## Introduction

The relation between triangulated categories and their higher categorical enhancements—pretriangulated dg, pretriangulated $A_{\infty }$ or stable $\infty $-categorical—has been under investigation for several years now. One reason is that while triangulated categories have grown remarkably important in representation theory and algebraic geometry, many of the constructions one wants to make rely on the functoriality that comes with an enhancement. Many instances of this phenomenon appeared in the recent developments of derived algebraic geometry: for example, in relation to deformation theory and moduli problems.

In this paper, we stick to the language of dg (differential graded) categories. We recall that, roughly, a dg enhancement (or simply an enhancement) of a triangulated category ${\mathscr {T}}$ is a pretriangulated dg category whose homotopy category is equivalent to ${\mathscr {T}}$. It is relevant to note that the ‘natural’ triangulated categories of algebra and geometry come with ‘natural’ dg enhancements. For example, the derived categories ${\mathbf {D}}^?({\mathscr {A}})$ of an abelian category ${\mathscr {A}}$, where $?=\emptyset ,b,+,-$ (i.e., where the cohomology is assumed unbounded, bounded, bounded below or bounded above), as well as the category $\mathrm {Perf}(X)$ of perfect complexes on a quasi-compact and quasi-separated scheme, all have ‘natural’ dg enhancements by construction. This existence does not hold in general. Indeed, there are well-known examples of ‘topological’ triangulated categories that do not admit dg enhancements (see, for instance, [Reference Canonaco and Stellari9, Section 3.2]). More recently, it has been proved in [Reference Rizzardo and Van den Bergh40] that there exist triangulated categories that are linear over a field and without a dg enhancement.

A priori, there is no good reason to expect different enhancements of the same triangulated category to be ‘comparable’. This is important because constructions that take place in an enhancement may depend on the choice of enhancement. And the right notion of ‘comparability’ turns out to be that two enhancements are declared equivalent if they agree up to isomorphism in the homotopy category ${\mathbf {Hqe}}$ of the category of (small) dg categories. Consequently, one says that a triangulated category has a unique enhancement if any two enhancements are isomorphic in ${\mathbf {Hqe}}$. As ${\mathbf {Hqe}}$ is the localisation of the category of dg categories by quasi-equivalences, two dg categories that are isomorphic in ${\mathbf {Hqe}}$ have equivalent homotopy categories, but the converse need not be true. So far, very few examples of triangulated categories admitting non-unique enhancements have been produced. A ‘classical’ one is reported in [Reference Canonaco and Stellari9, Section 3.3] (see also Corollary 6.12). If one requires that everything be linear over a field, the first example was recently found by Rizzardo and Van den Bergh [Reference Rizzardo and Van den Bergh39].

Back to ${\mathbf {Hqe}}$: one can describe all morphisms in this category thanks to the seminal work of Toën [Reference Toën46] (see also [Reference Canonaco, Ornaghi and Stellari8, Reference Canonaco and Stellari11]). Indeed, for the natural enhancements of geometric categories, such as the bounded derived categories of coherent sheaves on smooth projective schemes, the morphisms in ${\mathbf {Hqe}}$ between them are all lifts of exact functors of a special form: the so-called Fourier–Mukai functors (see [Reference Lunts and Schnürer28, Reference Toën46] and [Reference Canonaco and Stellari9, Reference Canonaco and Stellari10] for a survey).

The way the triangulated and dg sides of this picture should be related was pinned down, in the geometric setting, in the seminal work [Reference Bondal, Larsen and Lunts6] by Bondal–Larsen–Lunts, where it is conjectured that

(C1) The geometric triangulated categories ${\mathbf {D}}^b({\mathbf {Coh}}(X))$, ${\mathbf {D}}({\mathbf {Qcoh}}(X))$ and ${\mathbf {Perf}}(X)$ should have a unique dg enhancement when

*X*is a quasi-projective scheme (i.e., any two dg enhancements should be isomorphic in ${\mathbf {Hqe}}$).(C2) If $X_1$ and $X_2$ are smooth projective schemes, then all exact functors between ${\mathbf {D}}^b({\mathbf {Coh}}(X_1))$ and ${\mathbf {D}}^b({\mathbf {Coh}}(X_2))$ should lift to morphisms in ${\mathbf {Hqe}}$.

Conjecture (C2) has recently been disproved in [Reference Rizzardo, Van den Bergh and Neeman38], and even when a lift exists, it is not unique in general by [Reference Canonaco and Stellari12]. On the other hand, special cases of (C1) have been proved to be correct, in increasing generality, by several authors over the last decade. Let us briefly go through this part of the story; after all, this article belongs to this string of results.

The first breakthrough in the direction of (C1) came from the beautiful work by Lunts and Orlov [Reference Lunts and Orlov27], which proved, among other things, that ${\mathbf {D}}({\mathscr {G}})$ has a unique enhancement when ${\mathscr {G}}$ is a Grothendieck abelian category with a small set of compact generators. This result implies that (C1) holds true for ${\mathbf {D}}({\mathbf {Qcoh}}(X))$ when *X* is a quasi-compact and separated scheme with enough locally free sheaves (see [Reference Lunts and Orlov27, Theorem 2.10]). This was extended to all Grothendieck abelian categories in [Reference Canonaco and Stellari13] by using the theory of well-generated triangulated categories. Hence (C1) holds for ${\mathbf {D}}({\mathbf {Qcoh}}(X))$ when *X* is any scheme or algebraic stack.

As for ${\mathbf {D}}^b({\mathbf {Coh}}(X))$ and ${\mathbf {Perf}}(X)$, Lunts and Orlov show in [Reference Lunts and Orlov27] that they have unique enhancements when *X* is a quasi-projective scheme (see Theorems 2.12 and 2.13 in [Reference Lunts and Orlov27]). Clearly, together with the previous result, this implies that (C1) holds even in greater generality. Actually, an additional improvement of the argument in [Reference Lunts and Orlov27] allowed the first and third author to prove that ${\mathbf {D}}^b({\mathbf {Coh}}(X))$ and ${\mathbf {Perf}}(X)$ have unique enhancements when *X* is any noetherian scheme with enough locally free sheaves (see Corollaries 6.11 and 7.2 in [Reference Canonaco and Stellari13]).

Fresh air was brought into the subject with the advent of the powerful theory of $\infty $-categories. Specifically, Antieau [Reference Antieau1] reconsidered the problem of uniqueness of enhancements, taking a completely different approach employing Lurie’s work on prestable $\infty $-categories (see [Reference Lurie30, Appendix C]). Using this amazing machinery, he proved the beautiful result that ${\mathbf {D}}^?({\mathscr {A}})$ has a unique enhancement when $?=b,+,-$ and ${\mathscr {A}}$ is any small abelian category. It should be noted that restricting to small categories is a minor issue, as explained in Section 3.5.

If ${\mathscr {A}}$ is not only abelian but also a Grothendieck category, one can, following Lurie, construct three interesting triangulated categories out of ${\mathscr {A}}$: the usual derived category ${\mathbf {D}}({\mathscr {A}})$, the *unseparated derived category* $\check {{\mathbf {D}}}({\mathscr {A}})$ and the *left completed derived category* $\widehat {{\mathbf {D}}}({\mathscr {A}})$. We know all about ${\mathbf {D}}({\mathscr {A}})$, and, in particular, we know that it has a unique enhancement by [Reference Canonaco and Stellari13]. The triangulated category $\check {{\mathbf {D}}}({\mathscr {A}})$ is nothing but the homotopy category of injectives in ${\mathscr {A}}$ and has been extensively studied by Krause in [Reference Krause23]. The uniqueness of enhancements for $\check {{\mathbf {D}}}({\mathscr {A}})$, when ${\mathscr {A}}$ is not only Grothendieck but also locally coherent, is one of the main results of Antieau (see [Reference Antieau1, Theorem 1]). The triangulated category $\widehat {{\mathbf {D}}}({\mathscr {A}})$ is more mysterious. It does not seem to have a purely triangulated description, and it should be thought of as a remedy to the fact that, in general, ${\mathbf {D}}({\mathscr {A}})$ is not left complete (see [Reference Neeman32]). In [Reference Antieau1], the uniqueness of the enhancement for $\widehat {{\mathbf {D}}}({\mathscr {A}})$ is stated as an open and challenging problem (see [Reference Antieau1, Question 8.1]).

Antieau’s striking achievements offered what appeared to be conclusive evidence of the superiority of the $\infty $-category machine, and our initial, humble goal was to try to find out how much of his opus could be obtained by more primitive methods. The authors were surprised by the outcome of what started as a modest project: beginning with a few simple new ideas, we ended up not only improving on Antieau’s results, but also solving most of the open problems in the literature.

Our first precise statement is the following:

Theorem A. Let ${\mathscr {A}}$ be an abelian category:

(1) The triangulated category ${\mathbf {D}}^?({\mathscr {A}})$ has a unique dg enhancement when $?=b,+,-,\emptyset $.

(2) If ${\mathscr {A}}$ is a Grothendieck abelian category, then $\check {{\mathbf {D}}}({\mathscr {A}})$ and $\widehat {{\mathbf {D}}}({\mathscr {A}})$ have unique dg enhancements.

The striking and new part of (1) is the uniqueness for ${\mathbf {D}}({\mathscr {A}})$ for *every* abelian category ${\mathscr {A}}$. Nonetheless, our approach will uniformly and harmlessly produce uniqueness of enhancements for ${\mathbf {D}}^?({\mathscr {A}})$, for $?=b,-,+$, thus recovering Antieau’s results in a completely different way. Part (2) of Theorem A on the one hand generalises [Reference Antieau1, Theorem 1] and, on the other hand, answers the questions in [Reference Antieau1] about $\widehat {{\mathbf {D}}}({\mathscr {A}})$ that we mentioned above. In particular, Theorem A (2) gives a positive answer to Question 4.7 in [Reference Canonaco and Stellari9].

