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 ith 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

$$ \begin{align*} \mathrm{Ob}({\mathbf{K}}^b({\mathscr{A}}))=\left\{A^{*}\in{\mathbf{K}}({\mathscr{A}}): A^i=0\text{ for all }|i|\gg0\right\}\\ \mathrm{Ob}({\mathbf{K}}^+({\mathscr{A}}))=\left\{A^{*}\in{\mathbf{K}}({\mathscr{A}}): A^i=0\text{ for all }i\ll0\right\}\\ \mathrm{Ob}({\mathbf{K}}^-({\mathscr{A}}))=\left\{A^{*}\in{\mathbf{K}}({\mathscr{A}}): A^i=0\text{ for all }i\gg0\right\}. \end{align*} $$

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

$$\begin{align*}\bigoplus_{i\in\mathbb{Z}}{A^i}[-i] \end{align*}$$

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

$$\begin{align*}\bigoplus_{i\in\mathbb{Z}}{A^i}[-i]\cong\prod_{i\in\mathbb{Z}}{A^i}[-i]\cong\coprod_{i\in\mathbb{Z}}{A^i}[-i]. \end{align*}$$

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

(1.1)$$ \begin{align} \mathsf{Q}:{\mathbf{K}}^?({\mathscr{A}})\longrightarrow{\mathbf{D}}^?({\mathscr{A}}). \end{align} $$

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

$$\begin{align*}{\mathbf{B}}^?({\mathscr{A}}):=\mathsf{Q}({\mathbf{V}}^?({\mathscr{A}})). \end{align*}$$

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

$$\begin{align*}\check{{\mathbf{D}}}({\mathscr{G}}):={\mathbf{K}}(\mathrm{Inj}({\mathscr{G}})). \end{align*}$$

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

$$\begin{align*}{\mathbf{K}}_{\mathrm{acy}}(\mathrm{Inj}({\mathscr{G}}))\overset{\mathsf{J}}{\longrightarrow}\check{{\mathbf{D}}}({\mathscr{G}})\overset{\mathsf{Q}}{\longrightarrow}{\mathbf{D}}({\mathscr{G}}). \end{align*}$$

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.

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.

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

$$\begin{align*}\cdots\longrightarrow N^{-n-2}\longrightarrow N^{-n-1}\longrightarrow K\longrightarrow0\longrightarrow0\longrightarrow\cdots, \end{align*}$$

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

$$\begin{align*}\bigoplus_{n=0}^{\infty} A_n^{*}\longrightarrow\bigoplus_{n=0}^{\infty} M_n^{\le-n}\longrightarrow\widetilde B^{*}, \end{align*}$$

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

The following is then a straightforward consequence.

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

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

2. (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

$$\begin{align*}\mathsf{F}\left(\prod_{i=1}^{\infty} T_i\right)\longrightarrow\prod_{i=1}^{\infty} \mathsf{F}(T_i) \end{align*}$$

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

$$\begin{align*}A_n:=\mathsf{F}\,\left(\prod_{i=m(n)}^{m(n+1)-1} T_i\right)\cong\prod_{i=m(n)}^{m(n+1)-1}\mathsf{F}(T_i). \end{align*}$$

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

$$\begin{align*}\prod_{i=1}^{\infty}\mathsf{F}(T_i)\cong\left(\prod_{i=1}^{m(1)-1}\mathsf{F}(T_i)\right)\prod\left(\prod_{n=1}^{\infty} A_n\right) \end{align*}$$

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

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

$$\begin{align*}\theta\colon\mathsf{F}\left(\prod_{i=1}^{\infty} T_i\right)\longrightarrow\prod_{i=1}^{\infty}\mathsf{F}(T_i) \end{align*}$$

can be identified with the product of the two natural maps

$$\begin{align*}\theta_1\colon\mathsf{F}\left(\prod_{i=1}^{m-1} T_i\right)\longrightarrow\prod_{i=1}^{m-1}\mathsf{F}(T_i), \qquad \theta_2\colon\mathsf{F}\left(\prod_{i=m}^{\infty} T_i\right)\longrightarrow\prod_{i=m}^{\infty}\mathsf{F}(T_i). \end{align*}$$

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

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

2. (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

$$\begin{align*}\coprod_{i=1}^{\infty} \mathsf{F}(T_i)\longrightarrow\mathsf{F}\left(\coprod_{i=1}^{\infty} T_i\right) \end{align*}$$

is an isomorphism.

