The left heart and exact hull of an additive regular category

Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $\mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $\mathcal{E}$, and can be constructed as the heart $\mathcal{LH}(\mathcal{E})$ of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$ or as the localization of the category of monomorphisms in $\mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $\operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. Additive regular categories can be characterized as those subcategories of abelian categories which are closed under subobjects. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $\mathcal{E}$ can be found as the heart of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$, or as the localization of the category of monomorphisms of $\mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.


Introduction
Quasi-abelian categories are a well-behaved class of additive categories, generalizing the notion of an abelian category.They are preabelian categories such that the class of all kernel-cokernel pairs satisfies the axioms of a Quillen exact category.Quasi-abelian categories occur often in functional analysis, and motivating examples include the categories of Banach spaces and of Fréchet spaces [57].
In [8,62,68], a characterization of a quasi-abelian category is given as follows: quasi-abelian categories are precisely those categories which occur as a cotilting torsionfree class in an abelian category.For a quasi-abelian category E, such an ambient abelian category A is (essentially) unique.A construction is given in [68, § 1.2]: one can obtain the category A as the heart of a t-structure on the bounded derived category D b (E).This t-structure is called the left t-structure by Schneiders in [68] and the associated heart is then called the left heart, denoted by LH(E).Furthermore, Schneiders shows that the embedding E → LH(E) lifts to a triangle equivalence D(E) ≃ → D(LH(E)), essentially reducing homological properties of the quasi-abelian category E to those of the abelian category LH(E).Schneiders also shows that LH(E) is equivalent to a localization of the monomorphism category of E with respect to the bicartesian squares (see [68,Corollary 1.2.21]).
For several quasi-abelian categories arising in functional analysis, such as the category of Banach spaces, this last construction of the left heart can already be found in [72], before the introduction of t-structures in [6].Indeed, it is noted in [6,Exemple 1.3.24]that Waelbroeck's construction can be recovered using t-structures.
Waelbroeck's approach nonetheless suggests similar ambient abelian categories could also be found in non quasi-abelian settings.Indeed, in [73] this was done for what was there called Waelbroeck categories.These include categories such as the non quasi-abelian category LB of LB-spaces (see section 9).This leads to the following natural question: how similar is the situation to that of quasi-abelian categories.Specifically, one can ask the following questions.
(1) Can these ambient abelian categories be obtained as the heart of a t-structure on some natural triangulated category?(2) What characterizes the embedding E → LH(E)?
When trying to solve the above question for LB, one might be tempted to search an appropriate exact structure on LB (so that the derived category is well-defined) such that the ambient abelian category is obtained as the heart of a natural t-structure.In fact, the category LB has several natural exact structures; we will recall some of these in Section 9.However, we show in Theorem 9.6 that this approach cannot be successful: none of these exact structures yield a well-suited derived category.Instead, we relax the conditions of an exact category and take the derived category of such a weaker structure.Our starting point is the recent observation in [31] that the category LB is left quasi-abelian and, as such, carries a natural one-sided exact structure.It is possible to construct the derived category of LB with respect to this one-sided exact structure.In this paper, we show that this derived category provides a good framework to answer the above questions.
Before addressing the above questions or describing the results in this paper, we sketch the setting more accurately.We work with a slight generalization of a left quasi-abelian category, namely with additive regular categories.An additive regular category is an additive category where (i) every morphism has a cokernel-monomorphism factorization, and (ii) cokernels have pullbacks and the pullback of a cokernel is again a cokernel.The difference between a left quasi-abelian category and an additive regular category is that the latter need not have cokernels.An additive regular category is an additive category which is regular in the sense of [4,11].
Whereas a quasi-abelian category is an exact category, an additive regular category (or even a left quasi-abelian category) need only satisfy those axioms of an exact category that pertain to the cokernelside of the exact sequences: it is a deflation-exact category (see Section 2.2 for a detailed definition).Even though the axioms of a deflation-exact category are weaker than those of an exact category, deflationexact categories still satisfy many attractive homological properties similar to those of an exact category, such as the Short Five Lemma, the Snake Lemma, and the Nine Lemma (see [5,37]).One possible explanation for this nice behavior is given by the existence of the exact hull ( [34,61]): a deflation-exact category E can be embedded in a 2-universal way in an exact category E ex ; this embedding lifts to a derived equivalence D b (E) → D b (E ex ).
The exact hull of a left quasi-abelian category need not be quasi-abelian (or even pre-abelian).In contrast, we show that the exact hull of an additive regular category is again additive regular.Similarly, we show that additive regular categories are stable under taking quotients (in the sense of [35]).Furthermore, the following proposition (see Proposition 4.14 in the text) gives a straightforward source of examples.Proposition 1.2.Let E be an additive regular category.Any full subcategory E ′ ⊆ E which is closed under subobjects, is also additive regular.
As abelian categories are additive regular, the previous proposition gives an easy way to find additive regular categories inside an abelian category.In fact, it follows from Theorem 1.3 below that every additive regular category occurs in this way.
We mentioned that an additive regular category E has a natural structure of a deflation-exact category.As such, one can consider the bounded derived category D b (E); we recall the construction in §2. 3.In this setting, the construction of the left heart for quasi-abelian categories given in [68] generalizes to the setting of additive regular categories.We obtain the following theorem directly generalizing the properties we mentioned for quasi-abelian categories.
Theorem 1.3.Let E be an additive regular category.There is an embedding of E into an abelian category LH(E), characterized by the following properties: (1) E is closed under subobjects in LH(E), (2) every object in LH(E) is a quotient of an object in E.
The characterization in this theorem follows from combining Propositions 5.2 and 5.7; the last statement follows from Proposition 5.9.
In section 8, we show that the left heart LH(E) of an additive regular category E can be obtained by localizing the category of monomorphisms hMon(E) in E (up to homotopy) at the bicartesian squares.This recovers Waelbroeck's construction as well as the construction of the left heart of the LB-spaces as in [73] (see section 9 for more details on the latter).The following theorem provides a construction of the left heart LH(E) that does not refer to the derived category D b (E).
Theorem 1.4.Let hMon(E) be the category whose objects are monomorphisms δ E : E −1 ֒→ E 0 in E and whose morphisms are commutative squares δF up to homotopy (meaning that there is a morphism t : The set S of all morphisms which are bicartesian squares is a multiplicative system in hMon(E).
Our proof of Theorem 1.4 is based on Auslander's Formula (see Section 7 and Section 8) and follows [62].We consider the category mod E of finitely presented functors on E. As E has kernels, mod E is an abelian category.We show that the subcategory eff E of effaceable functors is a hereditary torsion class in mod E; the corresponding torsionfree class is the category mod 1 (E) of objects of projective dimension at most one.Using the Yoneda embedding E → mod E, it is straightforward to show that hMon(E) ≃ mod 1 (E).The proof of Theorem 1.4 then follows from studying the composition hMon(E) ≃ mod 1 (E) → mod(E) → mod(E)/ eff(E) ≃ LH(E).
Acknowledgments.The authors thank Luisa Fiorot and Michel Van den Bergh for helpful discussions.The second author would like to thank the Hausdorff Institute for Mathematics in Bonn, since parts of the paper were written during his stay at the junior trimester program "New Trends in Representation Theory".The third author gratefully acknowledges the support received from the Research Foundation-Flanders (FWO), 12.M33.16N.

Preliminaries
This section is preliminary in nature.We summarize some results of [5,34,35] in a convenient form.Throughout the paper, all categories are assumed essentially small.Furthermore, all categories and functors will be additive.
2.1.The category of finitely presented functors.Let E be an additive category.We denote by Mod(E) the category Fun(E • , Ab) of contravariant additive functors E → Ab.We write Y : E → Mod E for the contravariant Yoneda functor E → Y(E) = Hom(−, E).
We say that M is finitely presented if M ∼ = coker Y(f ) where f is a morphism in E. We write mod(E) for the category of finitely presented objects in Mod (E).If E has weak kernels, then mod(E) is abelian.The category of finitely presented objects satisfies the following universal property (see [46, Universal Property 2.1]).
Theorem 2.1.Let F : E → A be a functor between additive categories.Assume that A has cokernels.There exists, up to a natural equivalence, a unique right exact functor F : mod(E) → A such that F = F • Y. Proposition 2.2.Let A be an abelian category and let E be an additive category with kernels.If a functor F : E → A commutes with kernels, then the lift F : mod(E) → A is exact.
Proof.Following [46,Lemma 2.5], it suffices to show the following property: for each exact sequence X → Y → Z of projective objects in mod(E), the corresponding sequence F (X) → F (Y ) → F (Z) is exact.
As E has kernels (and hence is idempotent complete), every projective in mod(E) is of the form Y(E), for some E ∈ E. Hence, the sequence X → Y → Z is isomorphic to a sequence of the form . As E has kernels, we find ker Y(g) ∼ = Y(ker g).In particular, ker Y(g) is projective.Hence, Y(A) → Y(ker g) is a split epimorphism.
We find a split epimorphism A → ker g and hence a split epimorphism F (A) → F (ker g).As F commutes with kernels, we also find an exact sequence 0 → F (ker g) → F (B) For any functor F : E → F between additive categories, there is an obvious restriction functor which sends an M ∈ Mod(F ) to M • F ∈ Mod(E).The restriction functor has a left adjoint which is the (essentially unique) cocontinuous functor which sends the projective generators Y(E) of Mod(E) to Y(F (E)).Note that − ⊗ E F : Mod(E) → Mod(F ) restricts to a functor mod(E) → mod(F ).
Let E be an additive category with kernels (in particular, mod E is abelian).We write mod 1 (E) for the subcategory of mod(E) consisting of all objects of global dimension at most one.The following description of the objects of mod 1 (E) is standard.
Proposition 2.3.Let E be an additive category with kernels.The following are equivalent for an object M ∈ mod(E): (1) M has projective dimension at most one, (2) there is a monomorphism The following proposition (see [33,Proposition 3.4]) will be used multiple times throughout the text.
Proposition 2.4.Let E be an additive category and write Y : E → Mod(E) for the Yoneda embedding.