Back to the geometric setting. If *X* is a quasi-compact and quasi-separated scheme, then the category ${\mathbf {D}}({\mathbf {Qcoh}}(X))$ is in general not equivalent to ${\mathbf {D}}_{{\mathbf {qc}}}(X)$, the full triangulated subcategory of the category of complexes of $\mathcal {O}_X$-modules consisting of all complexes with quasi-coherent cohomology. Thus the uniqueness of dg enhancements for ${\mathbf {D}}_{{\mathbf {qc}}}(X)$ cannot directly be deduced from Theorem A. Nonetheless, this category is of primary interest: for example, because the category of perfect complexes on *X* coincides with the subcategory of compact objects in ${\mathbf {D}}_{{\mathbf {qc}}}(X)$.

The uniqueness of dg enhancements for ${\mathbf {D}}_{{\mathbf {qc}}}(X)$ and ${\mathbf {Perf}}(X)$ was formulated as an open problem by Antieau in [Reference Antieau1, Question 8.16], and our second main result positively answers his question in the context of dg enhancements:

Theorem B. Let *X* be a quasi-compact and quasi-separated scheme. Then the categories ${\mathbf {D}}^?_{{\mathbf {qc}}}(X)$ and ${\mathbf {Perf}}(X)$ have a unique dg enhancement for $?=b,+,-,\emptyset $.

The case of ${\mathbf {Perf}}(X)$ is covered by [Reference Antieau1, Corollary 9] under the stronger assumption that the scheme is quasi-compact, quasi-separated and $0$-complicial. The latter condition, which we do not need to make explicit here, roughly refers to a property of ${\mathbf {Perf}}(X)$ induced by the t-structure on ${\mathbf {D}}_{{\mathbf {qc}}}(X)$ and predicts how perfect complexes with only non-negative cohomologies are generated by perfect quasi-coherent sheaves. As we will explain below, part of the interest of Theorem B is that the proof introduces a new technique to study the uniqueness of enhancements based on homotopy limits.

### Applications: past, present and future

We have already said that the key to our approach lies in a few simple new ideas. One can ask if these ideas might be relevant or useful in other contexts. Hence we should mention that a linchpin to our approach, namely Proposition 2.9 and its proof, has already been used by the second author in [Reference Neeman31] to provide a counterexample to conjectures by Schlichting [Reference Schlichting43] and by Antieau, Gepner and Heller [Reference Antieau, Gepner and Heller2] about vanishing in negative *K*-theory. Note that this application is totally unrelated to the content of the current article.

Within the present paper, there are further, easy applications of our main results. In particular, we deduce that the following triangulated categories have unique enhancements:

• ${\mathbf {D}}^?({\mathbf {Qcoh}}(X))$, for

*X*any scheme or algebraic stack, and ${\mathbf {D}}^?({\mathbf {Coh}}(X))$, for any scheme or any locally noetherian algebraic stack, for $?=\emptyset ,b,+,-$ (see Corollary 5.9);• ${\mathbf {D}}({\mathscr {G}})^{\alpha }$, where ${\mathscr {G}}$ is a Grothendieck abelian category and $\alpha $ is a large regular cardinal (see Corollary 5.8). Recall here that ${\mathbf {D}}({\mathscr {G}})$ is a well-generated triangulated category and ${\mathbf {D}}({\mathscr {G}})^{\alpha }$ is the full triangulated subcategory consisting of its $\alpha $-compact objects;

• ${\mathbf {K}}^?({\mathscr {A}})$, for any abelian category ${\mathscr {A}}$ and for $?=\emptyset ,b,+,-$ (see Corollary 5.6).

The first two items together yield a complete positive answer to Question 4.8 in [Reference Canonaco and Stellari10].

Another interesting and surprising application, discussed in Section 5.5, is that we recover and generalise the construction of the *realisation functor* of Beĭlinson, Bernstein and Deligne [Reference Beĭlinson, Bernstein and Deligne4]. This functor plays a key role in the study of triangulated categories with t-structures. The original, involved proof is replaced in Section 5.5 by a different approach, combining the vital Proposition 2.9 with the techniques in Section 4 to deliver the results easily.

In future work, we will analyse how our uniqueness results apply to study the liftability of exact functors along the lines of (C2) above. More specifically, we will investigate a classical conjecture by Rickard asserting that all autoequivalences of ${\mathbf {D}}(\mathrm {Mod}(R))$, where *R* is a commutative ring, are liftable. We have already (briefly) discussed the case of projective schemes. The affine case is still very challenging and essentially open. We will explain how our techniques provide simplifications and generalisations of the existing results.

Still with an eye to the future, we conclude this discussion by pointing out that there remain a couple of situations of high algebro-geometric interest where the (non-)uniqueness of the enhancements needs to be fully understood: the categories of matrix factorisations and the case of admissible subcategories of triangulated categories admitting a unique enhancement. If we work with categories and functors linear over $\mathbb {Z}$, then there are examples of categories of matrix factorisations with non-unique enhancements (see [Reference Dugger and Shipley17, Reference Schlichting42]). Similarly, in Section 6.4, we provide an example of a $\mathbb {Z}$-linear triangulated category with a unique enhancement (by Theorem A) but with an admissible subcategory with non-unique $\mathbb {Z}$-linear enhancements (see Corollary 6.12). It remains open to understand if similar examples can be found for categories linear over a field and if one can find admissible subcategories with non-unique enhancements in ${\mathbf {D}}^b({\mathbf {Coh}}(X))$ when *X* is a smooth projective scheme.

### The strategy of the proofs

The one-sentence summary of the proof of Theorem A (1) would be that it is an elaborate study of special generators for ${\mathbf {D}}^?({\mathscr {A}})$, coupled with a suitable description of ${\mathbf {D}}^?({\mathscr {A}})$ as a Verdier quotient. The same principle underlies all the existing papers in the literature proving the uniqueness of enhancements of derived categories of abelian categories. The many papers differ in which generators they use and what quotient they study.

In the current article, we realise ${\mathbf {D}}^?({\mathscr {A}})$ as a quotient of the homotopy category ${\mathbf {K}}^?({\mathscr {A}})$ and show that ${\mathbf {K}}^?({\mathscr {A}})$ is generated in $3$ steps by objects that are direct sums of shifts of objects in ${\mathscr {A}}$ (Proposition 2.9 and Corollary 2.10). The key fact here, namely that ${\mathbf {K}}^?({\mathscr {A}})$ is generated in $3$ steps by the simple objects described above, can be seen by combining Krause’s [Reference Krause23, Lemma 3.1 and its proof] with Max Kelly’s old result [Reference Kelly22] (see [Reference Neeman37, Theorem 7.5 and its proof] for a modern account). But we include a full proof in this article because we make use of the explicit three steps that suffice.

This is different from and, in a sense, more natural than the point of view of [Reference Lunts and Orlov27] and [Reference Canonaco and Stellari13, Reference Canonaco and Stellari14]. In those earlier papers, to prove that ${\mathbf {D}}({\mathscr {G}})$ has a unique enhancement for ${\mathscr {G}}$ a Grothendieck abelian category, one uses the strong and special property that ${\mathscr {G}}$ has a generator. Thus one can take generators for ${\mathbf {D}}({\mathscr {G}})$, which all live in degree $0$, and ${\mathbf {D}}({\mathscr {G}})$ is a suitable quotient of the derived category of modules over the category formed by these generators.

The technical complications of our approach involve, at the triangulated level, a careful analysis of certain special products and coproducts. It is discussed in Section 1 and reverberates at the dg level where one has to construct suitable dg enhancements of ${\mathbf {K}}^?({\mathscr {A}})$ and ${\mathbf {D}}^?({\mathscr {A}})$ and an intricate zigzag of dg functors linking them. The dg work is carried out in Section 4. Again, the dg part of the argument is simpler in [Reference Lunts and Orlov27, Reference Canonaco and Stellari13] and involves a short zigzag diagram consisting of one roof of dg functors. The last part of the proof in Section 5 is then very close in spirit to the argument in Section 4 and 5 of [Reference Lunts and Orlov27] (and thus in Section 4 of [Reference Canonaco and Stellari13]).

The proof of the uniqueness of dg enhancements for $\check {{\mathbf {D}}}({\mathscr {A}})$ in Section 6.1 is a reduction to Theorem C in [Reference Canonaco and Stellari13, Reference Canonaco and Stellari14] (see Theorem 6.2). It uses the work of Krause [Reference Krause23] to show that $\check {{\mathbf {D}}}({\mathscr {A}})$ is a well-generated triangulated category and can be written as a quotient of the derived category of the abelian category of modules over the abelian subcategory ${\mathscr {A}}^{\alpha }$ of ${\mathscr {A}}$, which consists of the $\alpha $-presentable objects in ${\mathscr {A}}$ (here, $\alpha $ is a sufficiently large regular cardinal).

Finally, the case of $\widehat {{\mathbf {D}}}({\mathscr {A}})$ is studied in Section 6.3, and the proof makes use of the natural t-structure induced on $\widehat {{\mathbf {D}}}({\mathscr {A}})$ by ${\mathbf {D}}({\mathscr {A}})$. With this t-structure, we have a natural equivalence $\widehat {{\mathbf {D}}}({\mathscr {A}})^+\cong {\mathbf {D}}^+({\mathscr {A}})$. We can invoke Theorem A (1) and then deduce the result by a careful analysis of the compatibility with homotopy colimits. It should be noted that here we need to use that ${\mathbf {D}}^?({\mathscr {A}})$ has a semi-strongly unique dg enhancement (see Remark 5.4). Roughly, this means if ${\mathscr {C}}_1$ and ${\mathscr {C}}_2$ are two dg enhancements of ${\mathbf {D}}^?({\mathscr {A}})$ (i.e., there are exact equivalences $\mathsf {E}_i\colon H^0({\mathscr {C}}_i)\xrightarrow {\sim }{\mathbf {D}}^?({\mathscr {A}})$), then the isomorphism $f\colon {\mathscr {C}}_1\xrightarrow {\sim }{\mathscr {C}}_2$ in ${\mathbf {Hqe}}$ provided by Theorem A (1) is such that $H^0(f)(X)\cong \mathsf {E}_2^{-1}\circ \mathsf {E}_1(X)$, for all *X* in $H^0({\mathscr {C}}_1)$.