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 $.

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.

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

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).

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.

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

$$\begin{align*}\check{{\mathbf{D}}}({\mathscr{G}})^{\alpha}\longrightarrow{\mathbf{D}}({\mathscr{G}})^{\alpha}. \end{align*}$$

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

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]).

In our special case, we can prove the following.

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

(2.1)

and $\psi ^i$ as below

(2.2)

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.

3.1 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,

$$\begin{align*}\mathrm{Hom}_{{\mathbf{C}}_{{\mathbf{dg}}}^?({\mathscr{A}})}(A^{*},B^{*})^n:=\prod_{i\in\mathbb{Z}}\mathrm{Hom}_{\mathscr{A}}(A^i,B^{n+i}) \end{align*}$$

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.

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

(3.1)$$ \begin{align} H^0(\mathrm{dgMod}({\mathscr{C}}))=\langle H^0(\mathrm{dgAcy}({\mathscr{C}})),H^0(\mathrm{h}\text{-}\mathrm{proj}({\mathscr{C}})\rangle. \end{align} $$

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.

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

(3.2)$$ \begin{align} H^0({\mathscr{C}})/H^0({\mathscr{D}})\longrightarrow H^0({\mathscr{C}}/{\mathscr{D}}), \end{align} $$

which need not be an equivalence, in general.

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

(3.3)

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

$$\begin{align*}{\mathscr{C}}_0\hookrightarrow{\mathscr{C}}_1\hookrightarrow{\mathscr{C}}_2\hookrightarrow{\mathscr{C}}_3\hookrightarrow\cdots,\qquad\qquad {\mathscr{C}}^{\prime}_0\hookrightarrow{\mathscr{C}}^{\prime}_1\hookrightarrow{\mathscr{C}}^{\prime}_2\hookrightarrow{\mathscr{C}}^{\prime}_3\hookrightarrow\cdots, \end{align*}$$

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

$$\begin{align*}\mathrm{Hom}_{{\mathscr{C}}_0}(A,B):= \begin{cases} \Bbbk\cdot\mathrm{id}_A & \text{if } A=B \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

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

$$\begin{align*}D^1_{\mathscr{C}}(A,B):=\left\{(f,0)\in\mathrm{Hom}_{{\mathscr{C}}}(A,B)\times\mathrm{Hom}_{{\mathscr{C}}_{0}}(A,B): d(f)=0\right\}. \end{align*}$$

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 $A,B\in \mathrm {Ob}({\mathscr {C}})$,

$$\begin{align*}D^n_{\mathscr{C}}(A,B):=\left\{(f,b)\in\mathrm{Hom}_{{\mathscr{C}}}(A,B)\times\mathrm{Hom}_{{\mathscr{C}}_{n-1}}(A,B): d(f)=\mathsf{I}_{n-1}(b)\right\}. \end{align*}$$

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

$$\begin{align*}\widehat D^n_{\mathscr{C}}(A,B):=\left\{(f,b)\in\mathrm{Hom}_{{\mathscr{C}}}(A,B)\times\mathrm{Hom}_{{\mathscr{C}}^{\prime}_{n-1}}(A,B): d(f)=\mathsf{I}^{\prime}_{n-1}(b)\right\}. \end{align*}$$

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

$$\begin{align*}Z\left(\mathrm{Hom}_{{\mathscr{C}}^{\prime}_{n-1}}(A,B)\right)\longrightarrow\mathrm{Hom}_{{\mathscr{T}}}(A,B), \end{align*}$$

allowing us to choose a subset $\overline D^n_{\mathscr {C}}(A,B)\subset \widehat D^n_{\mathscr {C}}(A,B)$ so that the composite

(3.4)

is an isomorphism. And now we are ready: ${\mathscr {C}}_n$ is the graded $\Bbbk $-linear category freely generated over ${\mathscr {C}}_{n-1}$ by the sets $D^n_{\mathscr {C}}(A,B)$, while ${\mathscr {C}}^{\prime }_n$ is the graded $\Bbbk $-linear category freely generated over ${\mathscr {C}}^{\prime }_{n-1}$ by the sets $\overline D^n_{\mathscr {C}}(A,B)$. It is easy to show that ${\mathscr {C}}_n,{\mathscr {C}}^{\prime }_n$ become dg categories if we extend the differential on ${\mathscr {C}}_{n-1}$ by sending a generator $(f,b)$, in either $D_n(A,B)$ or $\overline D^n_{\mathscr {C}}(A,B)$, to b.