Consider a commutative diagram
where ECBD is a pullback square and β = β ′ β ′′ .Applying Y and taking the cokernel of the vertical maps induces the epi-mono factorization It will be convenient to state the following corollary.(where g ′ = m and β ′′ = p), we find that 0 2.2.One-sided exact categories.One sided-exact categories are obtained via a weakening of the axioms of a Quillen exact category [5,28,64].
Definition 2.6.A conflation category is an additive category E together with a chosen class of kernelcokernel pairs, called conflations, such that this class is closed under isomorphisms.The kernel part of a conflation is called an inflation and the cokernel part of a conflation is called a deflation.We depict inflations by the symbol and deflations by ։.Moreover, we depict monomorphisms by ֒→.A morphism X → Y is called admissible if it admits a deflation-inflation factorization X ։ Z Y.
An additive functor F : C → D between conflation categories is called (conflation-)exact if conflations are mapped to conflations.We say that F is left (conflation-)exact if any conflation → F (C) where F (g) is admissible and F (f ) = ker(F (g)).
Definition 2.7.A deflation-exact category E is a conflation category satisfying the following axioms: R0 For each X ∈ E, the map X → 0 is a deflation.R1 The composition of two deflations is a deflation.R2 The pullback of a deflation along any morphism exists and deflations are stable under pullbacks.Dually, an inflation-exact category is a conflation category E satisfying the following axioms: L0 For each X ∈ E, the map 0 → X is an inflation.L1 The composition of two inflations is an inflation.L2 The pushout of an inflation along any morphism exists and inflations are stable under pushouts.Definition 2.8.Let E be a conflation category.In addition to the properties listed in Definition 2.7, we will also consider the following axioms: The axioms L3 and L3 + are defined dually.A deflation-exact category satisfying R3 is called strongly deflation-exact.Dually, an inflation-exact category satisfying axiom L3 is called a strongly inflation-exact category.
(1) Inflation-exact and deflation-exact categories are called left or right exact categories in the literature.However, as the use of left and right is not consistent, we prefer to use the above terminology to avoid possible confusion.
The following theorem highlights the importance of axioms R3 and R3 + .
(1) The category E satisfies axiom R3 if and only if the Nine Lemma holds.
(2) The category E satisfies axiom R3 + if and only if the Snake Lemma holds.
The following observation is essentially contained in [62].
Proposition 2.11.Let E be a conflation category.If every kernel-cokernel pair is a conflation and E satisfies axiom R2, then E is deflation-exact.
Proof.As all kernel-cokernel pairs are conflations, E satisfies axiom R0.That E satisfies axiom R1 follows from [43,Proposition 5.11] (in the terminology of [43] and assuming axiom R2, axiom R1 is equivalent to saying that the composition of totally regular epimorphisms is again a totally regular epimorphism).
Definition 2.12.We recall that a pre-abelian category is an additive category where every morphism has a kernel and a cokernel.We say that a pre-abelian category is deflation quasi-abelian (or left quasi-abelian) if the class of all kernel-cokernel pairs endow it with the structure of a deflation-exact category.Dually, a pre-abelian category is inflation quasi-abelian (or right quasi-abelian) if the class of all kernel-cokernel pairs endow it with the structure of an inflation-exact category.
(1) A quasi-abelian category is called an almost abelian category in [62].
(2) For a pre-abelian category to be deflation quasi-abelian, it suffices that the pullback of a cokernel is a cokernel, see [  [5,28,34].We recall the definition of the derived category, starting with the notion of an acyclic complex.
Definition 2.14.Let E be a conflation category.A complex ; ; where the deflation p n−1 is the cokernel of d n−2 X and the inflation i n−1 is the kernel of d n X .A complex X • is called acyclic or exact if it is acylic in each degree.We write Ac C (E) for the full subcategory of C(E) consisting of acyclic complexes.We write Ac K (E) for the full subcategory of K(E) given by those complexes which are homotopy equivalent to an acyclic complex (thus, Ac K (E) is the closure of Ac C (E) under isomorphisms in K(E)).We simply write Ac(E) for either Ac C (E) or Ac K (E) if there is no confusion.The bounded versions are defined by The subcategory Ac C (E) of K(E) is not replete, i.e. it is not closed under isomorphisms in K(E).Nonetheless, it is a triangulated subcategory of K(E).Lemma 2.15 ([5, Lemma 7.2]).For each map f : In particular, the category Ac C (E) is a triangulated subcategoy of K(E) which is not necessarily closed under isomorphisms.
Analogously to exact categories, one can define the derived category D(E) as the Verdier localization K(E)/ Ac(E) thick of the bounded homotopy category by the thick closure of the triangulated subcategory of acyclic complexes.The bounded versions are defined analogously.The following theorem summarizes some useful properties of the derived category.
(1) The natural embedding i : With regard to the derived category, axioms R3 and R3 + have useful interpretations.
Proposition 2.17 ([34, Propositions 3.11 and 6.2] and [37, Theorem 1.1.(4)and 1.2.(2)]).Let E be a deflation-exact category. ( Moreover, for each i ≤ 0, the sequence P i X P i Y ։ P i Z is a split kernel-cokernel pair.Proof.The proof of [14,Theorem 12.8] for exact categories holds verbatim in the deflation-exact setting.One can replace the reference to [14, Corollary 3. 2.4.Preresolving subcategories.This subsection is a brief summary of [38].We recall the following definition.
Definition 2.20.Let E be a deflation-exact category.A full additive subcategory A ⊆ E is called preresolving if the following two conditions are met: PR1 For every E ∈ E, there exists a deflation A ։ E with A ∈ E. PR2 The subcategory A ⊆ E is deflation-closed, i.e,.for every conflation X Y ։ Z in E with Y, Z ∈ A, we have X ∈ A as well.If A ⊆ E satisfies PR1, we define the A-resolution dimension of an object E ∈ E, denoted by res.dim A (E), as the smallest integer n ≥ 0 for which there exists an exact sequence where all A k ∈ A. If such an n does not exist, we write res.dim A (E) = ∞.Furthermore, we set res. dim A (E) = sup E∈E res.dim A (E).
A preresolving subcategory A ⊆ E is called finitely preresolving if for all E ∈ E, res.dim A (E) < ∞ and is called uniformly preresolving if res.dim A (E) < ∞.
Deflation-closed subcategories of deflation-exact categories inherit a deflation-exact structure.Proposition 2.21 ([38,Proposition 3.6]).Let E be a deflation-exact category and let A ⊆ E be a full additive subcategory.If A ⊆ E is deflation-closed, then A inherits a deflation-exact structure from E (the conflations are precisely the conflations in E with terms in A).Furthermore, if E satisfies axioms R3 or R3 + , then so does A.
The following theorem is an extension of [29,Lemma I.4.6].The conflation structure on E ex is given as follows (based on [24]): a sequence With this conflation structure, the canonical embedding j : E → E ex is conflation-exact.Theorem 2.24 ([34, Section 7]).Let E be a deflation-exact category.
(1) The embedding j : E ֒→ E ex is fully faithful, and is 2-universal among conflation-exact functors to exact categories.(2) The embedding j lifts to a triangle equivalence (3) For every Z ∈ E ex , there is a conflation X Y ։ Z in E ex with X, Y ∈ i(E).Furthermore, if E satisfies axiom R3, then the embedding j reflects conflations.
When working with the exact hull, it is often useful to describe objects of the exact hull E ex as iterated extensions of objects in E. For this, the following notation will be useful.Lemma 2.26.Let E be a deflation-exact category.Let f : X → Y be a morphism in E. Then f is a monomorphism (resp.epimorphism) if and only if j(f ) is a monomorphism (resp.epimorphism).
Proof.As j is fully faithful, it is clear that j reflects epimorphisms and monomorphisms.We first show that j preserves monomorphisms.For this, consider a monomorphism f : X → Y in E. Let t : T → X be a map in E ex such that f • t = 0 in E ex .As T ∈ E ex , there exists an n such that T ∈ E n .We proceed by induction on n.If n = 0, then t = 0 as f is a monomorphism in E 0 .Assume now that n ≥ 1.By construction, there is a conflation By the induction hypothesis, we have t • i = 0.It follows that there is a unique map u : B → X such that u • p = t.Note that f • t = f • u • p = 0 and thus f • u = 0 as p is a deflation (and hence an epimorphism).The induction hypothesis implies that u = 0 and thus t = u • p = 0.This shows that j(f ) is a monomorphism.
To show that j preserves epimorphisms, consider an epimorphism f : Using that f is an epimorphism in E ≃ E 0 , we obtain from p • t • f = 0 that p • t = 0 and hence t : Y → T factors through i : A → Y.Using an induction argument as before, one can show that t = 0.This shows that j(f ) is an epimorphism in E ex .
If E satisfies axiom R3 + , the embedding j : E → E ex satisfies additional properties.
Theorem 2.27 ([38,Theorem 5.7]).Let E be a deflation-exact category.If E satisfies axiom R3 + , E ⊆ E ex is a uniformly preresolving subcategory such that res.dim E (E ex ) ≤ 1.In particular, the derived equivalences of Theorem 2.22 hold.
2.6.t-Structures and their hearts.Let T be a triangulated category with suspension functor Σ.A t-structure on T is a pair (T ≤0 , T ≥0 ) of full and replete (i.e.closed under isomorphisms) subcategories satisfying the following properties: (1) Hom We write T ≤i := Σ −i T ≤0 and T ≥i := Σ −i T ≥0 .Given a t-structure (T ≤0 , T ≥0 ) on T , the heart of T is defined as the subcategory T ♥ = T ≤0 ∩ T ≥0 .The following proposition is standard (see [6]).
Proposition 2.28.Given a t-structure (T ≤0 , T ≥0 ) on a triangulated category T .The categories T ≤0 and T ≥0 are closed under extensions and the heart T ♥ is an abelian subcategory.Moreover, a sequence

The left t-structure and left heart
The left t-structure and the left heart were introduced in [68] for quasi-abelian categories.In this section, we show that these constructions and many of the properties lift to a weaker setting, namely that of a deflation-exact structure on an additive category E with kernels.We follow the same outline as [68,Section 1.2].
We will assume that the deflation-exact structure is strong (that is, satisfies axiom R3).This is a purely technical condition: as E has kernels, one can take the closure of E under the axiom R3 without changing the derived category (see [37]).

A t-structure on K(E).
As in [68], we start by considering a t-structure on the homotopy category K(E).In this subsection, we only use the additive structure on E. We will use the following truncation functors.Definition 3.1.Let C • be a complex in E. As E has kernels, every differential where i n−1 is the kernel of d n C .The canonical truncation τ ≤n C • is a complex together with a morphism τ ≤n C • → C • given by: and the canonical truncation C • → τ ≥n+1 C • is similarly defined by: The following proposition is [34,Proposition 3.13].
For each n ∈ Z, the following triangle is a distinguished triangle in K * (E): In other words, C • is an extension of the canonical truncation τ ≥n+1 C • by τ ≤n C • in K * (E).
We can now consider the t-structure (K ≤0 (E), K ≥0 (E)) on the homotopy category K(E) where In other words, K ≤0 (E) is given by those complexes X • such that ker 3.2.Induced t-structures on the derived category.In this subsection, we show that the above tstructure on K(E) induces a t-structure on D(E) = K(E)/ Ac(E).For this, we use the following statement from [68, Lemma 1.2.17]Proposition 3.3.Let (T ≤0 , T ≥0 ) be a t-structure on a triangulated category T .Let N ⊆ T be a thick subcategory and write Q : T → T /N for the corresponding quotient.The pair Proof.The proof follows that of [5,Lemma 7.2] closely.As X • is acyclic in degree n, we know that i n−1 X : ker d n X X n is an inflation.By [37,Proposition 4.4], this implies that . We find the following commutative diagram where the morphisms g n−1 , g n , and g n+1 are uniquely determined.We can apply [35, Proposition 3.9] (or the dual of [5, Proposition 5.2]) to the maps (g n−1 , f n−1 , g n ) and (g n , f n , g n+1 ) between conflations to obtain the following commutative diagram (the squares marked with BC are bicartesian squares): . By the dual of [5, Propositions 5.4 and 5.5], we have the conflations ) and As m is a monomorphism, it follows from Let E be a strongly deflation-exact category with kernels.There is a t-structure on D(E) given by . Hence, to prove this proposition, it suffices to show that the conditions of Proposition 3.3 are satisfied for T = K(E) and N = Ac(E).As E has kernels and satisfies axiom R3, we know by Proposition 2.17 that Ac(E) is a thick subcategory of K(E).The rest follows directly from Proposition 3.4.Definition 3.6.Let E be a strongly deflation-exact category with kernels.We call the t-structure (D ≤0 (E), D ≥0 (E)) from Proposition 3.5 the left t-structure.We write LH(E) for the heart for the corresponding cohomology functors.
Remark 3.7.As an alternative description, we have Embedding into the left heart.We now turn our attention to the heart of the left t-structure (see Definition 3.6) on D(E).
Via the embedding i : E → D(E), the category E can be considered as a subcategory of the left heart LH(E).We write φ : E → LH(E) for the corresponding embedding.Proposition 3.9.The embedding φ : E → LH(E) commutes with kernels.
Proof.Let f : X → Y be any map in E. Let C be the cone of the corresponding morphism i(f ) : i(X) → i(Y ) in D(E).Applying the cohomology functors LH • , we find the following exact sequence in LH(E) : Corollary 3.10.An object C • ∈ D(E) belongs to the heart LH(E) of the left t-structure if and only if it is isomorphic to a complex of the form Proof.If C • belongs to the heart, then it must be isomorphic to LH 0 (C • ), for some C • ∈ D(E).By Proposition 3.8, it is isomorphic to a complex of the required form.Conversely, it is easy to see that any such complex must be in the heart.
Let D be the cone of the morphism i(f As in the previous proof, we find an exact sequence 0 Using the definition of the cohomology functors, we recover the exact sequence 0 → φ(ker f ) Proposition 3.11.Let E be a deflation-exact category with kernels.Assume that E satisfies axiom R3.
(1) The embedding φ is an exact and fully faithful embedding that reflects conflations.
(2) For every object Z ∈ LH(E), there exists an epimorphism Y → Z with Y ∈ E.
(1) By Theorem 2.16, the embedding φ is fully faithful and exact.Proposition 2.17 now shows that φ reflects exactness.
(2) This follows directly from the exact sequence in Corollary 3.10.
Theorem 3.12.Let E be a deflation-exact category with kernels and assume that E satisfies axiom R3.
The category E is a uniformly preresolving subcategory of LH(E) with res.dim E (LH(E)) ≤ 2. Consequently, the embedding lifts to a triangle equivalence Φ : Proof.By Proposition 3.9, we know that E ⊆ LH(E) is deflation-closed (hence, axiom PR2 is satisfied).Corollary 3.10 implies that axiom PR1 is satisfied, as well as res.dim E (LH(E)) ≤ 2. Hence, E ⊆ LH(E) is uniformly preresolving.That φ lifts to a derived equivalence now follows from Theorem 2.22.
Proposition 3.13.Let A be an abelian category.Let E ⊆ A be a full subcategory satisfying condition PR1 (thus, every object in A is a quotient of an object in E).If for any morphism f in E, we have Proof.As E satisfies axiom PR1 and is closed under kernels, we find that E is a uniformly preresolving subcategory of A (with res.dim E (A) ≤ 2).By Proposition 2.21, we know that E is a strongly deflation-exact category.It now follows from Theorem 2.22 that the natural functor D(E) → D(A) is an equivalence.In particular, every complex with terms in A is quasi-isomorphic to a complex with terms in E. Using the explicit form of the truncation functors on D(E) from Definition 3.1, we see that the equivalence D(E) → D(A) maps the left t-structure on D(E) to the standard t-structure on D(A).This now gives the equivalence LH(E) ≃ A.