The strategy of the proof of Theorem B is new and is based on the idea of realising a dg enhancement of ${\mathbf {D}}^?_{{\mathbf {qc}}}(X)$ (and of ${\mathbf {Perf}}(X))$ as the homotopy limit of dg enhancements of the derived category of the open subschemes in an affine open cover of *X*. More precisely: in Section 7, we prove that given any enhancement ${\mathscr {C}}$ of ${\mathbf {D}}^?_{{\mathbf {qc}}}(X)$ (or ${\mathbf {Perf}}(X))$, one can produce an isomorphism in ${\mathbf {Hqe}}$ between ${\mathscr {C}}$ and the homotopy limit of induced enhancements of the derived categories of the affine schemes in the cover (and of their finite intersections). This can be deduced from Theorem 7.4, which is a general criterion involving the simpler case of homotopy pullbacks.

This has the clear advantage that for each Zariski open subset *U* in the covering of *X* and all their finite intersections, one knows that ${\mathbf {D}}^?_{{\mathbf {qc}}}(U)\cong {\mathbf {D}}^?({\mathbf {Qcoh}}(U))$, since *U* is quasi-compact and separated. Thus the uniqueness of their enhancements is guaranteed by Theorem A (1). The hard work comes in showing that the constructions in Section 4, and thus the proof of Theorem A (1) in Section 5, are compatible with restriction to appropriate open subschemes. In Section 8, we show this compatibility with the special homotopy limits we are considering, concluding the proof.

### Related work

Throughout this article, we work with dg enhancements. Since Antieau’s enhancements are stable $\infty $-categories, a comparison requires one to invoke results like [Reference Antieau1, Meta Theorem 13] and [Reference Cohn15]. These results assert that for most of the triangulated categories we study, the difference is immaterial; the uniqueness of enhancement problems in the two different settings are equivalent. This is clear for the triangulated categories in Theorem A. As for the categories ${\mathbf {Perf}}(X)$ and ${\mathbf {D}}^?_{{\mathbf {qc}}}(X)$ in Theorem B, the situation is a bit more delicate. Indeed, as pointed out in Section 8.2 of [Reference Antieau1], Meta Theorem 13 may not automatically apply, and, by [Reference Cohn15], Theorem B only implies that the two triangulated categories mentioned above have unique $\mathbb {Z}$-linearised stable $\infty $-enhancements. We leave this to the interested reader.

The reason we stick to (pretriangulated) dg categories is for convenience—it makes the key Proposition 2.9 and Corollary 2.10 easier and cleaner to use. In the triangulated category ${\mathbf {D}}^?({\mathscr {A}})$, the generators given by Proposition 2.9 and Corollary 2.10 have (non-canonical) direct sum decompositions, and using this is easier in enhancements that are additive. In fact, the interested reader can check that for us, the case where $?=b$ is easy. The subtlety comes in dealing with $?\in \{+,-,\infty \}$, where issues of limits come up. And as far as we can tell, the $\infty $-category machine would not help much here.

Of course, Antieau [Reference Antieau1] has taught us that a clever application of the $\infty $-category machine can lead to spectacular progress. It is entirely possible that our new ideas, combined with the powerful machinery, will produce startling advances. For example, the local-to-global approach of Theorem B might readily be amenable to the $\infty $-category methods.

We leave this to the experts.

### Structure of the paper

In Section 1, we deal with some foundational questions about products and coproducts in the triangulated categories ${\mathbf {K}}^?({\mathscr {A}})$ and ${\mathbf {D}}^?({\mathscr {A}})$. We also discuss their behaviour with respect to exact functors.

Section 2 introduces various notions of generation for triangulated categories. In particular, in Section 2.1, we define well-generated triangulated categories, and we set the stage for our study of $\check {{\mathbf {D}}}({\mathscr {G}})$. Section 2.2 is about the approach to the generation of ${\mathbf {K}}^?({\mathscr {A}})$ and ${\mathbf {D}}^?({\mathscr {A}})$, which is crucial in the proof of Theorem A (1).

In Section 3, we introduce some standard material about dg categories and, at the same time, slightly extend known results and constructions such as Drinfeld’s notion of homotopy flat resolution (see Section 3.2). In Section 3.3, we study homotopy limits and homotopy pullbacks. We also reconsider localisations in the dg context (Section 3.4) and carefully define the notion of dg enhancement and why their uniqueness is independent of the universe (see Section 3.5).

Section 4 is devoted to the construction of appropriate enhancements for ${\mathbf {K}}^?({\mathscr {A}})$ and ${\mathbf {D}}^?({\mathscr {A}})$. This naturally leads to the proof of Theorem A (1) in Section 5. The second part of Theorem A is proved in the subsequent section and uses somewhat different techniques.

The proof of Theorem B occupies Section 7 and Section 8. The first of the two sections sets up the general technique and criterion that link homotopy pullbacks and limits to dg enhancements. The second combines this with Theorem A to finish the argument.

### Notation and conventions

All preadditive categories and all additive functors are assumed to be $\Bbbk $-linear for a fixed commutative ring $\Bbbk $. By a $\Bbbk $-linear category, we mean a category whose Hom-spaces are $\Bbbk $-modules and such that the compositions are $\Bbbk $-bilinear, not assuming that finite coproducts exist.

With the small exception of Section 3.5, throughout the paper, we assume that a universe containing an infinite set is fixed. Several definitions concerning dg categories need special care because they may, in principle, require a change of universe. The major subtle logical issues in this sense can be overcome in view of [Reference Lunts and Orlov27, Appendix A] and Section 3.5. A careful reader should look at them; but in this paper, after these delicate issues are appropriately discussed, we will not explicitly mention the universe we are working in. The members of this universe will be called small sets. Unless stated otherwise, we always assume that the Hom-spaces in a category form a small set. A category is called *small* if the isomorphism classes of its objects form a small set.

If ${\mathscr {T}}$ is a triangulated category and ${\mathscr {S}}$ a full triangulated subcategory of ${\mathscr {T}}$, we denote by ${\mathscr {T}}/{\mathscr {S}}$ the Verdier quotient of ${\mathscr {T}}$ by ${\mathscr {S}}$. In general, ${\mathscr {T}}/{\mathscr {S}}$ is not a category according to our convention (meaning the Hom-spaces in ${\mathscr {T}}/{\mathscr {S}}$ need not be small sets), but it is in many common situations, for instance when ${\mathscr {T}}$ is small.

## 1 The triangulated categories

In this section, we discuss some properties of most of the triangulated categories whose dg enhancements are studied in this paper. The focus is on the existence of (co)products of special objects and the commutativity of such (co)products with exact functors.

### 1.1 The categories

Let us recall that when ${\mathscr {A}}$ is a small additive category, then ${\mathbf {K}}({\mathscr {A}})$ denotes the homotopy category of complexes. Namely, its objects are cochain complexes of objects in ${\mathscr {A}}$, while its morphisms are homotopy equivalence classes of morphisms of complexes. For $A^{*}\in \mathrm {Ob}({\mathbf {K}}({\mathscr {A}}))$, we denote by $A^i$ its *i*th component. We can then define the full subcategories ${\mathbf {K}}^b({\mathscr {A}})$, ${\mathbf {K}}^+({\mathscr {A}})$, ${\mathbf {K}}^-({\mathscr {A}})$ of the category ${\mathbf {K}}({\mathscr {A}})$ whose objects are

For $?=b,+,-,\emptyset $, we single out the full subcategory ${\mathbf {V}}^?({\mathscr {A}})\subseteq {\mathbf {K}}^?({\mathscr {A}})$ consisting of objects with zero differentials. The properties of such a subcategory will be studied in Section 2 and will be crucial in the rest of this paper. Here we just point out that for an object $A^{*}\in {\mathbf {V}}^?({\mathscr {A}})$, we will use the shorthand

to remind that the object $A^i\in {\mathscr {A}}$ is placed in degree *i*.

Remark 1.1. It is not difficult to prove that $\bigoplus _{i\in \mathbb {Z}}{A^i}[-i]$ satisfies both the universal property of product and coproduct in ${\mathbf {K}}^?({\mathscr {A}})$. Namely, there are canonical isomorphisms

When ${\mathscr {A}}$ is an abelian category, the full triangulated subcategory ${\mathbf {K}}^?_{\mathrm {acy}}({\mathscr {A}})\subseteq {\mathbf {K}}^?({\mathscr {A}})$ consists of *acyclic complexes*: that is, objects in ${\mathbf {K}}({\mathscr {A}})$ with trivial cohomology. The triangulated category ${\mathbf {D}}^?({\mathscr {A}})$ is then the Verdier quotient of ${\mathbf {K}}^?({\mathscr {A}})$ by ${\mathbf {K}}^?_{\mathrm {acy}}({\mathscr {A}})$, and it comes with a quotient functor

We can then consider the full subcategory ${\mathbf {B}}^?({\mathscr {A}})\subseteq {\mathbf {D}}^?({\mathscr {A}})$ as

Remark 1.2. By the definition of the Verdier quotient, ${\mathbf {B}}^?({\mathscr {A}})$ has the same objects as ${\mathbf {V}}^?({\mathscr {A}})$. Thus we will freely denote them by $\bigoplus {A^i}[-i]$. But since the morphisms in ${\mathbf {D}}^?({\mathscr {A}})$ differ from those in ${\mathbf {K}}^?({\mathscr {A}})$ in a significant way, we should not expect $\bigoplus {A^i}[-i]$ to automatically satisfy the universal properties of either product or coproduct in ${\mathbf {D}}^?({\mathscr {A}})$. This will be discussed later.