For $n\ge 1$, the dg functor $\mathsf {I}_n\colon {\mathscr {C}}_n\to {\mathscr {C}}$ is determined by setting $\mathsf {I}_n|_{{\mathscr {C}}_{n-1}}:=\mathsf {I}_{n-1}$ and $\mathsf {I}_n((f,b)):=f$ for all $(f,b)\in D^n_{\mathscr {C}}(A,B)$. Similarly, the dg functor $\mathsf {I}^{\prime }_n\colon {\mathscr {C}}^{\prime }_n\to {\mathscr {C}}$ is given by setting $\mathsf {I}^{\prime }_n|_{{\mathscr {C}}^{\prime }_{n-1}}:=\mathsf {I}^{\prime }_{n-1}$ and $\mathsf {I}^{\prime }_n((f,b)):=f$ for all $(f,b)\in \overline D^n_{\mathscr {C}}(A,B)$.

The following are easy consequences of the definitions.

Proof. Part (1) has the same proof as in [Reference Drinfeld16, Lemma B.5] since properties (i)–(iv) in the construction of [Reference Drinfeld16, Lemma B.5] are obviously satisfied by our construction as well (note that (iii) is verified starting from $n=2$). The functoriality and naturality in (2) are clear; the construction is choice-free. Indeed, $\mathsf {F}$ induces a natural dg functor $\mathsf {F}_0\colon {\mathscr {C}}_0\to {\mathscr {D}}_0$ acting as $\mathsf {F}$ on the objects and as the identity on the morphisms. Similarly, $\mathsf {F}$ induces a dg functor $\mathsf {F}_n\colon {\mathscr {C}}_n\to {\mathscr {D}}_n$ extending $\mathsf {F}_{n-1}$ and determined by the assignment

$$\begin{align*}(f,b)\in D^n_{\mathscr{C}}(A,B)\mapsto (\mathsf{F}(f),\mathsf{F}_{n-1}(b))\in D^n_{\mathscr{D}}(\mathsf{F}(A),\mathsf{F}(B)). \end{align*}$$

We set ${\mathsf {F}}^{\mathrm {hf}}$ to be the colimit of the dg functors $\mathsf {F}_n$. Part (3) is easy from (1).

Suppose now that ${\mathscr {C}}$ is a dg category and ${\mathscr {D}}\subseteq {\mathscr {C}}$ is a full dg subcategory. Moreover, let ${\mathscr {D}}'$ be the full dg subcategory of ${{\mathscr {C}}}^{\mathrm {hf}}$ with the same objects as ${\mathscr {D}}$. When ${\mathscr {C}}$ and ${\mathscr {D}}$ are pretriangulated, there are natural exact equivalences $H^0({\mathscr {C}})/H^0({\mathscr {D}})\cong H^0({{\mathscr {C}}}^{\mathrm {hf}})/H^0({\mathscr {D}}')\cong H^0\left ({{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'\right )$. Furthermore, not only are $\widetilde {\mathscr {C}}$ and ${{\mathscr {C}}}^{\mathrm {hf}}$ isomorphic in ${\mathbf {Hqe}}$, but by the universal property of Drinfeld quotients mentioned above, the dg categories $\widetilde {\mathscr {C}}/\widetilde {\mathscr {D}}$ and ${{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'$ are isomorphic in ${\mathbf {Hqe}}$.