3.4.
Universal properties of the left heart.The left heart of a strongly deflation-exact category E with kernels can be characterized via a universal property.The first universal property we give is analogous to [62, Proposition 12].
Proposition 3.14.The embedding φ : E → LH(E) is 2-universal among conflation-exact functors to abelian categories that preserve kernels, that is to say, the functor − • φ : is a fully faithful functor whose essential image consists of those functors E → A that preserve kernels.
Here, Fun ex (−, −) stands for the category of conflation-exact functors.
Proof.The required fully faithful functor is given by this diagram: Let F : E → A be any exact functor.Deriving this functor gives a triangle functor F : given by taking a complex E • ∈ D b (E) and applying F pointwise.As F preserves kernels, it maps the heart LH(E) of the left t-structure to the heart of the standard t-structure on D b (A).That is, F restricts to an exact functor LH(E) → A. Thus, for any exact F : E → A, we find an exact functor Finally, we show that − • φ : Fun ex (LH(E), A) → Fun ex (E, A) is full.For this, we consider the arrow category A [1] of A, that is, the objects of A [1] are arrows A → B in A and morphisms are given by commutative diagrams.An exact functor E → A [1] is given by two exact functors F, G : E → A, together with a natural transformation η : E, A) is full follows from the lifting property of − • φ : Fun ex (LH(E), A [1] ) → Fun ex (E, A [1] ).Remark 3.15.If E is left quasi-abelian, the above proposition and [62, Proposition 12] imply that LH(E) ≃ Q l (E) where Q l (E) is the left abelian cover as defined in [62].
The following proposition is a generalization of [68,Proposition 1.2.34].For conflation categories E, F , we write Rex(E, F ) for the category of right exact functors E → F .Proposition 3.16.Let E be a strongly deflation-exact category with kernels.For any abelian category A, the inclusion functor φ : E → LH(E) induces an equivalence of categories Under this equivalence, conflation-exact functors correspond to conflation-exact functors.
Proof.The proof of [68, Proposition 1.2.34] carries over to this setting.We only note that, since A is abelian, that a right exact functor E → A maps admissible morphisms to admissible morphisms.Remark 3.17.If E is an exact category with kernels, then Proposition 3.16 shows that φ : E → LH(E) is the right abelian envelope of E in the sense of [7, Definition 4.2].

3.5.
The left heart as a localization.Our final result in this section is a description of the left heart of E as a quotient of the category mod(E).To describe this quotient, we first recall the notion of an effaceable functor.Definition 3.18.Let E be a deflation-exact category.We say that an object M ∈ mod(E) is effaceable if M ∼ = coker Y(f ) for a deflation f in E. We write eff(E) for the category of effaceable functors.Proposition 3.19.Let E be a strongly deflation-exact category.If E has kernels, then the category eff(E) is a Serre subcategory of mod(E).
Proof.This is similar to [53,Lemma 2.3].Alternatively, using the Horseshoe lemma, one readily verifies that eff(E) is extension-closed in Mod(E).It follows from Proposition 2.4 that eff(E) is closed under subobjects and quotients in mod(E).
We start with the embedding φ : E → LH(E).By the universal property of mod(E), we find a natural functor φ : mod(E) → LH(E).Theorem 3.20.Let E be strongly deflation-exact category with kernels.The natural right exact functor Proof.Write Q : mod(E) → mod(E)/ eff(E) for the quotient functor.We show that Q • Y : E → mod(E)/ eff(E) satisfies the universal property of φ : E → LH(E) given in Proposition 3.14.
Let F : E → A be a conflation-exact functor, preserving kernels, to an abelian category A. We consider the lift F : mod(E) → A given by the universal property (Theorem 2.1).As F commutes with kernels, we know that F is exact (Proposition 2.2).
Since F (eff(E)) ∼ = 0, we find that F factors through Q : mod(E) → mod(E)/ eff(E).It remains to show that Q • Y preserves kernels and conflations.As both Q and Y commute with kernels, so does the Remark 3.21.In [53, Theorem 2.9], Ogawa shows that mod(E)/ eff(E) ≃ lex(E) for any extriangulated category E with weak kernels (any exact category is extriangulated in the sense of [52]).In [25, Theorem 6.11], Fiorot shows that Theorem 3.20 holds for any (n-)quasi-abelian category E. Hence, for any quasi-abelian category E, we have the following equivalent characterizations of the left heart LH(E): where Q l (E) is the left abelian cover as defined in [62,65].
(1) If E is abelian, then the left t-structure is the standard t-structure; the heart of the standard t-structure is E itself.
(2) If E is quasi-abelian, then the t-structure given here is the left t-structure from [68, Definition 1.2.18].(3) If E is equipped with the split conflation structure (thus, the only conflations are the split kernelcokernel pairs), then D(E) = K(E) ≃ D(mod E) where this last equivalence uses that objects in mod E have projective dimension at most two (as E has kernels).The left t-structure is the canonical t-structure on D(mod E).We see that the heart is equivalent to the category mod E of finitely presented functors (see also Theorem 3.20).

Additive regular categories, admissible kernels, and the admissible intersection property
In this section, we consider additive regular categories (see Definition 4.1 below).We show that, endowing an additive regular category E with the class of conflations consisting of all kernel-cokernel pairs, E has the structure of a deflation-exact category.In fact, the conflations of a deflation-exact category are given by the kernel-cokernel pairs of a regular category if and only if one of the following equivalent conditions hold: E has admissible kernels (Definition 4.6) or E has admissible intersections (Definition 4.7).This will be shown in Proposition 4.12.
As a deflation-exact category, E admits a derived category D(E) and the construction of the left heart LH(E) as in Section 3 goes through: we show that E is a uniformly preresolving subcategory of LH(E) so that the embedding E ֒→ LH(E) lifts to a derived equivalence.

4.1.
Additive regular categories and their conflation structure.We start by defining an additive regular category.This definition is an additive version of a regular category, as defined in [4,11].(1) In [19] (based on [59,69]), a cokernel c was called semi-stable if pullbacks along c exist and the pullback of c is a cokernel.With this terminology, one can reformulate axiom Reg2 as: all cokernels are semi-stable.
(2) In [4, p.122], a regular category is a (not necessarily additive) finitely complete category where the class of regular epimorphisms satisfies the following properties: (i) every morphism f has a factorization f = m • p where p is a regular epimorphism and m is a monomorphism, and (ii) the pullback of a regular epimorphism is a regular epimorphism.As additive regular categories are finitely complete (see Proposition 4.4(1) below), we see that additive categories are precisely those categories which are both additive and regular.Note that a regular epimorphism is an epimorphism that occurs as the coequalizer of a pair of parallel morphisms [9, Definition 4.3.1].Hence, in a (pre)additive category, regular epimorphisms are precisely the cokernel maps.
(3) Let E be an additive regular category.We write E for the class of cokernels and M for the class of monomorphism.As an additive regular category is regular (see item 2 above), it follows from [44] that the pair (E, M) defines a factorization system on E (in the sense of [9, Definition 5. (1) Each morphism in E admits a kernel.
(3) Each cokernel is the cokernel of its kernel, and each kernel is the kernel of its cokernel.
(4) The cokernel-monomorphism factorization in axiom Reg1 is unique up to isomorphism.
Proof.As cokernels have pullbacks in E, every cokernel p : X → Y admits a kernel; this kernel can be found as the pullback along 0 → Y. Let f : X → Y be any morphism, and let f = m•p be a cokernel-mono factorization.We find that ker p = ker f , so that f does admit a kernel.Moreover, p = coker(ker f ) so that all kernel maps have cokernels.
The third statement is standard (see, for example, [51, Proposition I.13.3] together with its dual).For the last statement, let f : X → Y be any morphism in E with cokernel-mono factorization By (3), we see that p = coker(ker p).As ker p = ker f , the uniqueness follows.
Proposition 4.5.Any additive regular category is a deflation-exact category (where the conflations are given by all kernel-cokernel pairs) satisfying axiom R3 + .
Proof.Choosing the class of all kernel-cokernel pairs as conflations, every cokernel is a deflation; this follows from Proposition 4.4.That this conflation structure satisfies axiom R2 is just axiom Reg2.
Hence, by Proposition 2.11, this conflation structure gives a deflation-exact category.It follows from [43,Proposition 5.12] that axiom R3 + is satisfied as well.4.2.On admissible kernels and admissible intersections.In the previous subsection, we started with an additive regular category and endowed it with a conflation structure.In this subsection, we start with a deflation-exact category E and find two properties which are equivalent to E being additive regular.The first property we consider has already been mentioned in [6, §1.3.22] for exact categories.Definition 4.6.Let E be a conflation category.We say that E has admissible kernels if every morphism admits a kernel and kernels are inflations.Having admissible cokernels is defined dually.
In [56], the admissible intersection property is introduced for exact categories (see [15] for some corrections), and in [30,13] for pre-abelian exact categories.It is shown in [31,Theorem 6.1] that a pre-abelian exact category satisfying the admissible intersection property is quasi-abelian.However, the admissible intersection property can be defined for general conflation categories.Definition 4.7.Let E be a conflation category.The category E satisfies the admissible intersection property if for any two inflations f : X Z and g : Y Z, the pullback of f along g exists and is of the following form: The admissible cointersection property is defined dually.
The following lemma (based on [13, Proposition 4.8]) shows that the property of having admissible kernels and the admissible intersection property coincide for conflation categories.Lemma 4.8.Let E be a conflation category such that all split kernel-cokernel pairs are conflations.The following are equivalent: (1) The category E satisfies the admissible intersection property. ( The category E has admissible kernels.
Proof.Assume that the admissible intersection property holds.Let g : Y → Z be a morphism in E. As all split kernel-cokernel pairs are conflations, the sequences As the bottom row is a kernel-cokernel pair and the square is a pullback, it follows that f = ker(( 0 1 ) 1 g ) = ker(g).
The reverse implication follows immediately from [51, Proposition I.13.2],where it is shown that f is the kernel of ( 0 1 ) 1 g and f ′ is the kernel of ( −g 1 )( 1 0 ).
Remark 4.9.The conflations of a conflation category E having admissible kernels, are given by all kernel-cokernel pairs.Moreover, as every cokernel is the cokernel of its kernel, all cokernels are deflations.
For deflation-exact categories, the above lemma can be extended (the proof is an adaptation of [68, Proposition I.1.4]).
Proposition 4.10.Let E be a deflation-exact category.The following are equivalent: (1) The admissible intersection property holds. ( The category E has admissible kernels. (3) Every morphism has a deflation-mono factorization, i.e. any morphism g : Y → Z factors as Y ։ coim(g) ֒→ Z.Moreover, a factorization as in (3) is unique up to isomorphism.Proof.By Remark 2.9, all split kernel-cokernel pairs are conflations in E. The equivalence (1) ⇔ (2) now follows from Lemma 4.8.
Assume (2).Let g : Y → Z be a map.As g admits a kernel which is an inflation, we find a sequence ker(g) We claim that k is a monomorphism, to that end, let t : T → coim(g) be a map such that k • t = 0.By axiom R2, the pullback of t along the deflation h exists and we obtain the following commutative diagram: ker(g) By commutativity of the diagram, g • t ′ = 0 holds and thus there exists a unique map u : a deflation, it is epic, and thus t • h ′ = 0 implies that t = 0.This shows that k is monic and thus (3) holds.The implication (3) ⇒ (2) and the uniqueness of a deflation-mono factorization are straightforward to show.
(1) Every left quasi-abelian category is a deflation-exact category having admissible kernels, or equivalent, satisfying the admissible intersection property.Despite [31, Theorem 6.1], such a category need not be quasi-abelian as it might fail to be exact.Such an example is given by the category LB (see [31,Theorem 3.4] or Section 9).( 2) Despite [31, Theorem 6.1], an exact category with the admissible intersection property might fail to be quasi-abelian as well.Indeed, in [34,Example 7.18] it shown that the exact hull I ex of the Isbell category I need not be pre-abelian.On the other hand, I ex is exact and has the admissible intersection property by Proposition 4.10 and Corollary 5.12.
Proposition 4.12.The following are equivalent for an additive category E.
(1) E is an additive regular category, (2) E is a deflation-exact category with admissible kernels, and (3) E is a deflation-exact category with admissible intersections, where the conflation structure is given by the class of all kernel-cokernel pairs.Proof.This follows from Proposition 4.5 and Proposition 4.10.
Remark 4.13.Being an additive regular category is a property of an (additive) category.In contrast, being a deflation-exact category with admissible kernels (or equivalently, admissible intersections) is a property of a conflation category.We have shown that an additive regular category endowed with the maximal conflation structure is deflation-exact with admissible kernels.
Later in this article, for example in Proposition 4.14, we consider results which produce a deflationexact category having admissible kernels.This is slightly stronger than producing an additive regular category.Indeed, the former means that we get an additive regular category with a conflation structure, and states on top, that this conflation structure is maximal.