Remark 1.3. It is interesting to observe that if ${\mathscr {A}}$ is a small abelian category, then [Reference Krause23, Lemma 3.1] implies that there is a (small) abelian category ${\mathscr {B}}$ and an exact equivalence ${\mathbf {K}}^?({\mathscr {A}})\cong {\mathbf {D}}^?({\mathscr {B}})$ for $?=b,+,-,\emptyset $. To be precise, the category ${\mathscr {B}}$ is the abelian category of coherent ${\mathscr {A}}$-modules. The result follows from [Reference Verdier47, Proposition III 2.4.4 (c)], once we observe that any coherent ${\mathscr {A}}$-module has a projective resolution of length at most $2$ (see the proof of [Reference Krause23, Lemma 3.1]).

If ${\mathscr {G}}$ is a Grothendieck abelian category, then it contains enough injectives, and one can take the full subcategory $\mathrm {Inj}({\mathscr {G}})\subseteq {\mathscr {G}}$ of injective objects. According to Lurie’s terminology [Reference Lurie29], the *unseparated derived category* of ${\mathscr {G}}$ is the triangulated category

This category has been extensively studied by the second author [Reference Neeman35] and Krause [Reference Krause23]. Some of its properties will be recalled later. For the moment, we content ourselves with the simple observation that it fits in the following localisation sequence

Namely, $\mathsf {Q}$ and $\mathsf {J}$ have right adjoints $\mathsf {Q}^{\rho }$ and $\mathsf {J}^{\rho }$, respectively. Under some additional assumptions (e.g., if ${\mathbf {D}}({\mathscr {G}})$ is compactly generated), $\mathsf {Q}$ and $\mathsf {J}$ have left adjoints $\mathsf {Q}^{\lambda }$ and $\mathsf {J}^{\lambda }$ as well.

Example 1.4. A case where the latter situation is realised can be obtained as follows. Let *p* be a prime number and $\mathbb {F}_p=\mathbb {Z}/p\mathbb {Z}$ the field with *p* elements. Consider the rings $R_1:=\mathbb {Z}/p^2\mathbb {Z}$ and $R_2:=\mathbb {F}_p[\varepsilon ]$ (where $\varepsilon ^2=0$). For $i=1,2$, set ${\mathscr {T}}_i:=\check {{\mathbf {D}}}(\mathrm {Mod}(R_i))$, and denote by ${\mathscr {S}}_i$ the full subcategory of ${\mathscr {T}}_i$ consisting of acyclic complexes. The ring $R_i$ is noetherian and ${\mathbf {D}}(\mathrm {Mod}(R_i))$ is compactly generated. Thus the inclusion of ${\mathscr {S}}_i$ has left adjoint, and ${\mathscr {S}}_i$ is a localising and admissible subcategory of ${\mathscr {T}}_i$.

The last triangulated category we study in this paper is the completed derived category of a Grothendieck category. Since, to the best of our knowledge, its definition intrinsically involves pretriangulated dg categories, this discussion is postponed to Section 6.2.

### 1.2 More on products and coproducts

As we observed in Remark 1.2, objects with zero differentials need not always agree with the obvious products or coproducts in ${\mathbf {D}}^?({\mathscr {A}})$. In this section, we provide sufficient conditions for agreement.

Let us introduce some notation, slightly generalising the problem. Let ${\mathscr {A}}$ be a small abelian category, and let $\{A_n^{*}\}_{n\ge 0}$ be a sequence of objects in ${\mathbf {D}}^?({\mathscr {A}}))$. If either $A_n^i=0$ for all $i>-n$ or $A_n^i=0$ for all $i<n$, we can consider the complex $\bigoplus _{n=0}^{\infty } A_n^{*}\in {\mathbf {D}}({\mathscr {A}})$, which is the termwise direct sum of the complexes $A_n$. Note that this makes sense because under our assumptions, each term of the complex $\bigoplus _{n=0}^{\infty } A_n^{*}\in {\mathbf {D}}({\mathscr {A}})$ consists of a finite direct sum.

With this in mind, we can prove the following.

Lemma 1.5. Let ${\mathscr {A}}$ be a small abelian category, and let $\{A_n^{*}\}_{n\ge 0}\subseteq \mathrm {Ob}({\mathbf {D}}^?({\mathscr {A}}))$:

(1) If $A_n^i=0$ for all $i>-n$ and $?=-,\emptyset $, then $\bigoplus _{n=0}^{\infty } A_n^{*}$ is a coproduct in ${\mathbf {D}}^?({\mathscr {A}})$: that is, there is a canonical isomorphism

$$\begin{align*}\bigoplus_{n=0}^{\infty} A_n^{*}\cong\coprod_{n=0}^{\infty} A_n^{*}. \end{align*}$$(2) If $A_n^i=0$ for all $i<n$ and $?=+,\emptyset $, then $\bigoplus _{n=0}^{\infty } A_n^{*}$ is a product in ${\mathbf {D}}^?({\mathscr {A}})$: that is, there is a canonical isomorphism

$$\begin{align*}\bigoplus_{n=0}^{\infty} A_n^{*}\cong\prod_{n=0}^{\infty} A_n^{*}. \end{align*}$$

Proof. The statements in (1) and (2) are obtained one from the other by passing to the opposite categories. Thus we just need to prove (1) and show that $\bigoplus _{n=0}^{\infty } A_n^{*}$ satisfies the universal property of a coproduct in ${\mathbf {D}}^?({\mathscr {A}})$ for $?=-,\emptyset $.

Suppose therefore that $B^{*}$ is an object of ${\mathbf {D}}^?({\mathscr {A}})$ and that we are given morphisms $\varphi _n:A_n^{*}\to B^{*}$. For $n\geq 0$, $\varphi _n$ can be represented in ${\mathbf {K}}^?({\mathscr {A}})$ by a roof

This means $\widetilde A_n^{*}\to A_n^{*}\to N_n^{*}$ is a distinguished triangle in ${\mathbf {K}}^?({\mathscr {A}})$ with $N_n^{*}\in {\mathbf {K}}_{\mathrm {acy}}({\mathscr {A}})$, and $\varphi _n=\beta \circ \alpha ^{-1}$ in ${\mathbf {D}}^?({\mathscr {A}})$.

As $A_n^i=0$ for all $i>-n$, the cochain map $A_n^{*}\to N_n^{*}$ must factor as $A_n^{*}\to N_n^{\le -n}\to N_n^{*}$, where $N_n^{\le -n}$ is still acyclic but vanishes in degrees $>-n$. Here $N_n^{\le -n}$ is the truncation

where *K* is the kernel of the map $N^{-n}\to N^{-n+1}$.

Now we form in ${\mathbf {K}}^?({\mathscr {A}})$ the morphism of distinguished triangles

which allows us to represent the morphism $\varphi _n:A_n^{*}\to B^{*}$ by the roof

with $\widehat A_n^i=0$ for all $i>-n+1$. And now the roof

is a well-defined diagram in ${\mathbf {K}}^?({\mathscr {A}})$ since in each degree, the sums are finite. Hence we obtain a morphism $\bigoplus _{n=0}^{\infty } A_n^{*}\to B^{*}$ in ${\mathbf {D}}^?({\mathscr {A}})$, and obviously the composite $A_n^{*}\to \bigoplus _{n=0}^{\infty } A_n^{*}\to B^{*}$ is $\varphi _n$ for every *n*.

It remains to prove that such a morphism is unique. This is equivalent to showing that given a morphism $\varphi :\bigoplus _{n=0}^{\infty } A_n^{*}\to B^{*}$ in ${\mathbf {D}}^?({\mathscr {A}})$ such that the composites $A_n^{*}\to \bigoplus _{n=0}^{\infty } A_n^{*}\to B^{*}$ vanish for every *n*, $\varphi $ vanishes.

Such a $\varphi $ may be represented in ${\mathbf {K}}^?({\mathscr {A}})$ by a roof

meaning $N^{*}\to B^{*}\to \widetilde B^{*}$ is a distinguished triangle in ${\mathbf {K}}^?({\mathscr {A}})$ with $N^{*}\in {\mathbf {K}}_{\mathrm {acy}}({\mathscr {A}})$ and $\varphi =\beta ^{-1}\circ \alpha $ in ${\mathbf {D}}^?({\mathscr {A}})$.

If the composite $A_n^{*}\to \bigoplus _{n=0}^{\infty } A_n^{*}\to B^{*}$ vanishes in ${\mathbf {D}}^?({\mathscr {A}})$, then the morphism $A_n^{*}\to \bigoplus _{n=0}^{\infty } A_n^{*}\to \widetilde B^{*}$ is a morphism in ${\mathbf {K}}^?({\mathscr {A}})$ whose image in ${\mathbf {D}}^?({\mathscr {A}})$ vanishes, and hence it must factor in ${\mathbf {K}}^?({\mathscr {A}})$ as $A_n^{*}\to M_n^{*}\to \widetilde B^{*}$, with $M_n^{*}\in {\mathbf {K}}_{\mathrm {acy}}({\mathscr {A}})$. As in the first part of the proof, the map $A_n^{*}\to M_n^{*}$ must factor as $A_n^{*}\to M_n^{\le -n}\to M_n^{*}$, where $M_n^{\le -n}$ is acyclic and vanishes in degrees $>-n$. But then $\bigoplus _{n=0}^{\infty } A_n^{*}\to \widetilde B^{*}$ factors in ${\mathbf {K}}^?({\mathscr {A}})$ as

showing that $\varphi $ vanishes in ${\mathbf {D}}^?({\mathscr {A}})$.

The following is then a straightforward consequence.

Corollary 1.6. Let ${\mathscr {A}}$ be a small abelian category, and let $\{A^i\}_{i\ge 0}\subseteq \mathrm {Ob}({\mathscr {A}})$. Then $\bigoplus _{i=0}^{\infty }{A^i}[k+i]$ is a coproduct in ${\mathbf {D}}^-({\mathscr {A}})$ and ${\mathbf {D}}({\mathscr {A}})$, while $\bigoplus _{i=0}^{\infty }{A^i}[k-i]$ is a product in ${\mathbf {D}}^+({\mathscr {A}})$ and ${\mathbf {D}}({\mathscr {A}})$ for all $k\in \mathbb {Z}$.