Remark 3.11. Now let ${\mathscr {C}}$ be a pretriangulated dg category and ${\mathscr {D}}$ a full pretriangulated dg subcategory of ${\mathscr {C}}$. Denoting by $\mathsf {Q}\colon {{\mathscr {C}}}^{\mathrm {hf}}\to {{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'$ the natural dg functor, assume that we have a dg functor $\mathsf {F}\colon {{\mathscr {C}}}^{\mathrm {hf}}\to {\mathscr {C}}'$ of pretriangulated dg categories such that $H^0(\mathsf {F})$ sends the objects of ${\mathscr {D}}'$ to zero. Then the universal property pictured in equation (3.3) can be made more explicit: there exists a dg functor $\overline {\mathsf {F}}\colon {{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'\to {\mathscr {C}}'$ such that the diagram

is commutative in ${\mathbf {dgCat}}$. The existence of $\overline {\mathsf {F}}$ is simple enough to see from the construction of ${{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'$ and $\mathsf {Q}$ in [Reference Drinfeld16, Section 3.1]. Indeed, as ${{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'$ has the same objects as ${{\mathscr {C}}}^{\mathrm {hf}}$, one sets $\overline {\mathsf {F}}(C):=\mathsf {F}(C)$ for all C in ${{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'$. If $D\in {\mathscr {D}}'$ is an object, then $\mathrm {Hom}_{{\mathscr {C}}'}(\mathsf {F}(D),\mathsf {F}(D))$ is acyclic, allowing us to choose a degree -1 morphism $f(D)\colon \mathsf {F}(D)\to \mathsf {F}(D)$ with $d(f(D))=\mathrm {id}$. Then the dg functor $\overline {\mathsf {F}}$ extends $\mathsf {F}$ on morphisms and takes each degree $-1$ morphism $f_D\colon D\to D$, in the definition of the category ${{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'$, to the degree -1 morphism $f(D)$ in ${\mathscr {C}}'$.

The construction of the dg functor $\overline {\mathsf {F}}$ in the paragraph above depends on making choices and is not unique; the uniqueness is only up to homotopy, meaning in the category ${\mathbf {Hqe}}$. But if ${\mathscr {C}}_1$ and ${\mathscr {C}}_2$ are pretriangulated dg categories with full pretriangulated dg subcategories ${\mathscr {D}}_1\subset {\mathscr {C}}_1$ and ${\mathscr {D}}_2\subset {\mathscr {C}}_2$, and $\mathsf {G}\colon {\mathscr {C}}_1\to {\mathscr {C}}_2$ is a dg functor with $\mathsf {G}({\mathscr {D}}_1)\subseteq {\mathscr {D}}_2$, then ${\mathsf {G}}^{\mathrm {hf}}$ induces a natural dg functor ${{\mathscr {C}}}^{\mathrm {hf}}_1/{\mathscr {D}}^{\prime }_1\to {{\mathscr {C}}}^{\mathrm {hf}}_2/{\mathscr {D}}^{\prime }_2$. Both the passage from ${\mathscr {C}}$ to ${{\mathscr {C}}}^{\mathrm {hf}}$ and Drinfeld’s construction of the quotient ${{\mathscr {C}}}^{\mathrm {hf}}/{\mathscr {D}}'$ are manifestly functorial.

In the rest of the paper, we will sometimes sloppily denote by ${\mathscr {C}}/{\mathscr {D}}$ either the Drinfeld quotient of ${\mathscr {C}}$ by ${\mathscr {D}}$ when ${\mathscr {C}}$ is h-flat or of ${{\mathscr {C}}}^{\mathrm {hf}}$ by ${\mathscr {D}}'$ otherwise.

3.3 The model structure and homotopy pullbacks

Let us recall that the pullback of a diagram

(3.5)$$ \begin{align} {\mathscr{C}}_1\xrightarrow{\mathsf{F}_1}{\mathscr{D}}\overset{\mathsf{F}_2}{\longleftarrow}{\mathscr{C}}_2 \end{align} $$

in ${\mathbf {dgCat}}$ is given by a dg category ${\mathscr {C}}_1\times _{\mathscr {D}}{\mathscr {C}}_2$ defined in the obvious way. Unfortunately, this notion of pullback does not behave well with respect to quasi-equivalences.

To overcome this issue, one has to note that by the work of Tabuada [Reference Tabuada44], ${\mathbf {dgCat}}$ has a model category structure whose weak equivalences are the quasi-equivalences. We refer to [Reference Hovey21] for an exhaustive introduction to model categories. Here we content ourselves with some remarks about the special case of ${\mathbf {dgCat}}$. In particular, in Tabuada’s model structure, all dg categories are fibrant, but not all are cofibrant. Furthermore, such a model structure is right proper: that is, every pullback of a weak equivalence along a fibration is a weak equivalence, thanks to the fact that all objects are fibrant (see [Reference Hirschhorn20, Corollary 13.1.3]). Finally, ${\mathbf {Hqe}}$ can be reinterpreted as the homotopy category of ${\mathbf {dgCat}}$ with respect to such a model structure.

