additive category which is a weak cokernel of a map g and an epimorphism. Then h is a cokernel of g.

Properties of C/X T
In this section, we consider the factor category C/X T . The objects in C/X T are the same as those in C. For objects X, Y in C, We will show the following result, which can be regarded as a generalisation of [11,Theorem 3.3]. Our proof is inspired by the proof in [11].
Theorem 3.1. The factor category C/X T is preabelian.
In order to prove Theorem 3.1, we will need the following lemmas.

Lemma 3.2. Consider a commutative diagram
For a map f : Proof. (c) follows by definition from (a) and (b). We prove only (b), the proof of (a) being dual.
We consider first the case when Z is in X T . We then need to show that f is an epimorphism in C/X T .
Let p : Y → M be a map such that pf = 0. Then there is an object U ′ in X T , and a commuting square: where the rows are triangles. By Wakamatsu's Lemma (Lemma 1.1), we have that Σ −1 N is in X T ⊥ . Therefore the map Z → N is zero, using that Z is by assumption in X T . By commutativity, the composition Σ −1 Z → X → U vanishes. Hence, we have by Lemma 3.2, that there are maps v 1 : Y → U , and v 2 : Now consider the general case, so assume g factors through an object V in X T and consider the induced commutative diagram where the rows are triangles. Now f ′ = f r and hence f ′ = f r. Note that since V is in X T , we have that f ′ is an epimorphism. It follows that f r, and hence f , is an epimorphism.
Conversely, if f is an epimorphism then, since gf = 0 we have gf = 0, so g = 0.
Lemma 3.4. For any map f : X → Y in C, the map f : X → Y has a kernel and a cokernel.
Proof. We construct a cokernel of f . The construction of a kernel is dual.
Consider a minimal right add T -approximation a : T 0 → X. Compose this with f , and complete the composition f a to a triangle By the octahedral axiom, there is a commutative diagram Extend this to a commuting diagram of triangles, and compose with the previous map of triangles, to obtain the diagram This shows that c is a weak cokernel for f . It is also clear that c is an epimorphism, using the triangle (2). It then follows from Lemma 2.5 that c is actually a cokernel for f . Proof of Theorem 3.1. The additivity of C/X T follows directly from the additivity of C. By Lemma 3.4, we have that for any map f : X → Y , the induced map f has both a kernel and a cokernel.
In order to show that C/X T is also integral, we need to study its projective objects. According to [13], an object P in a preabelian category (indeed, in any category) is said to be projective if, for any epimorphism c : B → C, any morphism f : P → C factors through c: We shall use this definition. But we note that Rump [17, p170] uses a different definition; the above diagram should commute only for any cokernel c. Such objects are referred to as quasi-projectives in [15, Defn. 7.5.2] and we shall use this terminology. Note that the two notions are the same in an abelian category, as then epimorphisms and cokernels coincide. The dual objects will be referred to as quasi-injectives.
In the following three proofs, we use some arguments based on the proof of [11,Theorem 4.3].
Lemma 3.5. Every object in add T , when regarded as an object in C/X T , is projective.
Proof. Let f : X → Y be an epimorphism in C/X T , and u : T 0 → Y any morphism, where T 0 lies in add T . Completing f to a triangle in C, we have the diagram: Since f is an epimorphism, by Lemma 3.3 we have that g factors through X T . Hence gu = 0, so u factors through f and thus u factors through f as required.
Lemma 3.6. The category C/X T has enough projectives.
Proof. Let X be an object in C/X T . and let f : T 0 → X be a minimal right add T approximation of X in C. Complete it to a triangle: Hence, by Lemma 3.3, f : T 0 → X is an epimorphism, as required. It now follows from Lemma 3.5 that C/X T has enough projectives.
Dually, it can be shown that: (a) Every object in add Σ 2 T , regarded as an object in C/X T , is injective. (b) The category C/X T has enough injectives.
We recall: If A is a preabelian category with enough quasi-projectives (respectively, quasi-injectives), then A is left (respectively, right) semi-abelian.
It already follows from Proposition 3.8 that C/X T is semi-abelian, using Lemmas 3.6 and 3.7. However, a modification of the argument in the proof of this result allows us to show: Proposition 3.9. Let A be a preabelian category.
(1) Suppose that A has enough projectives. Then A is left integral.
(2) Suppose that A has enough injectives. Then A is right integral.
(3) Suppose that A has enough projectives and enough injectives. Then A is integral.
Proof. As remarked above, we use an approach similar to the proof of [17,Cor. 2]. For (a), suppose we are given a pullback diagram: Since A has enough projectives, there is an epimorphism a ′ : Since d is an epimorphism, there is a map b ′ : P → C in C/X T such that ca ′ = db ′ . Since the diagram (3) is a pullback, there is a map e : P → A such that ae = a ′ and be = b ′ . Since a ′ is an epimorphism, so is a, and (a) follows.
The proof of (b) is dual to the proof of (a), and (c) follows from (a) and (b).
We also remark that, for a semi-abelian category, left integrality is equivalent to right integrality (see [17,Cor. p173]). Proof. This follows from Lemmas 3.5, 3.6 and 3.7, together with Proposition 3.9.