Some examples.
We now provide some examples of deflation-exact categories with admissible kernels.We start with an easy criterion.Proposition 4.14.Let E be a deflation-exact category having admissible kernels.If F ⊆ E is a subcategory closed under subobjects, then F is deflation-exact and has admissible kernels.
Proof.Assume that F ⊆ E is closed under subobjects.In particular, F ⊆ E is deflation-closed and thus inherits a deflation-exact structure by Proposition 2.21.Let f : X → Y be a morphism in F .By Proposition 4.10, f admits a deflation-mono factorization X By assumption, ker(f ), coim(f ) ∈ F .The result then follows from Proposition 4.10 as ker(f ) X ։ coim(f ) is a conflation in F and the map f ′′ is a monomorphism in F .
Example 4.15.For any category A, a preradical functor T is a subfunctor of the identity functor on A. Let A be a conflation category.Consider a preradical functor T with embedding η : T → 1 A .Assume now that for each A ∈ A, the given monomorphism η A : T (A) → A is an inflation in A. To any such a preradical functor T , one assigns the full subcategory T consisting of those objects C ∈ A such that η C : T (C) → C is an isomorphism.Using the naturality of T → 1 A , one readily verifies that T ⊆ A is closed under epimorphic quotients (see, for example, [16,Proposition 2]).Indeed, let f : A → B be an epimorphism in A with A ∈ T .Naturality of η gives the following commutative diagram: As η A : T (A) → A is an isomorphism, we find that the composition f • η A = η B • T (f ) is an epimorphism and, hence, so is η B : T (B) B. This shows that η B is an isomorphism so that B ∈ T .
If A is inflation-exact with admissible cokernels and T is a normal preradical functor on A (that is, the monomorphisms η A : T (A) A are inflations), then the dual of Proposition 4.14 yields that T is an inflation-exact category having admissible cokernels.Example 4.17.Let E be a deflation-exact category and let J be any small category.The category E J := Fun(J , E) inherits a deflation-exact structure from E in the following way: a sequence F → G → H in E J is a conflation if and only if F (J) → G(J) → H(J) is a conflation, for every J ∈ Ob(J ).If E has admissible kernels, then so does E J .

The left heart and the exact hull
Let E be a deflation-exact category with admissible kernels.In this section, we have a closer look at the bounded derived category D b (E) and study two subcategories of D b (E): the left heart and the exact hull.
5.1.The left heart.In Section 3, we described the left heart of a deflation-exact category with kernels.We did not require any compatibility between the exact structure and the kernels.In this section, we narrow the scope and consider only those cases where the kernels are inflations.This allows us to strengthen some results presented in Section 3.
Throughout this section, let E be a deflation-exact category with admissible kernels.
The following proposition is a straightforward adaptation of [68, Proposition 1.2.19] and strengthens Proposition 3.8.Proposition 5.1.Let E be a deflation-exact category with admissible kernels.Let We interpret this diagram as a morphism between complexes: the complexes here are given by the rows in the previous diagram.As the top row is an acyclic complex, the morphism . with ker(d n ) in degree 0, up to homotopy.
Proposition 5.2.Let E be a deflation-exact category with admissible kernels.Let φ : E → LH(E) be the canonical embedding.
(2) For every object Z ∈ LH(E), there exists a short exact sequence X Y ։ Z with X, Y ∈ E. Proof.
(2) As Z ∈ LH(E), we know by Proposition 5.1 that Z can be represented by a complex , where i : E → D(E) is the canonical embedding.The long exact sequence coming from the cohomology functors LH i now give the required short exact sequence.Proof.The only improvement over Theorem 3.12 is that res.dim E (LH(E)) ≤ 1.This follows from Proposition 5.2.Proposition 5.5.Let E be a deflation-exact category with admissible kernels and let A be any abelian category.For a conflation-exact functor F : E → A, the following are equivalent: (1) F commutes with kernels, (2) F maps monomorphisms to monomorphisms.
Proof.A morphism is a monomorphism if and only if the kernel is zero.This shows the implication (1) ⇒ (2).For the other implication, let f : X → Y be any morphism in E. Let X p ։ coim f i ֒→ Y be the deflation-mono factorization.We have ker where we have used that F preserves monomorphisms (*) and that F is conflation-exact (**).This shows that F commutes with kernels, as required.
The previous proposition allows for a reformulation of the 2-universal property of the left heart (see Proposition 3.14).
Corollary 5.6.Let E be a deflation-exact category with admissible kernels.The embedding φ : E → LH(E) is 2-universal among conflation-exact functors to abelian categories that preserve monomorphisms.
The following proposition is somewhat of a converse to Corollary 5.4.Proposition 5.7.Let A be an abelian category.Let E ⊆ A be a full subcategory satisfying condition PR1 (thus, every object in A is a quotient of an object in E).If E is closed under subobjects, then E has admissible kernels and LH(E) ≃ A.
Proof.As E is closed under subobjects, we know that E is a uniformly preresolving subcategory of A. By Proposition 4.14, we know that E is a deflation-exact category with admissible kernels.The rest follows from Proposition 3.13.Remark 5.8.In the language of [48, Definition 1.1], the previous result, together with Proposition 5.2 implies that additive regular categories axiomatize subcategories of abelian categories which are generating (that is, satisfying axiom PR1) and are closed under subobjects.5.2.The exact hull.Recall from Proposition 4.12 that an additive regular category E is a deflationexact category with admissible kernels.As a deflation-exact category, it admits an exact hull E ex (see Section 2.5).In this subsection, we show that the exact hull E ex has admissible kernels as well.In other words, the property of having admissible kernels inherits to taking the exact hull.This means that the exact hull of an additive regular category is still an additive regular category.
The exact hull of E is defined as the extension-closure of E in the derived category D b (E).In the following proposition, we describe the exact hull as a subcategory of the left heart of E. Proposition 5.9.There is a fully faithful conflation-exact functor k : E ex → LH(E) for which the diagram is essentially commutative.In particular, the category E ex is a full and extension-closed subcategory of LH(E).Furthermore, φ, j and k all lift to derived equivalences Proof.By the universal property of the embedding j : E → E ex (see Theorem 2.24), the functor φ factors (essentially uniquely) as where k is an exact functor.As φ and j are fully faithful, so is k.To see that E ex is a full and extension-closed subcategory of LH(E), it suffices to note that LH(E) is an extension-closed subcategory of D(E) and that E ex is the extension-closure of E ⊆ LH(E) in D(E).Furthermore, as the embeddings φ and j lift to triangle equivalences D * (E) → D * (LH(E)) and D * (E) → D * (E ex ) for * ∈ {∅, b, −} (see Theorem 2.27 and Corollary 5.4), so does k.
In the following proposition, we use the categories E n from Notation 2.25.
Proposition 5.10.Let E be a deflation-exact category with admissible kernels.The subcategory E ex is closed under subobjects in LH(E).
Proof.Consider a monomorphism X ֒→ Y in LH(E) and assume that Y ∈ E ex .We need to show that X ∈ E ex .By construction, Y ∈ E n for some n ≥ 0. We show, by induction on n, that X ∈ E n as well.For n = 0, Proposition 5.2.(1) yields that X ∈ E 0 .Now assume that n ≥ 1.By definition, there is a conflation A Y ։ B in E ex with A ∈ E n−1 and B ∈ E 0 .Consider the following commutative diagram in LH(E): Here, I is the image of the composition X ֒→ Y → B and P → X is the kernel of X → I.In particular, the top line in this diagram is an exact sequence in LH(E), and thus corresponds to a triangle in D b (E).By [51, Proposition I.13.2], the left square is a pullback and hence the induced map P → A is a monomorphism (as the pullback of a monomorphism is a monomorphism, see [51, Proposition I.7.1]).By the induction hypothesis, P ∈ E n−1 and the base case yields that I ∈ E 0 .It follows that X ∈ E n as required.
Theorem 5.11.Let E be a deflation-exact category with admissible kernels.The exact hull E ex of E also has admissible kernels.
Proof.This follows from Proposition 4.14 and Proposition 5.10.
(3) The embedding j : E → E ex commutes with kernels.(4) A morphism X → Y in E is a deflation if and only if it is a deflation in E ex .
(5) Exact categories with admissible kernels are precisely the extension-closed subcategories of abelian categories that are closed under subobjects. Proof.
(1) Consider a monomorphism f : X ֒→ Y in E ex ⊆ LH(E).Take a morphism t : T → X such that f • t = 0.By Proposition 3.11.(2),there is an epimorphism p : Z → T with Z ∈ E ⊆ E ex .As f is a monomorphism in E ex , it follows from f • t • p that t • p = 0.As p is an epimorphism, we find that f = 0.This shows that f is a monomorphism in LH(E).
(2) Consider a monomorphism f : X ֒→ Y in E ex with Y ∈ E. We have shown that f is also a monomorphism in LH(E).It follows from Proposition 5.2.(1) that X ∈ E. (3) Follows directly from the fact that φ : E → LH(E) preserves kernels.(4) Consider the conflation K X ։ Y in E ex .As E ⊆ E ex is closed under subobjects, we find K ∈ E. We can now use that j : E → E ex reflects conflations (see Theorem 2.24).( 5) Clearly any extension-closed subcategory of an exact category is exact.Combining this fact with Proposition 4.14 yields that any extension-closed subcategory of an abelian category closed under subobjects is an exact category with admissible kernels.Conversely, any exact category E equals its hull E ∼ = E ex .Additionally, if E has admissible kernels, then E ⊆ LH(E) is closed under subobjects by Proposition 5.10.By construction, E ex lies extension-closed in LH(E).This concludes the proof.
It follows from Lemma 2.26 that a morphism f : X → Y in a deflation-exact category that becomes an inflation in E ex is necessarily a monomorphism.However, it gives no criterion for which monomorphisms become inflations.The following provides such a criterion for deflation-exact categories with kernels; these kernels need not be admissible.Proposition 5. 13.Let E be a deflation-exact category satisfying axiom R3.Assume that E admits all kernels.Any inflation f : X Y in E ex with X, Y ∈ E is a finite composition of inflations in E.
Proof.We show that for any conflation X Y ։ Z in E ex with X, Y ∈ E, the map X → Y is a finite composition of inflations in E. As Z ∈ E ex , there is an n ≥ 0 such that Z ∈ E n .We proceed by induction on n ≥ 0. If n = 0, X Y is an inflation in E as the embedding j : E → E ex reflects exactness (see Theorem 2.24).If n ≥ 1, then there exists a conflation A Z ։ B in E ex such that A ∈ E n−1 and B ∈ E. Consider the following commutative diagram in E ex where the upper-right square is bicartesian.By [38, Proposition 5.5], we know that E lies deflationclosed in E. As Y, B ∈ E, we find that P ∈ E. The induction hypothesis now shows that X P is a finite string of inflations.