### 1.3 (Co)products and functors

We continue with some technical results that will be used later. In particular, in this section, we investigate when special exact functors commute with products and, dually, with coproducts.

If ${\mathscr {A}}$ is an abelian category and $n\in \mathbb {Z}$, we denote by ${\mathbf {D}}^?({\mathscr {A}})^{\geq n}$ (respectively, ${\mathbf {D}}^?({\mathscr {A}})^{\leq n}$) the full subcategory of ${\mathbf {D}}^?({\mathscr {A}})$ consisting of objects with trivial cohomologies in degrees $<n$ (respectively, $>n$).

Proposition 1.7. Let ${\mathscr {A}}$ be a small abelian category and $\mathsf {F}:{\mathscr {T}}\to {\mathbf {D}}^?({\mathscr {A}})$ an additive functor, where $?=+,\emptyset $. Assume that $\{T_i\}_{1\leq i<\infty }\subseteq \mathrm {Ob}({\mathscr {T}})$ is such that

(i) The product $\prod _{i=1}^{\infty } T_i$ exists in ${\mathscr {T}}$.

(ii) For every integer $n>0$, there exists an integer $m(n)>0$ such that $\mathsf {F}\left (\prod _{i=m(n)}^{\infty } T_i\right )\in {\mathbf {D}}({\mathscr {A}})^{\geq n}$.

Then the product $\prod _{i=1}^{\infty }\mathsf {F}(T_i)$ exists in ${\mathbf {D}}^?({\mathscr {A}})$ and the canonical map

is an isomorphism.

Proof. Given $k>0$ and $j\geq k$, the object $\mathsf {F}(T_j)$ is a direct summand of $\mathsf {F}\left (\prod _{i=k}^{\infty } T_i\right )$. Thus, assumption (ii) implies that for every $n>0$, there exists $m(n)>0$ with $F(T_j)\in {\mathbf {D}}({\mathscr {A}})^{\geq n}$ for all $j\geq m(n)$. For $n>0$, we set

By the previous discussion, $A_n\in {\mathbf {D}}({\mathscr {A}})^{\geq n}$, and Lemma 1.5 implies that $\prod _{n=1}^{\infty }A_n$ exists in ${\mathbf {D}}^?({\mathscr {A}})$. Thus

exists in ${\mathbf {D}}^?({\mathscr {A}})$.

Let us move to the second part of the statement. For $m\geq 2$, the natural map

can be identified with the product of the two natural maps

This clearly implies that $\mathrm {Cone}\left (\theta \right )\cong \mathrm {Cone}\left (\theta _1\right )\oplus \mathrm {Cone}\left (\theta _2\right )$. Since $\mathsf {F}$ (being additive) commutes with finite products, $\theta _1$ is an isomorphism, whence $\mathrm {Cone}\left (\theta _1\right )\cong 0$. On the other hand, condition (ii) tells us that when $m\gg 0$, the object $\mathrm {Cone}\left (\theta _2\right )$ will belong to ${\mathbf {D}}^?({\mathscr {A}})^{\geq n}$ with *n* arbitrarily large. Hence $\mathrm {Cone}\left (\theta \right )\cong \mathrm {Cone}\left (\theta _2\right )$ must vanish, and $\theta $ must be an isomorphism.

Clearly, Proposition 1.7 has the following dual version whose proof simply consists in reducing to the previous result by passing to the opposite category.

Proposition 1.8. Let ${\mathscr {A}}$ be a small abelian category and $\mathsf {F}:{\mathscr {T}}\to {\mathbf {D}}^?({\mathscr {A}})$ an additive functor, where $?=-,\emptyset $. Assume that $\{T_i\}_{1\leq i<\infty }\subseteq \mathrm {Ob}({\mathscr {T}})$ is such that

(i) The coproduct $\coprod _{i=1}^{\infty } T_i$ exists in ${\mathscr {T}}$.

(ii) For every integer $n>0$, there exists an integer $m(n)>0$ such that $\mathsf {F}\left (\coprod _{i=m(n)}^{\infty } T_i\right )\in {\mathbf {D}}({\mathscr {A}})^{\leq -n}$.

Then the coproduct $\coprod _{i=1}^{\infty }\mathsf {F}(T_i)$ exists in ${\mathbf {D}}^?({\mathscr {A}})$ and the canonical map

is an isomorphism.

## 2 Generation

The key idea pursued in this paper is that uniqueness of dg enhancements is tightly related to suitable notions of generations. Those that are of interest in this paper are explained in this section.

### 2.1 Well-generated triangulated categories

Let ${\mathscr {T}}$ be a triangulated category with small coproducts. For a cardinal $\alpha $, an object *S* of ${\mathscr {T}}$ is $\alpha $*-small* if every map $S\to \coprod _{i\in I}X_i$ in ${\mathscr {T}}$ (where *I* is a small set) factors through $\coprod _{i\in J}X_i$ for some $J\subseteq I$ with $\lvert J\rvert <\alpha $. A cardinal $\alpha $ is called *regular* if it is not the sum of fewer than $\alpha $ cardinals, all of them smaller than $\alpha $.

Definition 2.1. The category ${\mathscr {T}}$ is *well-generated* if there exists a small set ${\mathscr {S}}$ of objects in ${\mathscr {T}}$ satisfying the following properties:

(G1) An object $X\in {\mathscr {T}}$ is isomorphic to $0$ if and only if $\mathrm {Hom}_{\mathscr {T}}(S,{X}[j])=0$ for all $S\in {\mathscr {S}}$ and all $j\in \mathbb {Z}$.

(G2) For every small set of maps $\{X_i\to Y_i\}_{i\in I}$ in ${\mathscr {T}}$, the induced map $\mathrm {Hom}_{\mathscr {T}}(S,\coprod _iX_i)\to \mathrm {Hom}_{\mathscr {T}}(S,\coprod _i Y_i)$ is surjective for all $S\in {\mathscr {S}}$ if $\mathrm {Hom}_{\mathscr {T}}(S,X_i)\to \mathrm {Hom}_{\mathscr {T}}(S, Y_i)$ is surjective for all $i\in I$ and all $S\in {\mathscr {S}}$.

(G3) There exists a regular cardinal $\alpha $ such that every object of ${\mathscr {S}}$ is $\alpha $-small.

Remark 2.2. The above notion was originally developed in [Reference Neeman36]. Here we used the equivalent formulation in [Reference Krause24]. A nice survey on the subject is in [Reference Krause25].

Part of this paper is about enhancements of triangulated categories constructed out of Grothendieck categories. For the non-expert reader, let us recall that an abelian category ${\mathscr {G}}$ is a Grothendieck category if it is closed under small coproducts, has a small set of generators ${\mathscr {S}}$ and the direct limits of short exact sequences are exact in ${\mathscr {G}}$. The objects in ${\mathscr {S}}$ are generators in the sense that for any *C* in ${\mathscr {G}}$, there exists an epimorphism $S\twoheadrightarrow C$ in ${\mathscr {G}}$, where *S* is a small coproduct of objects in ${\mathscr {S}}$. By taking the coproduct of all generators in ${\mathscr {S}}$, we can assume that ${\mathscr {G}}$ has a single generator *G*.

(i) If

*X*is an algebraic stack, the abelian categories $\mathrm {Mod}(\mathcal {O}_X)$ and ${\mathbf {Qcoh}}(X)$ of $\mathcal {O}_X$-modules and quasi-coherent modules are Grothendieck categories.(ii) If ${\mathscr {A}}$ is a small, $\Bbbk $-linear category, we denote by $\mathrm {Mod}({\mathscr {A}})$ the Grothendieck category of additive functors ${\mathscr {A}}{^{\circ }}\to \mathrm {Mod}(\Bbbk )$. For later use, recall that if $\alpha $ is a regular cardinal, then we denote by $\mathrm {Lex}_{\alpha }({\mathscr {A}}{^{\circ }},\mathrm {Mod}(\Bbbk ))$ the full subcategory of $\mathrm {Mod}({\mathscr {A}})$ consisting of left exact functors that commute with $\alpha $-coproducts.

(iii) If ${\mathscr {A}}$ is an abelian category, we denote by $\mathrm {Ind}({\mathscr {A}})$ its Ind-category (see [Reference Grothendieck and Verdier19, Section 8]), which is a Grothendieck category. Roughly, it is obtained from ${\mathscr {A}}$ by formally adding filtered colimits of objects in ${\mathscr {A}}$.

The following states an important property for the derived category of a Grothendieck category.

#### Theorem 2.4 ([Reference Neeman33], Theorem 0.2).

If ${\mathscr {G}}$ is a Grothendieck category, then ${\mathbf {D}}({\mathscr {G}})$ is well-generated.

A full triangulated subcategory of ${\mathscr {T}}$ is $\alpha $*-localising* if it is closed under $\alpha $-coproducts and under direct summands (the latter condition is automatic if $\alpha>\aleph _0$). An $\alpha $-coproduct is a coproduct of strictly less than $\alpha $ summands. A full subcategory of ${\mathscr {T}}$ is *localising* if it is $\alpha $-localising for all regular cardinals $\alpha $.

When the category ${\mathscr {T}}$ is well-generated and we want to put emphasis on the cardinal $\alpha $ in (G3) and on ${\mathscr {S}}$, we say that ${\mathscr {T}}$ is $\alpha $*-well-generated* by the set ${\mathscr {S}}$. In this situation, following [Reference Krause24], we denote by ${\mathscr {T}}^{\alpha }$ the smallest $\alpha $-localising subcategory of ${\mathscr {T}}$ containing ${\mathscr {S}}$. By [Reference Krause24, Reference Neeman36], ${\mathscr {T}}^{\alpha }$ does not depend on the choice of the set ${\mathscr {S}}$, which well-generates ${\mathscr {T}}$.