Localisation
Let D be a category. A class R of morphisms in D is said to admit a calculus of right fractions [4, I.2] provided that the following holds: (RF1) The identity morphisms of D lie in R and R is closed under composition.
(RF2) Any diagram of the form: with r ∈ R has a completion to a commuting square of the following form: There is a dual set of axioms, (LF1-3), for left fractions. Let us assume that D is skeletally small, so that the Gabriel-Zisman localisation D R of D at R exists. In this situation, D R has a very nice description; see [4, I.2] The composition of two right fractions [r ′ , f ′ ] RF •[r, f ] RF is given by the right fraction [rr ′′ , f ′ f ′′ ] RF where r ′′ , a morphism in R, and f ′′ , a morphism in C, are obtained from an application of axiom (RF2) which gives rise to the following commutative diagram: C  (LF1-3), there is a dual description of D R by left fractions, and so every morphism in D R can be written in the form [g, s] LF = x s [g], where g is a morphism in D and s lies in R.
According to [17, p173], the following result holds. We include a proof for the convenience of the reader.

Proposition 4.1. [17, p173] Let A be a semi-abelian category. Then A is integral if and only if the class R of regular morphisms in A admits a calculus of right fractions and a calculus of left fractions.
Proof. We firstly note that it follows from the definitions that R satisfies (RF1). Let r, f, f ′ be as in (RF3) above and suppose that rf = rf ′ . Then r(f − f ′ ) = 0. Since r is regular, it is a monomorphism, so f − f ′ = 0 and f = f ′ . Thus we can just take r ′ to be the identity map on X and we see that (RF3) is satisfied.
We will now show that A is left integral if and only if (RF2) holds. Suppose first that A is left integral and we are given a diagram: be the pullback of this diagram. Since r is an epimorphism and A is left integral, a is also an epimorphism. Since r is a monomorphism, a is a monomorphism by Lemma 2.3. Hence a is also regular and we see that (RF2) holds. Conversely, suppose that (RF2) holds and consider a pullback diagram of the form:

there is a commuting diagram
with a ′ regular. Since the diagram (5) is a pullback, we have a map e : A ′ → A making the diagram: Since a ′ is regular, it is an epimorphism, so a is also an epimorphism. Again by Lemma 2.3, the fact that d is a monomorphism implies that a is also a monomorphism. Hence a is regular. We note that the proof shows that in fact:

Corollary 4.2. Let A be a semi-abelian category. Then A is integral if and only if the class R of regular morphisms in A admits a calculus of right fractions (respectively, a calculus of left fractions).
Proof. If A is integral then it is left integral. The proof of Proposition 4.1 shows that then RF1-3 are satisfied by R and conversely that if RF1-3 are satisfied then A is integral. The statement for right fractions follows, and a dual argument shows the statement for left fractions.
In the rest of this section, we assume that A is skeletally small, so that localisations exist.
We see that r ′′ = u = v is regular, and f u = 0. Since u is an epimorphism, f = 0 as required.
We note that: We recall that every morphism f : X → Y in a preabelian category A has a factorisation of the form:

Proof. Suppose first that [f ] is an epimorphism and g is a morphism in
and A is abelian if and only iff is an isomorphism for all morphisms f in A.

Lemma 4.7. [17, p167] Let A be a preabelian category. Then A is semi-abelian if and only iff is regular for all morphisms f in A.
Proof. By definition (see Section 2), if A is semi-abelian then every morphism f has a factorisation of the form ip where p = coim(f ) and i is a monomorphism. Comparing this with the factorisation above we see that i = vf and thus, since i is a monomorphism, so isf . Dually, we see thatf is an epimorphism, and hence regular. Conversely, suppose that for all morphisms f in A,f is regular. Then, in the factorisation above, vf must be a monomorphism as v andf are. Dually,f u is an epimorphism and we see that A is semi-abelian as required.
According to [17], we have the following theorem. Again we give details for the interested reader. Proof. As we have already observed, by [4, 3.3, Cor. 2], A R is an additive category. By Lemma 4.6, A R is preabelian. Since A is semi-abelian (by Proposition 2.2), it follows from Lemma 4.7 that in the factorisation: of any morphism f in A,f is regular. It is easy to check that applying the localisation functor to this factorisation gives the corresponding factorisation of [f ]. It follows that for morphisms of the form α = [f ] in A R ,α is invertible. Since every morphism in A R can be obtained by composing a morphism of this form with an invertible morphism in A R , it follows thatũ is invertible for all morphisms u in A R and hence that A R is abelian as required.