Quotients of additive regular categories
In [34,35], a quotient/localization theory for (one-sided) exact categories at percolating subcategories is studied.This localization theory simultaneously generalizes localization theories for exact categories developed in [18,67] and provides new examples (even for exact categories).As additive regular categories are deflation-exact, this framework allows to take quotients of additive regular categories.The main result is that a quotient of an additive regular category is again regular as an additive category, and that the induced deflation-exact structure on the quotient consists of all kernel-cokernel pairs.In addition, we provide an easy characterization of percolating subcategories for additive regular categories (see Proposition 6.6).
6.1.Basic definitions and results.We recall the basic definitions and results from [34,35].Definition 6.1.Let E be a conflation category.A non-empty full subcategory A of E is called a deflation-percolating subcategory of E if the following axioms are satisfied: P1 A is a Serre subcategory, meaning: P2 For all morphisms X → A with X ∈ E and A ∈ A, there exists a commutative diagram with A ′ ∈ A and where X ։ A ′ is a deflation.
P3 For any composition X / / i / / Y t / / T which factors through A, there exists a commutative diagram with A ∈ A and such that the square XY AP is a pushout square.
P4 For all maps X f → Y that factor through A and for all inflations A i X (with A ∈ A) such that f • i = 0, the induced map coker(i) → Y factors through A. By dualizing the above axioms one obtains a similar notion of an inflation-percolating subcategory or an inflation-percolating subcategory.Remark 6.2.If E is exact, axiom P3 in the above definition is redundant (see [35,Remark 4.4]).Definition 6.3.Let A be a full additive subcategory of E. A morphism f ∈ Mor(E) is called a weak A-isomorphism if it is a finite composable string of inflations with cokernels in A and deflations with kernels in A. The weak A-isomorphisms are denoted by S A .
The following theorem summarizes the results of [34,35].We write i : E → D b (E) for the canonical embedding.Theorem 6.4.Let E be a deflation-exact category and let A ⊆ E be a deflation-percolating subcategory.
(1) The set S A is a right multiplicative system.
(2) The smallest conflation structure on A ] ex = E/ /A to distinguish it from the one-sided quotient.6.2.Percolating subcategories of deflation-exact categories having admissible kernels.We start with the following proposition, stating that having admissible kernels is stable under quotients.Proposition 6.5.Let E be a deflation-exact category and let A ⊆ E be a deflation-percolating subcategory.If E has admissible kernels, so does E/A.Furthermore, E[S −1 A ] ex has admissible kernels as well.Proof.By Theorem 6.4 the quotient E/A is a deflation-exact category.By Proposition 4.10, it suffices to show that E/A admits kernels and that kernels are inflations.Since S A is a right multiplicative system, the localization functor A ] commutes with kernels.Hence every morphism in E/A has a kernel, moreover, as Q is a conflation-exact functor and every kernel in E is an inflation, kernels in E/A are inflations as well.The last part follows from Theorem 5.11.Proposition 6.6.Let E be a deflation-exact category having admissible kernels and let A ⊆ E be a strictly full additive subcategory.If either E is exact, or if E is pre-abelian, the following are equivalent: (1) A ⊆ E is a deflation-percolating subcategory.
(2) A ⊆ E is a Serre subcategory which is closed under subobjects.
Proof.Assume first that A ⊆ E is a deflation-percolating subcategory.In particular, A is a Serre subcategory.Consider a monomorphism X f ֒→ A in E with A ∈ A. By axiom P2, f factors as X ։ A ′ → A with A ′ ∈ A. As f is monic, so is X ։ A ′ and hence this map is an isomorphism, thus X ∈ A.
Conversely, assume that A is a Serre subcategory which is closed under subobjects.Axiom P1 holds by assumption.Axiom P2 follows immediately from Proposition 4.10.(3).
We now show axiom P4.Let f : X → Y be a map which factors through an object B ∈ A and let i : A X be an inflation such that f • i = 0. We first claim that we may assume X → B to be a deflation and B → Y to be a monomorphism.Indeed, by axiom P2, the map X → B factors as X ։ B ′ → B with B ′ ∈ A. By Proposition 4.10.(3),we find that the composition B ′ → B → Y factors as B ′ ։ B ′′ ֒→ Y .By axiom P1, B ′′ ∈ A and by axiom R1, the composition X ։ B ′ ։ B ′′ is a deflation.This shows the claim.Let p : X ։ X ′ be the cokernel of i : A X. As f • i = 0, we obtain a factorization X ։ X ′ → Y of f .Again, by Proposition 4.10.(3), the map X ′ → Y factors as X ′ ։ X ′′ ֒→ Y .By axiom R1 we obtain the deflation-mono factorization X ։ X ′′ ֒→ Y of f .As deflation-mono factorizations are unique, we conclude that X ′′ ∼ = B and thus axiom P4 holds.
It remains to verify axiom P3.If E is exact, axiom P3 is automatic (see Remark 6.2) and there is nothing to prove.Now assume that E is pre-abelian.Let i : X Y be an inflation and let t : Y → T be a map such that t • i factors as X → A → T with A ∈ A. By axiom P2 we may assume that X → A is a deflation.Write K X ։ A for the corresponding conflation.As E is pre-abelian, the cokernel P of the composition K X Y exists.Hence we obtain the following commutative diagram: Axiom R3, which is satisfied by Proposition 4.5, implies that P → Z is a deflation.Write L P ։ Z for the corresponding conflation.By [35,Proposition 3.7], the square XY LP is bicartesian.In particular, ker(X → L) ∼ = M ∼ = ker(Y → P ).As the map X → L factors through A, we find that coim(X → L) ∈ A.
We write B = coim(X → L) and we write Q = coim(Y → P ).We obtain the following commutative is a conflation by the Nine Lemma.It is now straightforward to check that XY BQ is the desired square for axiom P3.Example 6.7.Consider the category LCA of locally compact abelian groups and let LCA D ⊆ LCA be the full subcategory of discrete abelian groups.By [39], LCA is a quasi-abelian category.As LCA D ⊆ LCA is a Serre subcategory closed under subobjects, Proposition 6.6 yields that LCA D ⊆ LCA is a deflationpercolating subcategory.By Proposition 6.5, the quotient LCA/LCA D is a deflation-exact category having admissible kernels.In particular, it is an additive regular category.
Furthermore, [34,Corollary 6.6] yields that LCA/LCA D is in fact two-sided exact.Thus LCA/LCA D ≃ LCA/ /LCA D .By Pontryagin duality, the quotient LCA/LCA C is an exact category having admissible cokernels.Here, LCA C ⊆ LCA is the full subcategory of compact abelian groups.6.3.Admissibly percolating subcategories.We recall the following special kind of percolating subcategories from [35].This type of percolating subcategories will appear in the next section.Definition 6.8.Let E be a conflation category.An admissibly deflation-percolating subcategory is a subcategory A ⊆ E such that the following axioms hold: A1 A is a Serre subcategory (see Definition 6.1).A2 Every morphism X → A with A ∈ A is admissible with image in A (that is, the morphism f : X → A has a deflation-inflation factorization X ։ A ′ A with A ′ ∈ A.) A3 If f : X ։ A is a deflation with A ∈ A and g : X Y is an inflation, the pushout of f along g exists and is of the following form: An admissibly inflation-percolating subcategory is defined dually.A two-sided admissibly percolating subcategory is both admissibly inflation-percolating and admissibly deflation-percolating.
Example 6.10.Given a filtered ring F R, one can consider a type of filtered representation theory called glider representations as in [17].The category Glid(F R) of glider representations is obtained as a quotient of the quasi-abelian category Preglid(F R) of pregliders by the subcategory Mod(R) (see [36]).
Here, the subcategory Mod(R) ⊆ Glid(F R) is an admissibly deflation-percolating subcategory.It follows that Glid(F R) is a deflation-exact category having admissible kernels.Following [36], there is an embedding Glid(F R) → Mod(F R) of Glid(F R) into an abelian category Mod(F R) which reflects kernels and lifts to a derived equivalence (here, F R is the filtered companion category, see [36,Definition 3.1]).It follows that this lift restricts to an equivalence on the left hearts, i.e.LH(Glid(F R)) ≃ Mod(F R).This recovers and generalizes [66,Theorem 4.20].
The following proposition explains the terminology (see [35,Section 6]).Proposition 6.11.Let E be a deflation-exact category and let A ⊆ E be an admissibly deflationpercolating subcategory.The following properties hold.
(1) The category A is abelian and is a deflation-percolating subcategory of E.
(2) The weak A-isomorphisms are precisely the admissible morphisms f ∈ Mor(E) with ker f, coker f ∈ A.
(3) The set S A of weak isomorphisms satisfies the 2-out-of-3-property and is saturated.
We conclude this section by recalling two useful properties of two-sided admissibly percolating subcategories of an exact category.Theorem 6.12 ([33, Theorem 2.16]).Let E be an exact category and let A ⊆ E be a two-sided admissibly percolating subcategory.A map f : In other words, the exact localization functor Q reflects admissible morphisms.
We end this section with a criterion for percolating subcategories using the language of torsion pairs in a conflation category, which is a direct adaptation from the abelian [20], the exact [35,70], the extriangulated [32], and the homological [12] setting.Definition 6.13.Let C be a conflation category.A torsion pair or a torsion theory is a pair (T , F ) of full and replete subcategories of C such that (1) Hom(T, F ) = 0 for all T ∈ T and F ∈ F , (2) every object M ∈ E fits into a conflation T M ։ F with T ∈ T and F ∈ F .A torsion pair (T , F ) is said to be hereditary if T is closed under subobjects.Proposition 6.14 ([33, Proposition 2.22]).Let E be an exact category with a torsion pair (T , F ).If T ⊆ E satisfies axiom A2, then the subcategory T ⊆ E is two-sided admissibly percolating.