Let ${\mathscr {G}}$ be a Grothendieck category, and let $\alpha $ be a sufficiently large regular cardinal. We are interested in describing the category ${\mathbf {D}}({\mathscr {G}})^{\alpha }$. To this end, we denote by ${\mathscr {G}}^{\alpha }$ the full subcategory of ${\mathscr {G}}$ consisting of $\alpha $-presentable objects. An object *G* in ${\mathscr {G}}$ is $\alpha $*-presentable* if the functor $\mathrm {Hom}_{{\mathscr {G}}}(G,-)\colon {\mathscr {G}}\to \mathrm {Mod}(\Bbbk )$ preserves $\alpha $-filtered colimits (see, for example, [Reference Krause25, Section 6.4] for the definition of $\alpha $-filtered colimit).

#### Theorem 2.5 ([Reference Krause23], Corollary 5.5 and Theorem 5.10).

Let ${\mathscr {G}}$ be a Grothendieck category, and let $\alpha $ be a sufficiently large regular cardinal.

(1) The category ${\mathscr {G}}^{\alpha }$ is abelian.

(2) There is a natural exact equivalence ${\mathbf {D}}({\mathscr {G}})^{\alpha }\cong {\mathbf {D}}({\mathscr {G}}^{\alpha })$.

The objects in ${\mathscr {T}}^{\alpha }$ are called $\alpha $*-compact*. Thus we will sometimes say that ${\mathscr {T}}$ is $\alpha $*-compactly generated* by the set of $\alpha $*-compact generators* ${\mathscr {S}}$. A very interesting case is when $\alpha =\aleph _0$. Indeed, with this choice, ${\mathscr {T}}^{\alpha }={\mathscr {T}}^c$, the full triangulated subcategory of compact objects in ${\mathscr {T}}$. Recall that an object *C* in ${\mathscr {T}}$ is *compact* if the functor ${\mathscr {T}}(C,-)$ commutes with small coproducts. Notice that the compact objects in ${\mathscr {T}}$ are precisely the $\aleph _0$-small ones.

The analogue of Theorem 2.4 can be proven for the unseparated derived category.

#### Theorem 2.6 ([Reference Krause23], Theorem 5.12).

If ${\mathscr {G}}$ is a Grothendieck category, then $\check {{\mathbf {D}}}({\mathscr {G}})$ is well-generated.

A weaker form of Theorem 2.5 (2) is also available. Indeed, by combining [Reference Krause23, Theorem 5.12(3)] with Theorem 2.5(2) for $\alpha $ a sufficiently large regular cardinal, there is a quotient functor

We will not use such a general result in this paper, but we will elaborate more on the following easier case.

Example 2.7. If ${\mathscr {A}}$ is a small abelian category, then one takes the Ind-category ${\mathscr {G}}:=\mathrm {Ind}({\mathscr {A}})$ (see Example 2.3). By [Reference Krause23, Theorem 4.9], there is a natural exact equivalence $\check {{\mathbf {D}}}({\mathscr {G}})^c\cong {\mathbf {D}}^b({\mathscr {A}})$.

### 2.2 Generating derived categories

In the general case when ${\mathscr {A}}$ is any abelian category, not necessarily Grothendieck, we need a different approach to the generation of ${\mathbf {D}}^?({\mathscr {A}})$.

Let us first recall the following rather general definition (see [Reference Bondal and Van den Bergh7]).

Definition 2.8. Let ${\mathscr {T}}$ be a triangulated category, and let ${\mathscr {S}}\subset \mathrm {Ob}({\mathscr {T}})$. We define

(1) ${\left \langle \mathscr {S} \right \rangle }_{1}$ is the collection of all direct summands of finite coproducts of shifts of objects in ${\mathscr {S}}$.

(2) ${\left \langle \mathscr {S} \right \rangle }_{n+1}$ consists of all direct summands of objects $T\in {\mathscr {T}}$ for which there exists a distinguished triangle $T_1\to T\to T_2$ with $T_1\in {\left \langle \mathscr {S} \right \rangle }_{n}$ and $T_2\in {\left \langle \mathscr {S} \right \rangle }_{1}$.

We set ${\left \langle \mathscr {S} \right \rangle }_{\infty }$ for the full subcategory consisting of all objects *T* in ${\mathscr {T}}$ contained in ${\left \langle \mathscr {S} \right \rangle }_{n}$ for some *n*.

In our special case, we can prove the following.

Proposition 2.9. Let ${\mathbf {V}}^?({\mathscr {A}})\subset {\mathbf {K}}^?({\mathscr {A}})$ be as defined in the opening paragraph of Subsection 1.1. For $?=b,+,-,\emptyset $, we have that ${\left \langle {\mathbf {V}}^?({\mathscr {A}}) \right \rangle }_{3}={\mathbf {K}}^?({\mathscr {A}})$.

Proof. Let $A^{*}\in \mathrm {Ob}({\mathbf {K}}^?({\mathscr {A}}))$, which we write as a complex

Let $K^i$ be the kernel of the differential $A^i\to A^{i+1}$. Then the map $A^{i-1}\to A^i$ factors uniquely as $A^{i-1}\xrightarrow {\alpha ^i} K^i\hookrightarrow A^i$.

This yields the morphism

in ${\mathbf {V}}^?({\mathscr {A}})$. Denote by $C^{*}$ its mapping cone. It is clear that $C^{*}\in {\left \langle {\mathbf {V}}^?({\mathscr {A}}) \right \rangle }_{2}$, and it is the direct sum over $i\in \mathbb {Z}$ of the complexes

Now consider the cochain map

whose components, out of ${K^{i}}[-i]$, are (respectively) $\varphi ^i$ as below

and $\psi ^i$ as below

It can be easily checked that the mapping cone of the morphism $\varphi +\psi $ is isomorphic to the direct sum of the complex $A^{*}$ and of complexes of the form

In other words, $\mathrm {Cone}\left (\varphi +\psi \right )\cong A^{*}$ in ${\mathbf {K}}^?({\mathscr {A}})$. Therefore, as $C^{*}$ belongs to ${\left \langle {\mathbf {V}}^?({\mathscr {A}}) \right \rangle }_{2}$ and $\varphi +\psi $ is a morphism from an object of ${\mathbf {V}}^?({\mathscr {A}})$ to $C^{*}$, we have that $A^{*}\in {\left \langle {\mathbf {V}}^?({\mathscr {A}}) \right \rangle }_{3}$.

The reader might wish to compare the proof of Proposition 2.9 above with the proof of [Reference Rouquier41, Proposition 7.22]; there are similarities.

As a straightforward consequence of Proposition 2.9 and of the fact that ${\mathbf {D}}^?({\mathscr {A}})$ is a quotient of ${\mathbf {K}}^?({\mathscr {A}})$, as explained in Section 1.1, we obtain the following.

## 3 Dg categories and enhancements

We briefly introduce dg categories and some basic machinery in Section 3.1. Next, we describe some constructions that will be important in the rest of the paper: Drinfeld quotients and h-flat resolutions (Section 3.2), the model structure and homotopy pullbacks (Section 3.3) and, finally, localisations for dg categories (Section 3.4). Dg enhancements, their uniqueness and dependence on the universe where the categories are defined are the contents of Section 3.5.

### 3.1 Dg categories

A *dg category* is a $\Bbbk $-linear category ${\mathscr {C}}$ such that the morphism spaces $\mathrm {Hom}_{\mathscr {C}}\left (A,B\right )$ are complexes of $\Bbbk $-modules and the composition maps $\mathrm {Hom}_{\mathscr {C}}(B,C)\otimes _{\Bbbk }\mathrm {Hom}_{\mathscr {C}}(A,B)\to \mathrm {Hom}_{\mathscr {C}}(A,C)$ are morphisms of complexes for all $A,B,C$ in $\mathrm {Ob}({\mathscr {C}})$.

A *dg functor* $\mathsf {F}\colon {\mathscr {C}}_1\to {\mathscr {C}}_2$ between two dg categories is a $\Bbbk $-linear functor such that the maps $\mathrm {Hom}_{{\mathscr {C}}_1}(A,B)\to \mathrm {Hom}_{{\mathscr {C}}_2}(\mathsf {F}(A),\mathsf {F}(B))$ are morphisms of complexes for all $A,B$ in $\mathrm {Ob}({\mathscr {C}}_1)$.

The *underlying category* $Z^0({\mathscr {C}})$ (respectively, the *homotopy category* $H^0({\mathscr {C}})$) of a dg category ${\mathscr {C}}$ is the $\Bbbk $-linear category with the same objects of ${\mathscr {C}}$ and such that $\mathrm {Hom}_{Z^0({\mathscr {C}})}(A,B):=Z^0(\mathrm {Hom}_{\mathscr {C}}(A,B))$ (respectively, $\mathrm {Hom}_{H^0({\mathscr {C}})}(A,B):=H^0(\mathrm {Hom}_{\mathscr {C}}(A,B))$) for all $A,B$ in $\mathrm {Ob}({\mathscr {C}})$ (with composition of morphisms naturally induced from the one in ${\mathscr {C}}$). Two objects of ${\mathscr {C}}$ are *dg isomorphic* (respectively, *homotopy equivalent*) if they are isomorphic in $Z^0({\mathscr {C}})$ (respectively, $H^0({\mathscr {C}})$). One can also define $Z({\mathscr {C}})$ (respectively $H({\mathscr {C}})$) to be the graded (namely, dg with trivial differential) category whose objects are the same as those of ${\mathscr {C}}$, while $\mathrm {Hom}_{Z({\mathscr {C}})}(A,B):=\oplus _{i\in \mathbb {Z}}Z^i(\mathrm {Hom}_{\mathscr {C}}(A,B))$ (respectively, $\mathrm {Hom}_{H({\mathscr {C}})}(A,B):=\oplus _{i\in \mathbb {Z}}H^i(\mathrm {Hom}_{\mathscr {C}}(A,B))$) for all $A,B$ in $\mathrm {Ob}({\mathscr {C}})$.