5.
The localisation of C/X T is equivalent to mod Γ.
In this section we will show that (C/X T ) R is isomorphic to mod Γ, where as before Γ = End C (T ) op .
We have seen (Corollary 3.10) that C/X T is an integral category. Since we assume C is skeletally small, C/X T is also skeletally small. Applying Theorem 4.8 to the integral category C/X T we see that (C/X T ) R is abelian: Theorem 5.1. Let C be a skeletally small, Hom-finite, Krull-Schmidt triangulated category with Serre duality, containing a rigid object T . Let X T denote the class of objects X in C such that Hom C (T, X) = 0. Then the class R of regular morphisms in C/X T admits a calculus of left fractions and a calculus of right fractions. Furthermore, the localisation (C/X T ) R of C/X T at the class R is abelian.
Remark 5.2. We have seen that the localisation (C/X T ) R inherits an additive structure from C/X T (see Remark 4.3). The localisation (C/X T ) R also inherits a k-additive structure from C/X T : a scalar λ takes the fraction [r, f ] RF to [r, λf ] RF . It can be checked that this action is well defined and, together with the additive structure inherited from C/X T , gives a k-additive structure on (C/X T ) R for which the localisation functor is k-additive.
We will show that the projectives in (C/X T ) R are the objects in add T and that (C/X T ) R has enough projectives. From this it will follow that (C/X T ) R is equivalent to mod Γ. Lemma 5.3. Let A be an additive category and P a projective object in A. If r : U → P is a regular morphism then it is an isomorphism.
Proof. Since r is an epimorphism, the identity map on P factors through r, so there is a morphism s : P → U such that rs = id. Then r(sr − id) = (rs)r − r = 0. Since r is a monomorphism, sr − id = 0 and it follows that r is an isomorphism.
Lemma 5.4. Let A be a skeletally small integral category and R the class of regular morphisms in A. Suppose that P is a projective object in A. Then P , when regarded as an object in the localisation A R , is again a projective object.
Proof. Suppose P is projective in A and we have a diagram: Proof. By Lemmas 5.4 and 3.5, the objects in add T are projective in (C/X T ) R . Let X be an object in C/X T . Then, by Lemma 3.6, there is an epimorphism p : T 0 → X in C/X T , where T 0 lies in add T , and (b) follows, using Lemma 4.5. If X is projective, the identity map on X factors through [ p ], so [ p ] is a split epimorphism and X is isomorphic to a summand of T 0 , hence in add T , and (a) follows.
Proof. The localisation functor induces a morphism If [f ]x r is an arbitrary element of End (C/X T ) R (T ) with f , r morphisms in C/X T and r regular, then r is an isomorphism in C/X T by Lemma 5.3, so x r = [r −1 ] and we see that ϕ is surjective. By Lemma 4.4, it is also injective, so The result follows, since the only homomorphisms from T to objects in X T are zero by definition.
Theorem 5.7. Let C be a skeletally small, Hom-finite, Krull-Schmidt triangulated category with Serre duality, containing a rigid object T . Let X T denote the class of objects X in C such that Hom C (T, X) = 0. Let R denote the class of regular morphisms in C/X T and (C/X T ) R the localisation of the integral category C/X T at R. Then Proof. By Theorem 5.1, (C/X T ) R is an abelian category. By Lemma 5.5, (C/X T ) R has enough projectives, given by the objects in add T . The result follows, with an equivalence being given by the functor Hom (C/X T ) R (T, −), noting that T is a projective generator for (C/X T ) R .
We note that, by Remark 5.2, (C/X T ) R inherits a k-additive structure from C/X T . It is easy to see that the above equivalence preserves this structure (as well as the abelian structure).