Constructions using Auslander's formula
Throughout this section, let E be a deflation-exact category with admissible kernels.Auslander's formula (see [50, p. 1] or [47,Theorem 2.2]) states that any small abelian category A can be recovered as the quotient mod(A)/ eff(A).The description of the left heart given in Theorem 3.20 as LH(E) ≃ mod(E)/ eff(E) has the same flavor.In this section, we consider two subcategories, mod w.adm (E) and mod adm (E), and consider similar quotients by the subcategory of the effaceable functors.In addition, we show that the effaceable functors form a torsion subcategory of these categories.
More specifically, we show that eff E is a torsion class in mod E; the corresponding torsionfree class is the full subcategory mod 1 (E) consisting of all modules of projective dimension at most one.This observation will play a part in the next section.
The main idea is the following.By Corollary 2.5, the deflation-mono factorization f = m • p of a morphism f in E gives rise to a short exact sequence 0 → coker Y(p) → coker Y(f ) → coker Y(m) → 0 in mod(E).It will follow that Y(m) ∈ mod 1 (E) and Y(p) ∈ eff(E), so that this sequence gives the required decomposition of Y(f ) into a torsion submodule and a torsionfree quotient module.
We also identify two interesting subcategories of mod E by imposing further conditions on the presenting morphism f = m • p: we consider mod adm E of objects of the form coker Y(f ) where m is an inflation in E, and the subcategory mod w.adm E consisting of those objects of the form coker Y(f ) where m is the composition of inflations in E.
The torsion theory (eff(E), mod 1 (E)) in mod E then induces a torsion theory (eff(E), mod 1 adm (E)) in mod adm E and a torsion theory (eff(E), mod 1 w.adm (E)) in mod w.adm E. Finally, the categories mod adm E and mod w.adm E are not abelian, but one can nonetheless consider their quotients by the subcategory of eff E of effaceable functors.By taking these quotients, the inclusions mod adm E ⊆ mod w.adm E ⊆ mod E give a sequence E ⊆ E ex ⊆ LH(E).
7.1.Preparatory notions.We start by formally introducing the categories mod adm (E) and mod w.adm (E) mentioned before.Definition 7.1.
(1) A morphism in E which is the composition of a finite string of inflations is called a weak inflation.A morphism X → Y is called a weakly admissible morphism if it is the composition of a deflation X ։ Z and a weak inflation Z → Y.
(2) We write mod adm (E) for the full subcategory of mod(E) consisting of those functors where f is admissible in We write mod 1 adm (E) for the full subcategory of mod adm (E) consisting of those functors F ∼ = coker(Y(f )) where f is an inflation.
(3) We write mod w.adm (E) for the full subcategory of mod(E) consisting of those functors F ∼ = coker(Y(f )) such that f is weakly admissible in E. We write mod 1 w.adm (E) for the full subcategory of mod w.adm (E) consisting of those functors F ∼ = coker(Y(f )) where f is a weak inflation.Remark 7.2.We have eff(E) ⊆ mod adm (E) ⊆ mod w.adm (E) ⊆ mod(E).
The following proposition explains the notation of mod 1 adm (E) and mod 1 w.adm (E).Proposition 7.3.For any deflation-exact category E with admissible kernels, we have: (1) w.adm (E) = mod 1 (E) ∩ mod w.adm (E).Proof.We only show the first statement, the proof of the second statement is similar.We start by showing the inclusion mod As f is a monomorphism, we have M ∈ mod 1 (E) by Proposition 2.3 and, as f is admissible, we have M ∈ mod 1 (E).
For the inclusion mod 1 (E) ∩ mod adm (E) ⊆ mod 1 adm (E), let M ∈ mod adm (E), say M ∼ = coker Y(f ) for an admissible f = m • p in E. Here, p is a deflation and m is an inflation.Since M ∈ mod 1 (E), we know, by Proposition 2.3 and the uniqueness of a deflation-inflation factorization, that p is a retraction.Hence coker Y(f ) = coker Y(m) ∈ mod 1 adm (E).The following lemma is an adaptation of [33,Lemma 3.27].
. By the Comparison Theorem ([14, Theorem 12.4]) and the fact that the Yoneda embedding is fully faithful, the sequence 0 → ker(g) → B → C → 0 is homotopy equivalent to the acyclic sequence 0 → X Y ։ Z → 0.
If E satisfies axiom R3, then by Proposition 2.17, we find that g : B → C is a deflation.Without the assumption of axiom R3, we may still enlarge the conflation structure on E until it satisfies axiom R3 (as E is weakly idempotent complete, we obtain the closure under axiom R3 as the closure under axiom R3 − , see [37,Proposition 3.3 and Corollary 7.14]).We can now use Proposition 2.17 to see that 0 → ker(g) → B → C → 0 is a conflation in the new conflation structure.By [37,Proposition 7.18], we find that there exists an object B ′ ∈ B such that 0 → ker(g) ⊕ B ′ B ⊕ B ′ ։ C → 0 is a conflation in the original conflation structure.

7.2.
A torsion theory for exact categories.Let F be an exact category with admissible kernels.Our main example will be F = E ex where E is a deflation-exact category with admissible kernels.We show that the subcategory eff(F ) of effaceable functors is a torsion class in mod(F ).This serves as a starting point for the other torsion pairs we will give in this section.Proposition 7.5.Let F be an exact category with admissible kernels.There is a hereditary torsion pair (eff(F ), mod 1 (F )) on mod(F ).
Proof.It follows from Proposition 3.19 that eff(F ) is a Serre subcategory of mod(F ).Corollary 2.5 shows that every M ∈ mod(F ) is the extension of a torsion-free module by a torsion module.We only need to show that Hom(eff(F ), mod 1 (F )) = 0.For this, consider a morphism η : F → G with F ∈ eff(F ) and G ∈ mod 1 (F ).We can choose projective presentations of F and G as follows: Note that E admits all pullbacks as it has admissible kernels.Using the notation of Proposition 2.4, we find that g ′ is an isomorphism as it is both a monomorphism (as pullback of a monomorphism) and a deflation (by applying axiom R3 + to the composition f = g ′ • β ′′ ).It follows that im(η) = 0 and hence that Hom(F, G) = 0.

7.3.
A torsion theory on mod E. By Proposition 7.5, we know there is a hereditary torsion theory (eff(E ex ), mod 1 (E ex )) on mod(E ex ).We intersect this torsion theory with mod(E) ⊆ mod(E ex ) to find a torsion theory on mod(E).
Remark 7.6.For any E deflation-exact category with admissible kernels.As j : E → E ex commutes with kernels, the natural fully faithful functor Lemma 7.7.Let E be a deflation-exact category with admissible kernels.We have ( 1) Proof.
(1) The inclusion eff(E) ⊆ eff(E ex ) ∩ mod(E) uses only that j : E → E ex is conflation-exact.For the inclusion eff(E ex ) ∩ mod(E) ⊆ eff(E), consider an object M ∼ = coker Y(f ) ∈ mod(E) where f is a morphism in E. By Lemma 7.4, we know that f is a deflation in E ex .Hence, by Corollary 5.12.(4), we know that f is a deflation in E.
As M ∈ mod(E), we know that M ∼ = coker Y(f ) for a morphism f in E ⊆ E ex .It follows from Proposition 2.3 that f = m • p in E ex where p is a retraction and m a monomorphism.As E is closed under subobjects in E ex , we find that the factorization f = m • p also holds in E. The statement now follows from the isomorphism coker Y(f ) ∼ = coker Y(m).
Proposition 7.8.Let E be a deflation-exact category with admissible kernels.There is a hereditary torsion pair (eff(E), mod 1 (E)) on mod(E).
Proof.It follows from Corollary 2.5 that every object in mod(E) is the extension of a torsion-free object by a torsion object.The other properties (i.e. that eff(E) is a Serre subcategory and that Hom(eff(E), mod 7.4.The exact hull and a torsion theory on mod w.adm (E).In this subsection, we consider the category mod w.adm (E), given as the full subcategory of mod(E) consisting of those objects with are presentable by a weak admissible morphism in E. We will show in Proposition 7.11 that this is an extension-closed subcategory, and hence exact.Next, we show that (eff(E), mod 1 w.adm (E)) is a torsion theory on mod w.adm (E) by intersecting the torsion theory from Proposition 7.5 with mod w.adm (E).Finally, we consider the quotient mod w.adm (E)/ eff(E) and show that it is equivalent to the exact hull E ex .Lemma 7.9.Let E be a deflation-exact category with admissible kernels.We have (1) eff(E) = eff(E ex ) ∩ mod w.adm (E), and (2) mod 1 w.adm (E) = mod 1 (E ex ) ∩ mod w.adm (E).Proof.The same argument as in Lemma 7.7 works.
In addition, we have the following description of mod w.adm (E).Lemma 7.10.Let E be a deflation-exact category with admissible kernels.We have mod w.adm (E) = mod adm (E ex ) ∩ mod(E).
Proof.The inclusion mod w.adm (E) ⊆ mod adm (E ex ) ∩ mod(E) only uses that j : E → E ex is conflationexact and that E ex satsifies axiom L1.For the other inclusion, take M ∈ mod(E) ∩ mod adm (E ex ).As M ∈ mod adm (E ex ), we can write M ∼ = coker(Y(f )) where f ∈ Hom E ex (X, Y ) is an admissible morphism in E ex .As M ∈ mod(E), we can write F ∼ = coker(Y(g)) for some g ∈ Hom E (U, V ).By the Comparison Theorem (see [14,Theorem 12.4]), the two resolutions of M are homotopy equivalent in mod(E ex ).As the Yoneda embedding is fully faithful and left exact, we obtain the following commutative diagram in E ex which defines a homotopy equivalence between the rows: As the lower row is acyclic in E ex , [14, Proposition 10.14] (or Proposition 2.17) implies that the upper row is acyclic in E ex as well (this uses that E ex has kernels, see Theorem 5.11 and hence is weakly idempotent complete).In particular, g is an admissible morphism in E ex .As E ⊆ E ex is closed under subobjects (see Corollary 5.12), one sees that ker(g), coim(g) ∈ E. By Proposition 5.13, the map coim(g) → V is a finite composition of inflations in E. This shows that M ∼ = coker(Y(g)) ∈ mod w.adm (E).
Proposition 7.11.The category mod w.adm (E) lies extension-closed in mod(E).In particular, mod w.adm (E) is an exact category.
Proof.It follows from Corollary 2.5 that every object in mod w.adm (E) is the extension of an object in mod 1  w.adm (E) by an object in eff(E).The other properties of a torsion pair follow easily from combining Proposition 7.5 and Lemma 7.9 (taking F = E ex ).
Corollary 7.13.The quotient mod w.adm (E)/ eff(E) is an exact category.Moreover, the exact categories mod w.adm (E)/ eff(E) and E ex are equivalent.
Proof.By Proposition 7.12, Theorem 6.4 and its dual, mod w.adm (E)/ eff(E) is an exact category.We write Q : mod w.adm (E) → mod w.adm (E)/ eff(E) for the corresponding localization functor.
We first claim that the composition As the Yoneda embedding is left exact, we obtain an exact sequence Y(X) Y(Y ) → Y(Z) ։ coker(Y(g)) in mod w.adm (E).Applying Q to this sequence, we obtain the conflation QY(X) QY(Y ) ։ QY(Z) as coker(Y(g)) ∈ eff(E).This shows that Q • Y is conflation-exact.
We show that Q • Y satisfies the universal property of j : E → E ex and thus the desired equivalence.Let Φ : E → F be a conflation-exact functor to an exact category F .We construct an exact functor Φ : mod w.adm (E) → F as follows.By the universal property of the exact hull, there is a unique (up to isomorphism) exact functor Φ ex : E ex → F such that Φ ex • j = Φ.By [33,Theorem 3.9], there is a unique (up to isomorphism) exact functor Φ ex : mod adm (E ex ) → F such that Φ ex • Y ex = Φ ex .Here, Y ex : E ex → mod(E ex ) is the Yoneda embedding of E ex .Clearly Φ ex (eff(E ex )) ∼ = 0.The restriction of Φ ex to mod w.adm (E) is still an exact functor and maps eff(E) to zero.Hence this functor further factors through Q : mod w.adm (E) → mod w.adm (E)/ eff(E) via an exact functor Φ : mod w.adm (E)/ eff(E) → F as required.This concludes the proof.7.5.One-sided Auslander's formula.Let E be a deflation-exact category with admissible kernels.We start with the following straightforward observation.Lemma 7.14.
(1) Inflation and admissible morphisms are stable under pullbacks in E.
(2) Weak inflation and weak admissible morphisms are stable under pullbacks in E.
Proof.Since kernels are stable under pullbacks and E has admissible kernels, the pullback of an inflation is an inflation.That weak inflations are stable under pullbacks follows from the first statement together with the Pullback Lemma.That (weak) admissible morphisms are stable under pullbacks then follows from the Pullback Lemma, axiom R2, and the first statement.
Proposition 7.15.The subcategory mod adm (E) of mod(E) is closed under subobjects.In particular, mod adm (E) inherits a deflation-exact structure having admissible kernels.
Proof.Let η : F ֒→ G be a monomorphism in mod(E) and assume that G ∈ mod adm (E).Let f : A → B and g : C → D be morphisms in E such that F ∼ = coker(Y(f )) and G ∼ = coker(Y(g)).We may assume that g is admissible in E. The map η : F ֒→ G induces a commutative square where the right square is a pullback square and E ∼ = coker(Y(α)).As E ∈ eff(E), Lemma 7.4 shows that α is a deflation.Since h ′ is obtained from the admissible morphism h via a pullback, h ′ itself is admissible (see Lemma 7.14).Hence, using axiom R1, we see that h ′ •α is admissible and hence so is the composition is a deflation and p ′′ is an inflation.Since E satisfies axiom R3 + by Proposition 4.5, it follows from [37, Theorem 1.2] that p ′ : A → B ′ is a deflation.Since p = p ′ • p ′′ , this shows that p is admissible.
It remains to show that eff(E) ⊆ mod adm (E) is an admissibly deflation-percolating subcategory.Axiom A1 follows directly from Proposition 7.12.For axiom A2, consider a morphism f : F → E with F ∈ mod adm (E) and E ∈ eff(E).As eff(E) satisfies axiom A2 in mod w.adm (E), it suffices to show that ker f ∈ mod adm (E).This is automatic since mod adm (E) is closed under subobjects in mod(E), see Proposition 7.15.
It remains to show axiom A3.To that end, consider a conflation F G ։ H in mod adm (E) and a map F ։ E with E ∈ eff(E).We obtain the following commutative diagram in Mod(E) where the left square is a pushout.By Lemma 7.16, P ∈ mod adm (E) as required.This completes the proof.
(1) The Yoneda embedding Y : E → mod adm (E) is left conflation-exact (see Definition 2.6) and maps admissible morphisms to admissible morphisms; (2) If F is a deflation-exact category having admissible kernels and Φ : E → F is a left exact functor that preserves admissible morphisms, then there exists a functor Φ : mod adm (E) → F , unique up to isomorphism, which is exact and satisfies Φ • Y = Φ.
Proof.We show that the Yoneda embedding Y : E → mod adm (E) maps admissible morphisms to admissible morphisms; the remainder of the proof is then a straightforward adaptation of [33,Theorem 3.9].Let f : X → Y be any admissible morphism in E, and let k : K X be the kernel.Using Proposition 2.4, starting from the commutative square  As L • Q • Y ∼ = 1 E , we know that Q • Y is fully faithful.We only need to show that Q • Y is essentially surjective.For this, take an arbitrary M ∈ Ob(mod adm (E)) = Ob(mod adm (E)/ eff(E)).Let f : X → Y be an admissible morphism in E with M ∼ = coker Y(f ).
From the deflation-inflation factorization X As coker(h) ∼ = coker Y(q) ∈ eff(E), we see that h is a weak isomorphism as well.We find that Proof.
(1) As every object of mod(E) has projective dimension at most two, we can apply Theorem 2.22.
(2) Theorem 6.4 one obtains the upper row.The right equivalences follow from the above.By Proposition 2.17, Ac b (E) is a thick triangulated subcategory of K b (E) and thus D b eff(E) (mod adm (E)) is equivalent to Ac b (E) as both categories are obtained as the kernel of the same Verdier localization.7.6.Some derived equivalences.Let E be a deflation-exact category with admissible kernels.In Definition 7.1, we introduced mod adm (E) and mod w.adm (E), as well as their subcategories mod 1 adm (E) and mod 1 w.adm (E) of objects of projective dimension at most one.We now show that these categories are derived equivalent.We start with the following observation.