Example 3.1. If ${\mathscr {A}}$ is a $\Bbbk $-linear category, there is a natural dg category ${\mathbf {C}}_{{\mathbf {dg}}}^?({\mathscr {A}})$ such that $H^0({\mathbf {C}}_{{\mathbf {dg}}}^?({\mathscr {A}}))={\mathbf {K}}^?({\mathscr {A}})$ for $?=b,+,-,\emptyset $. Explicitly,

for every $A^{*},B^{*}\in \mathrm {Ob}({\mathbf {C}}_{{\mathbf {dg}}}^?({\mathscr {A}}))$ and for every $n\in \mathbb {Z}$. While the composition of morphisms is the obvious one, the differential is defined on a homogeneous element $f\in \mathrm {Hom}_{{\mathbf {C}}_{{\mathbf {dg}}}^?({\mathscr {A}})}(A^{*},B^{*})^n$ by $d(f):=d_B\circ f-(-1)^nf\circ d_A$.

A dg functor $\mathsf {F}\colon {\mathscr {C}}_1\to {\mathscr {C}}_2$ induces a $\Bbbk $-linear functor $H^0(\mathsf {F})\colon H^0({\mathscr {C}}_1)\to H^0({\mathscr {C}}_2)$. We say that $\mathsf {F}$ is a *quasi-equivalence* if the maps $\mathrm {Hom}_{{\mathscr {C}}_1}(A,B)\to \mathrm {Hom}_{{\mathscr {C}}_2}(\mathsf {F}(A),\mathsf {F}(B))$ are quasi-isomorphisms for all $A,B$ in $\mathrm {Ob}({\mathscr {C}}_1)$, and $H^0(\mathsf {F})$ is an equivalence.

If $\mathbb {U}$ is a universe, we denote by ${\mathbf {dgCat}}_{\mathbb {U}}$ (or simply by ${\mathbf {dgCat}}$, if there can be no ambiguity about $\mathbb {U}$) the category whose objects are $\mathbb {U}$-small dg categories and whose morphisms are dg functors. It is known (see [Reference Tabuada44]) that ${\mathbf {dgCat}}$ has a model structure whose weak equivalences are quasi-equivalences and such that every object is fibrant. We denote by ${\mathbf {Hqe}}$ (or ${\mathbf {Hqe}}_{\mathbb {U}}$, if needed) the corresponding homotopy category, namely the localisation of ${\mathbf {dgCat}}$ with respect to quasi-equivalences. Since $H^0$ sends quasi-equivalences to equivalences, for every morphism $f\colon {\mathscr {C}}_1\to {\mathscr {C}}_2$ in ${\mathbf {Hqe}}$ there is a $\Bbbk $-linear functor $H^0(f)\colon H^0({\mathscr {C}}_1)\to H^0({\mathscr {C}}_2)$, which is well-defined up to isomorphism.

Dg functors between two dg categories ${\mathscr {C}}_1$ and ${\mathscr {C}}_2$ form in a natural way the objects of a dg category $\underline {\mathcal {H}om}({\mathscr {C}}_1,{\mathscr {C}}_2)$. For every dg category ${\mathscr {C}}$, we set $\mathrm {dgMod}({\mathscr {C}}):=\underline {\mathcal {H}om}({\mathscr {C}}^{\circ },{\mathbf {C}}_{{\mathbf {dg}}}(\mathrm {Mod}(\Bbbk )))$ and call its objects (right) dg ${\mathscr {C}}$-modules. Observe that $\mathrm {dgMod}(\Bbbk )$ can be identified with ${\mathbf {C}}_{{\mathbf {dg}}}(\mathrm {Mod}(\Bbbk ))$.

For every dg category ${\mathscr {C}}$, the map defined on objects by $A\mapsto \mathrm {Hom}_{\mathscr {C}}(-,A)$ extends to a fully faithful dg functor $\mathsf {Y}^{{\mathscr {C}}}_{\mathrm {dg}}\colon {\mathscr {C}}\to \mathrm {dgMod}({\mathscr {C}})$ (the dg Yoneda embedding). It is easy to see that the image of $\mathsf {Y}^{{\mathscr {C}}}_{\mathrm {dg}}$ is contained in the full dg subcategory $\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})$ of $\mathrm {dgMod}({\mathscr {C}})$ whose objects are *h-projective* dg ${\mathscr {C}}$-modules. By definition, $M\in \mathrm {Ob}(\mathrm {dgMod}({\mathscr {C}}))$ is h-projective if $\mathrm {Hom}_{H^0(\mathrm {dgMod}({\mathscr {C}}))}(M,N)=0$ for every $N\in \mathrm {Ob}(\mathrm {dgAcy}({\mathscr {C}}))$, where $\mathrm {dgAcy}({\mathscr {C}})$ is the full dg subcategory of $\mathrm {dgMod}({\mathscr {C}})$ whose objects are acyclic (in the sense that $N(A)$ is an acyclic complex for every $A\in \mathrm {Ob}({\mathscr {C}})$).

If $\mathsf {F}\colon {\mathscr {C}}_1\to {\mathscr {C}}_2$ is a dg functor, composition with $\mathsf {F}^{\circ }$ yields a dg functor $\mathsf {Res}(\mathsf {F})\colon \mathrm {dgMod}({\mathscr {C}}_2)\to \mathrm {dgMod}({\mathscr {C}}_1)$. It turns out that there also exists a dg functor $\mathsf {Ind}(\mathsf {F})\colon \mathrm {dgMod}({\mathscr {C}}_1)\to \mathrm {dgMod}({\mathscr {C}}_2)$, which is left adjoint to $\mathsf {Res}(\mathsf {F})$ and such that $\mathsf {Ind}(\mathsf {F})\circ \mathsf {Y}^{{\mathscr {C}}_1}_{\mathrm {dg}}\cong \mathsf {Y}^{{\mathscr {C}}_2}_{\mathrm {dg}}\circ \mathsf {F}$. Moreover, $\mathsf {Ind}(\mathsf {F})$ preserves h-projective dg modules, and $\mathsf {Ind}(\mathsf {F})\colon \mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}_1)\to \mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}_2)$ is a quasi-equivalence if $\mathsf {F}$ is. This last fact clearly implies that a(n iso)morphism $f\colon {\mathscr {C}}_1\to {\mathscr {C}}_2$ in ${\mathbf {Hqe}}$ induces a(n iso)morphism $\mathsf {Ind}(f)\colon \mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}_1)\to \mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}_2)$ in ${\mathbf {Hqe}}$.

As is explained, for instance, in [Reference Bondal, Larsen and Lunts6], there is a notion of formal shift by an integer *n* of an object *A* in a dg category ${\mathscr {C}}$ (denoted, as usual, by ${A}[n]$). Similarly, one can define the formal cone of a morphism *f* in $Z^0({\mathscr {C}})$ (denoted, as usual, by $\mathrm {Cone}\left (f\right )$). Shifts and cones need not exist in an arbitrary dg category, but when they do, they are unique up to dg isomorphism and are preserved by dg functors. The following property of the cone of a morphism will be useful later.

Lemma 3.2. Let $A\xrightarrow {f}B\xrightarrow {g}C$ be morphisms in $Z^0({\mathscr {C}})$ such that $\mathrm {Cone}\left (f\right )$ exists and $g\circ f$ is a coboundary. Then there exists $h\colon \mathrm {Cone}\left (f\right )\to C$ in $Z^0({\mathscr {C}})$ such that $g=h\circ j$, where $j\colon B\to \mathrm {Cone}\left (f\right )$ is the natural morphism.

Proof. See [Reference Genovese18, Proposition 2.3.4].

Definition 3.3. A dg category ${\mathscr {C}}$ is *strongly pretriangulated* if ${A}[n]$ and $\mathrm {Cone}\left (f\right )$ exist (in ${\mathscr {C}}$) for every $n\in \mathbb {Z}$, every object *A* of ${\mathscr {C}}$ and every morphism *f* of $Z^0({\mathscr {C}})$.

A dg category ${\mathscr {C}}$ is *pretriangulated* if there exists a quasi-equivalence ${\mathscr {C}}\to {\mathscr {C}}'$ with ${\mathscr {C}}'$ strongly pretriangulated.

Remark 3.4. If ${\mathscr {C}}$ is a pretriangulated dg category, then $H^0({\mathscr {C}})$ is a triangulated category in a natural way. If *f* is a morphism in ${\mathbf {Hqe}}$ between two pretriangulated dg categories, then the functor $H^0(f)$ is exact.

If ${\mathscr {C}}$ is a dg category, $\mathrm {dgMod}({\mathscr {C}})$, $\mathrm {dgAcy}({\mathscr {C}})$ and $\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})$ are strongly pretriangulated dg categories. Moreover, the (triangulated) categories $H^0(\mathrm {dgMod}({\mathscr {C}}))$, $H^0(\mathrm {dgAcy}({\mathscr {C}}))$ and $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$ have arbitrary coproducts, and there is a semi-orthogonal decomposition

This clearly implies that there is an exact equivalence between $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$ and the Verdier quotient $\mathcal {D}({\mathscr {C}}):=H^0(\mathrm {dgMod}({\mathscr {C}}))/H^0(\mathrm {dgAcy}({\mathscr {C}}))$ (which is by definition the *derived category* of ${\mathscr {C}}$).

For every dg category ${\mathscr {C}}$, we will denote by $\mathrm {Pretr}({\mathscr {C}})$ (respectively, $\mathrm {Perf}({\mathscr {C}})$) the smallest full dg subcategory of $\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})$ containing $\mathsf {Y}^{{\mathscr {C}}}_{\mathrm {dg}}({\mathscr {C}})$ and closed under homotopy equivalences, shifts, cones (respectively, also direct summands in $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$). It is easy to see that $\mathrm {Pretr}({\mathscr {C}})$ and $\mathrm {Perf}({\mathscr {C}})$ are strongly pretriangulated and that ${\mathscr {C}}$ is pretriangulated if and only if $\mathsf {Y}^{{\mathscr {C}}}_{\mathrm {dg}}\colon {\mathscr {C}}\to \mathrm {Pretr}({\mathscr {C}})$ is a quasi-equivalence. Moreover, $\mathrm {Pretr}({\mathscr {C}})\subseteq \mathrm {Perf}({\mathscr {C}})$ and $H^0(\mathrm {Perf}({\mathscr {C}}))$ can be identified with the idempotent completion $H^0(\mathrm {Pretr}({\mathscr {C}}))^{\mathrm {ic}}$ of $H^0(\mathrm {Pretr}({\mathscr {C}}))$. Hence $\mathsf {Y}^{{\mathscr {C}}}_{\mathrm {dg}}\colon {\mathscr {C}}\to \mathrm {Perf}({\mathscr {C}})$ is a quasi-equivalence if and only if ${\mathscr {C}}$ is pretriangulated and $H^0({\mathscr {C}})$ is idempotent complete.