Cotorsion pairs
We recall the notion of a cotorsion pair in a triangulated category, considered in Nakaoka [14]. By [14, 2.3] this can be defined as a pair (U , V) of full additive subcategories satisfying (a) U ⊥ = V; (c) For any object C, there is a (not necessarily unique) triangle: with U ∈ U and V ∈ V. Nakaoka points out that (U , V) is a cotorsion pair in this sense if and only if (U , ΣV) is a torsion theory in the sense of [8, 2.2].
If C is a triangulated category as in Section 1 and T is a rigid object in C then (add T, T ⊥ ) is a cotorsion pair (e.g. see [2,Sect. 6]). One might then ask whether Theorem 3.1 can be generalised to this set-up. However, it is easy to see that this cannot be the case. Consider a triangulated category C, satisfying our usual assumptions. Assume that C has a non-zero nonisomorphism between two indecomposables. Then this map does not have a cokernel or kernel in C. But the pair (U , V) = (C, 0) is clearly a cotorsion pair. This gives many examples of cotorsion pairs (U , V) such that C/V is not preabelian.
More interesting examples where C/V is not preabelian also exist. Let C be the cluster category associated to the quiver: For a vertex i, let P i (respectively, I i , S i ) denote the corresponding indecomposable projective (respectively, injective, simple) module. Let U be the additive subcategory whose indecomposable objects are P 2 , P 3 and ΣP 3 . It is easy to check that U ⊥ is the additive subcategory with indecomposables given by P 1 , P 2 and S 2 and that (U , U ⊥ ) is a cotorsion pair. Note that the torsion pair (U , ΣU ⊥ ) appears in [7].
Let f be a non-zero map from P 3 to I 2 . Suppose that c : I 2 → C is a cokernel of f in C/U ⊥ . Since the only non-zero maps g : I 2 → Y with Y indecomposable such that gf = 0 have Y = ΣP 1 or Y = ΣP 2 , it follows that C is a direct sum of copies of ΣP 1 and ΣP 2 . Then dc = 0 for any non-zero map d from C to ΣP 3 , a contradiction to the fact that c is an epimorphism.

7.
The functor Hom C (T, −) : C → mod Γ Let T denote a rigid object in a triangulated category C, where C satisfies the same properties as earlier, and let Γ = End C (T ) op . We have seen that we can obtain mod Γ in a process consisting of two steps: first forming the preabelian factor category C/X T , and then localising this category with respect to the class of regular morphisms.
In [2] we considered the functor H = Hom C (T, −) : C → mod Γ. Let S = S T be the collection of maps f : X → Y in C with the property that in the induced triangle both g and h factor through X T . Let L S : C → C S denote the Gabriel-Zisman localisation. We proved in [2] that there is an equivalence G : C S → mod Γ such that GL S = H. We also proved that a map s belongs to S if and only if H(s) is an isomorphism in mod Γ.
In this section, we point out that the functor H is actually naturally equivalent to the composition of the quotient functor C → C/X T and the localisation functor with respect to regular morphisms.
Consider the set of maps S 0 = {X ∐ U → X | U ∈ X T , X ∈ C} and the Gabriel-Zisman localisation L S 0 : C → C S 0 . The quotient functor Q : C → C/X T inverts all maps in S 0 , so there is a functor G 0 : C S 0 → C/X T , making the following diagram commute. ?
The localisation functor L S 0 has the following elementary properties.
Proof. The proof is identical to the proof of Lemma 3.5 in [2].
By construction, G 0 is the identity on objects. It is clear that G 0 is full, since Q has this property.
We claim that G 0 is also faithful. Firstly, note that by Lemma 7.1(a), L S 0 is full. So let f and f ′ be maps in C with G 0 L S 0 (f ) = G 0 L S 0 (f ′ ). Then H(f ) = H(f ′ ), so f − f ′ factors through X T (by [2, Lemma 2.3]), and hence, by Lemma 7.1, we have that L S 0 (f ) = L S 0 (f ′ + f − f ′ ) = L S 0 (f ′ ). Hence, we have the following.
Proposition 7.2. The induced functor G 0 : C S 0 → C/X T is an isomorphism of categories.
By Lemma 3.3, a morphism f in C/X T is regular if and only if f lies in S. Combining this with Proposition 7.2 we see that the image L S 0 (S) in the preabelian category C S 0 consists of exactly the regular morphisms.
Let R denote the regular morphisms in C/X T . By the universal property of localisation, it follows that we also get an induced isomorphism of categories K : (C S 0 ) L S 0 (S) → (C/X T ) R making the following diagram commute.
It is clear that H factors uniquely through Q, and hence by the universal property of localisation, also uniquely through L R : C/X T → (C/X T ) R , so we have a commutative diagram of functors C/X T We have checked above that the diagram commutes apart from the rightmost square. We recall that, for a localisation functor L and two functors J, J ′ composable with it, JL = J ′ L implies that J = J ′ , by the universal property. We have that so GL ′ = H ′ K. It follows that G −1 H ′ is naturally equivalent to L ′ K −1 .
We remark that the fact that L ′ is an isomorphism implies that H ′ is an equivalence of categories, by the commutativity of the right hand square. Thus the equivalence in Theorem 5.7 can also be derived from the fact that G is an equivalence (i.e. [2,Theorem 4.3]) together with the above analysis.