The left heart as a localization of hMon(E)
Let E be an additive regular category.In Section 7, we showed that the left heart LH(E) can be obtained as the quotient mod(E)/ eff(E).When E is quasi-abelian, then it has been shown in [68,73] that the left heart of E can be described as a localization of the category hMon(E) of monomorphisms in E (up to homotopy).In this section, we give a similar description of the left heart of a deflation-exact category with admissible kernels.
Our approach is the following.Let (T , F ) be a hereditary torsion theory in an abelian category A. It follows from Proposition 8.1 below that the quotient A/T can be described as a localization of the torsionfree class F ; specifically, one formally inverts all bimorphisms in F .
Applying this to the torsion pair (eff E, mod 1 E) in mod E shows that the quotient mod E/ eff E(≃ LH(E)) can be obtained as a localization of mod 1 (E) at the class of all bimorphisms (= morphisms that are both epimorphisms and monomorphisms).All that is left, is then to study the map Mon(E) → mod 1 (E).
The following observation allows us to obtain Theorem 8.8 from the results in Section 7.
Proposition 8.1.Let (T , F ) be a hereditary torsion theory in abelian category A. Let Σ T ⊆ A be the set of all morphisms f such that ker f, coker f ∈ T . ( As T is a Serre subcategory of A, we know that Σ T is a multiplicative system.By [27, Proposition 3.1], the localization functor T ] commutes with kernels and cokernels (and thus is exact).Now write t : A → T for the torsion functor and write f : A → F for the torsion-free functor.For any object A ∈ A, the short exact sequence t Note that F has kernels (these coincide with kernels in A) and cokernels (these are given by f • coker A ). Hence, T ] commutes with kernels and cokernels.It now follows from [27,Proposition I.3.4] that Σ T ∩ Mor F is a multiplicative system in F .
Note that a morphism f ∈ Mor(F ) lies in Σ T if and only if ker A (f ), coker A (f ) ∈ T , which is equivalent to ker F (f ), coker F (f ) = 0.This is then equivalent to f being both a monomorphism and an epimorphism in F . The for which there exists a morphism t : E 0 → F −1 satisfying t • δ E = u −1 and δ F • t = u 0 .We define the category hMon(E) as Mon(E)/I.

Remark 8.3.
There is a natural full embedding Mon(E) → C(E), mapping a monomorphism δ E : E −1 ֒→ E 0 in E to a complex with E −1 and E 0 in degrees −1 and 0, respectively, and zero elsewhere.For the category hMon(E), there is a similar full embedding into K(E).
We show that this square is bicartesian.Since coker φ ∈ eff(E), it follows from Lemma 7.4 and Proposition 2.4 that g α : C ⊕ B → D is a deflation.Next we take the pullback of g along α and use the notation from Proposition 2.4.Using that φ is a monomorphism and that ker(g) = 0, we have that β ′′ : A → E is a retraction.Furthermore, using that g ′ • β ′′ = f is a monomorphism, we see that β ′′ is an isomorphism.This shows that the square ABCD is a pullback.Hence, A → C ⊕ B is the kernel of the deflation C ⊕ B ։ D, so that A C ⊕ B ։ D is a conflation and may conclude that the square ABCD is both a pullback and a pushout.The other implication, that a bicartesion square in E corresponds to a bimorphism in mod 1 (E), follows easily from Proposition 2.4.
The previous proposition motivates introducing the following notation.Notation 8.7.We write S for those morphisms u : δ E → δ F such that δF is a bicartesian square.Furthermore, we write θ : hMon(E) → K(E) for the embedding functor in Remark 8.3, mapping a monomorphism δ E : E −1 ֒→ E 0 in E to a complex with E −1 and E 0 in degrees −1 and 0, respectively, and zero elsewhere.
Theorem 8.8.In the category hMon(E), the class S of all bicartesian squares is a left and right multiplicative system.The natural functor hMon(E) → LH(E) induces an equivalence hMon(E)[S −1 ] → LH(E).
Proof.Consider a morphism u : δ E → δ F given by the commutative diagram: The cone of θ(u) is given by / / F 0 is a conflation (equivalently, a kernel-cokernel pair by Remark 4.9).The latter is clearly equivalent to requiring the above square to be bicartesian.This completes the proof.Remark 8.10.We turn our attention back to the category Mon(E).Let N be the class of all objects X ֒→ Y which are isomorphisms, and let [N ] be the ideal of Mon(E) consisting of all morphisms factoring through an object of N .It is straightforward to verify that hMon(E) ≃ Mon(E)/[N ].It follows from Proposition 8.6 that the set S ⊆ Mor(Mon(E)) of bicartesian squares is precisely the set of all bimorphisms in hMon(E).In this case, the localization Mon(E)[S −1 ] has also been denoted by Mon(E)/N in [63] (this notion differs from the one used in Section 6 as N is neither an inflation-nor deflation-percolating subcategory of Mon(E)).
With a small abuse of notation, we write S for the class of bicartesian squares in both Mon(E) and hMon(E), cf.Notation 8.7.Proof.Consider a map u : δ E → δ F in Mon(E) and assume that u is null-homotopic.Then there exists a map h : E 0 → F −1 such that the diagram commutes.It follows that u factors as follows: As the square / / 0 is bicartesian, u = 0 in Mon(E)[S −1 ].From this one readily deduces that Mon(E)[S −1 ] ≃ hMon(E)[S −1 ], and the result follows.
(1) The set S hWInf(E) := S ∩ Mor(WInf(E)) is a right multiplicative system in hWInf(E).Moreover, we have hWInf(E)[S −1 hWInf(E) ] ≃ E ex .(2) The set S hInf(E) := S ∩ Mor(Inf(E)) is a right multiplicative system in hInf(E).Moreover, we have hInf(E)[S −1 hInf(E) ] ≃ E. Proof.Following Lemma 7.14, we know that weak inflations are stable under pullbacks.Hence, for any morphism f : δ E → δ F in hMon(E), if f is a pullback square and δ F ∈ hWInf(E), we know that δ E ∈ hWInf(E).It now follows from [41, Proposition 7.2.1] that S hWInf(E) is a right multiplicative set and the induced functor hWInf(E)[S −1 hWInf(E) ] → hMon(E)[S −1 ] is fully faithful.It follows from Theorem 2.24 that every object Z ∈ E ex fits in a conflation X Y ։ Z in E ex with X, Y ∈ E. It follows from Proposition 5.13 that the inflation X Y in E ex is a finite composition of inflations in E. Therefore the restriction of the functor coker •Y : hMon(E) → LH(E) to the subcategory hWInf(E) gives an equivalence between hWInf(E)[S −1 hWInf(E) ] and the subcategory E ex of LH(E).The second statement is proven in a similar fashion.