Remark 3.5. Recall that an additive category ${\mathscr {A}}$ is *idempotent complete* if every idempotent (namely, a morphism $e\colon A\to A$ in ${\mathscr {A}}$ such that $e^2=e$) splits or, equivalently, has a kernel. Every additive category ${\mathscr {A}}$ admits a fully faithful and additive embedding ${\mathscr {A}}\hookrightarrow {\mathscr {A}}^{\mathrm {ic}}$, where ${\mathscr {A}}^{\mathrm {ic}}$ is an idempotent complete additive category, with the property that every object of ${\mathscr {A}}^{\mathrm {ic}}$ is a direct summand of an object from ${\mathscr {A}}$. The category ${\mathscr {A}}^{\mathrm {ic}}$ (or, better, the functor ${\mathscr {A}}\to {\mathscr {A}}^{\mathrm {ic}}$) is called the *idempotent completion* of ${\mathscr {A}}$. It can be proved (see [Reference Balmer and Schlichting3]) that if ${\mathscr {T}}$ is a triangulated category, then ${\mathscr {T}}^{\mathrm {ic}}$ is triangulated as well (and ${\mathscr {T}}\hookrightarrow {\mathscr {T}}^{\mathrm {ic}}$ is exact).

If ${\mathscr {C}}$ is a dg category, then $H^0(\mathrm {Perf}({\mathscr {C}}))$ is idempotent complete, and from this it is easy to deduce that $Z^0(\mathrm {Perf}({\mathscr {C}}))$ is also idempotent complete.

If ${\mathscr {A}}$ is an abelian category, it follows from [Reference Balmer and Schlichting3, Reference Schlichting43] that ${\mathbf {D}}^?({\mathscr {A}})$ is idempotent complete for $?=b,+,-,\emptyset $. More precisely, for $?\in \{-,+\}$, the result may be found in [Reference Balmer and Schlichting3, Lemma 2.4]; for $?=b$, see [Reference Balmer and Schlichting3, Lemma 2.8]; and for $?=\emptyset $, see Theorem 6 of Section 10 in [Reference Schlichting43] combined with Lemma 7 of Section 9 in the same paper.

Observe that by Remark 1.3 combined with the paragraph above, the categories ${\mathbf {K}}^?({\mathscr {A}})$ are also idempotent complete—as long as ${\mathscr {A}}$ is abelian and with $?=b,+,-,\emptyset $.

### 3.2 Drinfeld quotients and h-flat resolutions

Let ${\mathscr {C}}$ be a dg category and ${\mathscr {D}}\subseteq {\mathscr {C}}$ a full dg subcategory. As explained in [Reference Drinfeld16, Section 3.1], one can form the *Drinfeld quotient* of ${\mathscr {C}}$ by ${\mathscr {D}}$, which we denote by ${\mathscr {C}}/{\mathscr {D}}$. This is a dg category, and its construction goes roughly as follows: given $D\in \mathrm {Ob}({\mathscr {D}})$, we formally add a morphism $f_D\colon D\to D$ of degree $-1$, and we set $d(f_D)=\mathrm {id}_D$.

If ${\mathscr {C}}$ is pretriangulated and ${\mathscr {D}}$ is a full pretriangulated dg subcategory, then ${\mathscr {C}}/{\mathscr {D}}$ is pretriangulated. In this case, the natural dg functor ${\mathscr {C}}\to {\mathscr {C}}/{\mathscr {D}}$ induces an exact functor $H^0({\mathscr {C}})\to H^0({\mathscr {C}}/{\mathscr {D}})$, which sends to zero the objects of ${\mathscr {D}}$. Thus it factors through the Verdier quotient $H^0({\mathscr {C}})\to H^0({\mathscr {C}})/H^0({\mathscr {D}})$, yielding an exact functor

which need not be an equivalence, in general.

Definition 3.6. We remind the reader of the terminology of [Reference Drinfeld16]. A dg category ${\mathscr {C}}$ is *h-flat* if, for all $C_1,C_2\in \mathrm {Ob}({\mathscr {C}})$, the complex $\mathrm {Hom}_{\mathscr {C}}(C_1,C_2)$ is homotopically flat over $\Bbbk $. The homotopic flatness of $\mathrm {Hom}_{\mathscr {C}}(C_1,C_2)$ means for any acyclic complex *M* of $\Bbbk $-modules, $\mathrm {Hom}_{\mathscr {C}}(C_1,C_2)\otimes _{\Bbbk } M$ is acyclic.

Example 3.7. If $\Bbbk $ is a field, then every dg category is clearly h-flat.

As a special case of [Reference Drinfeld16, Theorem 3.4], we have that if ${\mathscr {C}}$ is an h-flat pretriangulated dg category and ${\mathscr {D}}$ is a full pretriangulated subcategory of ${\mathscr {C}}$, then equation (3.2) is an exact equivalence.

If ${\mathscr {C}}$ is not h-flat, Drinfeld shows in [Reference Drinfeld16, Lemma B.5] that one can construct an h-flat dg category $\widetilde {\mathscr {C}}$ with a quasi-equivalence $\mathsf {I}_{\mathscr {C}}\colon \widetilde {\mathscr {C}}\to {\mathscr {C}}$. One can then define $\widetilde {\mathscr {D}}$ to be the full dg subcategory $\mathsf {I}_{\mathscr {C}}^{-1}{\mathscr {D}}\subset \widetilde {\mathscr {C}}$ and take the morphism $q\in \mathrm {Hom}_{\mathbf {Hqe}}\left ({\mathscr {C}},\widetilde {\mathscr {C}}/\widetilde {\mathscr {D}}\right )$ represented as

where the dg functor on the right is the natural one mentioned above.

This construction satisfies the following universal property, which is a special instance of [Reference Drinfeld16, Main Theorem]. Assume that ${\mathscr {C}}'$ is a pretriangulated dg category and $f\in \mathrm {Hom}_{\mathbf {Hqe}}({\mathscr {C}},{\mathscr {C}}')$ is such that $H^0(f)$ sends the objects of ${\mathscr {D}}$ to zero. Then there is a unique $\overline {f}\in \mathrm {Hom}_{\mathbf {Hqe}}\left (\widetilde {\mathscr {C}}/\widetilde {\mathscr {D}},{\mathscr {C}}'\right )$ making the diagram

commute in ${\mathbf {Hqe}}$.

In the rest of this section, we describe two variants of $\widetilde {\mathscr {C}}$ with properties that we will need in the rest of the paper. We start with a dg category ${\mathscr {C}}$, we let ${\mathscr {T}}$ be the graded category ${\mathscr {T}}=H({\mathscr {C}})$, and the aim is to produce two sequences of dg categories and faithful dg functors

together with compatible dg functors $\mathsf {I}_n\colon {\mathscr {C}}_n\to {\mathscr {C}}$ and $\mathsf {I}^{\prime }_n\colon {\mathscr {C}}^{\prime }_n\to {\mathscr {C}}$, which are the identity on objects. Then we set ${{\mathscr {C}}}^{\mathrm {hf}}$ and ${{\mathscr {C}}}^{\mathrm {sm}}$ to be the respective colimits, with the induced dg functors ${\mathsf {I}}^{\mathrm {hf}}_{\mathscr {C}}\colon {{\mathscr {C}}}^{\mathrm {hf}}\overset {\sim }{\longrightarrow }{\mathscr {C}}$ and ${\mathsf {I}}^{\mathrm {sm}}_{\mathscr {C}}\colon {{\mathscr {C}}}^{\mathrm {sm}}\overset {\sim }{\longrightarrow }{\mathscr {C}}$.

We define ${\mathscr {C}}_0={\mathscr {C}}^{\prime }_0$ to be the discrete $\Bbbk $-linear category with the same objects as ${\mathscr {C}}$. This means

The dg functor $\mathsf {I}_0=\mathsf {I}^{\prime }_0$ is the obvious one that acts as the identity on objects and morphisms.

For $n=1$, we set

Now the composite

is surjective by construction; hence we may choose a splitting. We let $\overline D^1_{\mathscr {C}}(A,B)\subset D^1_{\mathscr {C}}(A,B)$ be a subset such that the composite $\overline D^1_{\mathscr {C}}(A,B)\to D^1_{\mathscr {C}}(A,B)\to \mathrm {Hom}_{{\mathscr {T}}}(A,B)$ is an isomorphism. And we define ${\mathscr {C}}_1$ so that $\mathrm {Hom}_{{\mathscr {C}}_1}^{}(A,B)$ is the graded $\Bbbk $-module freely generated by the basis $D^1_{\mathscr {C}}(A,B)-\{0\}$, with the composition being obvious on basis vectors. And ${\mathscr {C}}^{\prime }_1$ is the graded $\Bbbk $-linear category freely generatedFootnote ^{1} over ${\mathscr {C}}_0$ by the sets $\overline D^1_{\mathscr {C}}(A,B)$. The differentials of ${\mathscr {C}}_1$ and ${\mathscr {C}}^{\prime }_1$ are trivial.

We continue for $n\geq 2$ by defining inductively, for all $A,B\in \mathrm {Ob}({\mathscr {C}})$,

The definition of $\overline D^n_{\mathscr {C}}(A,B)$ is slightly more delicate. We begin by copying the procedure above with ${\mathscr {C}}^{\prime }_{n-1}$ in place of ${\mathscr {C}}_{n-1}$, setting

And then we observe that $\widehat D^n_{\mathscr {C}}(A,B)$ surjects to the kernel of the surjective map

allowing us to choose a subset