The heart of the LB-spaces
For this section, let k be either the field of real or the field of complex numbers.Let us denote by LB the category of LB-spaces.Its objects are by definition all those Hausdorff locally convex topological k-vector spaces (X, τ ) that can be represented by an N-indexed direct limit of Banach spaces, meaning that there are Banach spaces X ֒→ X 1 ֒→ X 2 ֒→ • • • with continuous injective linking maps such that X = ∞ n=1 X n holds as linear spaces and τ is the finest linear topology that makes all inclusion maps X n ֒→ (X, τ ) continuous.A morphism between LB-spaces is by definition a linear and continuous map.
It is well-known that LB is a pre-abelian category.Indeed, given a morphism f : X → Y , then its cokernel is given by coker(f ) : Y → Y /f (X) where f (X) is the topological closure of f (X) and where Y /f (X) is endowed with the locally convex quotient topology (this is then again of the LB-type explained above).Its kernel is given by ker(f ) : f −1 (0) → X where f −1 (0) carries the direct limit topology of the sequence X 0 ∩ f −1 (0) ֒→ X 1 ∩ f −1 (0) ֒→ X 2 ∩ f −1 (0) ֒→ • • • of Banach spaces.The latter can be strictly finer than the subspace topology, see [54,Example 6.8.13] for an example.To indicate that we are not using the subspace topology, we will write f −1 (0) ♭ for the kernel in LB.From this discussion it follows that a pair of composable morphisms is a kernel-cokernel pair in LB if and only if f is injective, g is surjective and f (X) = g −1 (0) holds as linear spaces.Observe that, in this case, f (X) ⊆ Y is automatically closed, but that f (X) (or, equivalently, g −1 (0)) endowed with the induced topology of Y is in general not an LB-space.We C all for the class of all kernel-cokernel pairs in LB.
Theorem 9.1.The category LB is a deflation-exact category with respect to the conflation structure C all of all kernel-cokernel pairs.In particular, (LB, C all ) has admissible kernels.The conflation structure C all is not exact.
Proof.By [31,Theorem 3.4], which had been mentioned without proof in [45, p. 540], the category LB is deflation quasi-abelian but not inflation quasi-abelian.This means explicitly, that in every pullback diagram in which a is a kernel but d is not.The latter statement implies immediately that C all cannot be an exact structure.
Remark 9.2.At first sight, and in light of Lemma 4.8, the previous result might appear to be inconsistent with [31, Theorem 6.1] which reads 'every pre-abelian category with the admissible intersection property is quasi-abelian'.Notice however, that in [31, Theorem 6.1] the admissible intersection property is required with respect to a conflation structure which is exact.
Applying our results from the previous sections, the category LB admits a heart which is by Theorem 8.8 equivalent to the localization of its monomorphism category modulo homotopy (denoted earlier in this paper by hMon(LB)) by the class of bicartesian squares.Writing the latter down explicitly for the LB-spaces gives LH(LB) ≃ (hMon LB) {bicartesian squares} −1   where the right hand side coincides with the category that was defined in an ad hoc fashion and established to be abelian in [73, Theorem 10 and Proposition 14] (see also [68]).In addition to recovering this ad hoc approach, our results show that the category defined in [73] is indeed derived equivalent to the category we started with.Proof.By Theorem 9.1 the category LB is deflation-exact and has admissible kernels.Thus, Theorem 3.12 and Proposition 5.9 imply the result.Remark 9.4.It is shown in Theorem 8.8 that the class S of all bicartesian squares is a multiplicative system in hMon(LB).It follows from Proposition 8.11 that LH(LB) ≃ Mon(LB)[S −1 ], so one can opt to describe LH(LB) starting from Mon(LB) instead of hMon(LB).However, the class of bicartesian squares S is not a multiplicative system in Mon(LB).Indeed, the localization Mon(LB) → Mon(LB)[S −1 ] does not commute with kernels as can be seen from the following example.Let E ∈ LB be a nonzero object and consider the following two objects in Mon(LB): the zero morphism δ : 0 → E and the identity δ ′ : E → E.
In addition to the natural, but non-exact, conflation structure C all , the category LB admits at least two natural conflation structures that are exact.Let us denote by E top the class of topologically exact sequences which consists by definition of all pairs (f, g) of composable morphisms that form an exact sequence of vector spaces in which f is closed and g is open.Notice that, due to the Open Mapping Theorem for LB-spaces, the second condition is satisfied automatically.On the other hand, let us write E max for the conflation structure given by all kernel-cokernel pairs (f, g) in which every pushout of f is again a kernel and every pullback of g is again a cokernel.By [59,69], the latter is the maximal exact structure.where we understand that g −1 (0) carries the subspace topology.(2) [21, Proposition 2.2.4 and Remark 2.2.6]The conflation category (LB, E max ) is exact, we have E max = (f, g) ∈ C all Hom(g −1 (0) ♭ , k) = Hom(g −1 (0), k) as vector spaces and E top ⊂ E max is a proper subclass.
Let us mention that the exact structure E top is inherited by LB from the category of all Hausdorff locally convex spaces, see [22].The latter category is quasi-abelian and its topologically exact sequences are precisely all kernel-cokernel pairs.Our final theorem shows however, that no exact structure on LB does induce a derived equivalence with (LB, C all ).Proof.As E ⊆ C all , the identity (LB, E) → (LB, C all ) lifts to a triangle functor D * (LB, E) → D * (LB, C all ).As (LB, C all ) is not exact, there is a conflation X Y ։ Z in (LB, C all ) which is not a conflation in (LB, E).Extending the above conflation to a complex U • , Proposition 2.17 implies that U • ∈ Ac(LB, C all ) but U • / ∈ Ac(LB, E) by Proposition 2.17.It follows that D * (LB, E) → D * (LB, C all ) is not faithful.
Remark 9.7.As the proof indicates, the statement of Theorem 9.6 holds after replacing LB with any additive regular category which is not an exact category (when endowed with the maximal conflation structure).The dual statement holds for additive coregular categories.
We conclude this article by outlining that the dual situation, i.e. inflation-exact categories having admissible cokernels (or, thus, additive coregular categories), appear naturally in the functional analytic context as well.
Example 9.8.The category COM of complete Hausdorff locally convex spaces, furnished with linear and continuous maps as morphisms, is inflation quasi-abelian and not deflation quasi-abelian, see [31,Theorem 3.3].As in the proof of Theorem 9.1 it follows that the latter category is inflation-exact and has admissible cokernels if endowed with the conflation structure consisting of all kernel-cokernel pairs.The latter contains the maximal exact structure as a proper subclass.Consequently, the embedding (COM, C all ) → RH(COM) lifts to an equivalence of bounded derived categories, whereas the functor (COM, E max ) → RH(COM) does not.
Example 9.9.Let Top sc Z be the category of complete and separated topological groups with linear topology.Likewise, for a field k, we write Top sc k for the category of complete and separated topological kvector spaces with linear topology.It is shown in [55, Proposition 8.3] that the categories Top sc Z and Top sc k are inflation quasi-abelian (thus, inflation-exact categories with admissible cokernels, or equivalently, satisfying the admissible cointersection property) but not quasi-abelian.
every morphism f in E for which M ∼ = coker Y(f ) factors as f = m • p where p is a retraction and m is a monomorphism.Proof.Straightforward adaptation of [2, Proposition 1.1].

Corollary 2 . 5 .
Let E be an additive category.Let f : A → B be any morphism in E with factorization f = m • p where m is a monomorphism.There is an associated exact sequence in mod(E) 0 → coker Y(p) → coker Y(f ) → coker Y(m) → 0. Proof.The given factorization gives the following commutative diagram in E : Applying the Yoneda embedding Y : E → mod(E) and then taking the cokernels of the vertical maps, we find a sequence F φ → G ψ → H in mod(E) where F = coker(Y(p)) and H = coker(Y(m)).By Proposition 2.4 2 and Corollary 3.6] by references to a deflation-exact version [37, Lemma 4.2.(2) and Theorem 4.1] (or the duals of [5, Lemma 5.10 and Proposition 5.11]).

Theorem 2 .( 1 )
22 ([38, Theorem 1.1]).Let E be a deflation-exact category and let A ⊆ E be a full additive subcategory.If A is preresolving, the embedding A → E lifts to a triangle equivalence D -(A) ≃ − → D -(E).(2) If A is finitely preresolving, the embedding A → E lifts to a triangle equivalence D b (A) ≃ − → D b (E).(3) If A is uniformly preresolving, the embedding A → E lifts to a triangle equivalence D(A) ≃ − → D(E).

2. 5 .
The exact hull.The following is based on[34, Section 7]; the exact hull of a one-sided exact category also appeared in [61, Proposition I.7.5].Definition 2.23.Let E be a deflation-exact category.The exact hull E ex of E is the extension closure of i(E) ⊆ D b (E).

Notation 2 . 25 .
For a deflation-exact category E, we write E 0 for the full subcategory of D b (E) consisting of stalk complexes concentrated in degree 0. The subcategories E n are recursively defined as all objects B which fit into a triangleA → B → C → ΣA with A ∈ E n−1 and C ∈ E 0 .With this notation, we have E ex = n≥0 E n ; this uses [6, Lemme 1.3.10].

Definition 4 . 1 .
An additive category is called additive regular if Reg1 every morphism f has a factorization f = m • p where p is a cokernel and m is a monomorphism, and Reg2 the pullback along every cokernel exists and the pullback of a cokernel is a cokernel.The dual of an additive regular category is called an additive coregular category.Remark 4.2.

1 )
−−−→ Z are conflations.By the admissible intersection property, we have the following pullback diagram: As a more specific example, let R be a ring and let I ⊳ R be a left ideal.Let Mod(R) be the category of right R-modules.The functor T mapping M ∈ Mod(R) to T (M ) = M I is a normal preradical functor.The corresponding subcategory T = {M ∈ Mod(R) | M = M I} of Mod(R) is an inflation-exact category having admissible cokernels.Example 4.16.Let R be a commutative artin ring and let A be an artin R-algebra.Let M ∈ mod(A) be a finitely generated module.Denote by fac(M ) the full additive subcategory of mod(A) consisting of factor modules of finite direct sums of M .It follows from Proposition 4.14 that fac(M ) is an inflationexact category with admissible cokernels.If Hom A (M, τ M ) = 0, that is, M is τ -rigid ([1, Definition 0.1]), then fac(M ) is an extension-closed subcategory of mod(A) and hence exact (see[3, Theorem 5.10]).
We consider the deflation-mono factorizationC n−1 β ։ coim(d n−1) γ ֒→ ker(d n ) from Proposition 4.10, giving us the following commutative diagram ) Let f : X ֒→ Y be a monomorphism in LH(E) with Y ∈ E. It follows from Proposition 3.11(2) that there is an epimorphism g : B → X in LH(E) with B ∈ E. Consider the deflation-mono factorization B ։ coim(f • g) ֒→ Y of the morphism f • g in E. Embedding this factorization in LH(E) gives a deflation-mono factorization of f • g in LH(E) (this uses Proposition 3.11(1) and (

Remark 5 . 3 .
For any map f : X → Y in E, the deflation-mono factorization in E (see Proposition 4.10) coincides with the epi-mono factorization of f in the abelian category LH(E).Corollary 5.4.Let E be a deflation-exact category with admissible kernels.The category E is a uniformly preresolving subcategory of LH(E) with res.dim E (LH(E)) ≤ 1.Consequently, the embedding lifts to a triangle equivalence Φ : D * (E) → D * (LH(E)) for * ∈ {−, b, ∅}.
is a deflation-exact structure.(3) The functor Q satisfies the 2-universal property of the quotient E/A of deflation-exact categories.(4) The localization sequence A → E → E/A induces a Verdier localization sequence D b A (E) → D b (E) → D b (E/A), here D b A (E) is the thick triangulated subcategory of D b (E) generated by i(A).If, in addition, E is two-sided exact, E[S −1 A ] ex satisfies the 2-universal property of a quotient of exact categories.We write E[S −1

Proof.
It follows from the above description of L : mod adm (E) → E that L(eff(E)) = 0. Hence, by the universal property of the quotient, L factors as mod adm (E) Q − → mod adm (E)/ eff(E) L − → E. We find that Q•Y is left adjoint to L (see [27, Lemma 1.3.1]with F = L, G = Q, and D = Y, or [36, Proposition 2.11.(1a)] with F = Y, G = Q, and H = L).

p։
X ′ m Y of f , we obtain the following conflation (see Corollary 2.5):0 → coker Y(p) → M g − → coker Y(m) → 0.As coker Y(p) ∈ eff(E), the map g : M → coker Y(m) is a weak isomorphism, i.e.Q(g) is an isomorphism.Consider now the conflationX ′ m Y q ։ Z in E.As the Yoneda embedding is left conflation-exact, we get the following diagram

a
is a cokernel whenever this is true for d, and that there exists a pushout diagram

Theorem 9 . 6 .
Let E be any exact structure on LB.Then (LB, E) → (LB, C all ) does not lift to a triangle equivalence D * (LB, E) → D * (LB, C all ).Consequently, none of the natural functors D * (LB, E top/max ) → D * (LH(LB, C all )) is a triangle equivalence, either.
Derived categories of one-sided exact categories.The derived category of a one-sided exact category was studied in 3) If E satisfies axiom R3 and is idempotent complete, then Ac C (E) is a thick triangulated subcategory of K(E).
16e result now follows from Lemma 7.14.Lemma 7.16.Let E P ։ H be a short exact sequence in mod(E).If E ∈ eff(E) and H ∈ mod adm (E), then P ∈ mod adm (E).Proof.Let p : A → B and h : C → D be maps in E such that P ∼ = coker(Y(p)), H ∼ = coker(Y(h)) and h is admissible.By Proposition 2.4, the map P ։ H induces the following commutative diagram in E: [33,llary 7.19.Let E be a deflation-exact category having admissible kernels.The Yoneda embedding Y : E → mod adm (E) has a left adjoint, sending an object M ∼ = coker Y(f ) ∈ mod adm (E) to coker(f ), for each admissible morphism f ∈ E.Proof.The proof is as in[33, Corollary 3.10].The left adjoint L : mod adm (E) → E is obtained by applying Theorem 7.18 to the identity E → E, that is, L • Y ∼ = 1.The explicit description is obtained using that L commutes with cokernels.Theorem 7.20.Let E be a deflation-exact category having admissible kernels.The Yoneda embedding Y : E → mod adm (E) induces an equivalence E ≃ mod adm (E)/ eff(E).

•
Combining Proposition 4.5 and Proposition 2.17, we see that u is a quasi-isomorphism if and only if cone