As is explained, for instance, in [Reference Hirschhorn20, Section 13.3], one can then consider the homotopy pullback ${\mathscr {C}}_1\times ^h_{\mathscr {D}} {\mathscr {C}}_2$ of equation (3.5). By definition ${\mathscr {C}}_1\times ^h_{\mathscr {D}} {\mathscr {C}}_2:={\mathscr {C}}^{\prime }_1\times _{\mathscr {D}}{\mathscr {C}}^{\prime }_2$ is the usual pullback of a diagram

(3.6)$$ \begin{align} {\mathscr{C}}^{\prime}_1\xrightarrow{\mathsf{F}^{\prime}_1}{\mathscr{D}}\overset{\mathsf{F}^{\prime}_2}{\longleftarrow}{\mathscr{C}}^{\prime}_2, \end{align} $$

where at least one among $\mathsf {F}^{\prime }_1$ and $\mathsf {F}^{\prime }_2$ is a fibration and (for $i=1,2$) $\mathsf {F}_i=\mathsf {F}^{\prime }_i\circ \mathsf {I}_i$ with $\mathsf {I}_i\colon {\mathscr {C}}_i\to {\mathscr {C}}^{\prime }_i$ a quasi-equivalence. Notice that such a factorisation of $\mathsf {F}_i$ always exists, and in fact one could choose $\mathsf {I}_i$ to be a cofibration as well. The homotopy pullback does not depend, up to isomorphism in ${\mathbf {Hqe}}$, on the choice of the diagram given by equation (3.6).

Let us spell out an explicit description of ${\mathscr {C}}_1\times ^h_{\mathscr {D}} {\mathscr {C}}_2$. We can take $\mathsf {F}^{\prime }_2=\mathsf {F}_2$ and factor only $\mathsf {F}_1$ as follows. Define ${\mathscr {C}}^{\prime }_1$ to be the dg category whose objects are triples, $(C_1,D,f)$, where $C_1\in \mathrm {Ob}({\mathscr {C}}_1)$, $D\in \mathrm {Ob}({\mathscr {D}})$ and $f\colon \mathsf {F}_1(C_1)\to D$ is a homotopy equivalence. A morphism of degree n from $(C_1,D,f)$ to $(C^{\prime }_1,D',f')$ in ${\mathscr {C}}^{\prime }_1$ is given by a triple $(a_1,b,h)$ with $a_1\in \mathrm {Hom}_{{\mathscr {C}}_1}(C_1,C^{\prime }_1)^n$, $b\in \mathrm {Hom}_{\mathscr {D}}(D,D')^n$ and $h\in \mathrm {Hom}_{\mathscr {D}}(\mathsf {F}_1(C_1),D')^{n-1}$. The differential is defined by

$$\begin{align*}d(a_1,b,h):=(d(a_1),d(b),d(h)+(-1)^n(f'\circ\mathsf{F}_1(a_1)-b\circ f)) \end{align*}$$

and the composition by

$$\begin{align*}(a^{\prime}_1,b',h')\circ(a_1,b,h):=(a^{\prime}_1\circ a_1,b'\circ b,b'\circ h+(-1)^nh'\circ \mathsf{F}_1(a_1)). \end{align*}$$

The dg functor $\mathsf {I}_1$ is defined by $\mathsf {I}_1(C_1):=(C_1,\mathsf {F}_1(C_1),\mathrm {id}_{\mathsf {F}_1(C_1)})$ on objects and $\mathsf {I}_1(a_1):=(a_1,\mathsf {F}_1(a_1),0)$ on morphisms. On the other hand, the dg functor $\mathsf {F}^{\prime }_1$ is defined as projection on the second component both on objects and on morphisms. It is not difficult to check that $\mathsf {I}_1$ is a quasi-equivalence and $\mathsf {F}^{\prime }_1$ is a fibration.

With the above choice, ${\mathscr {C}}_1\times ^h_{\mathscr {D}}{\mathscr {C}}_2$ can be identified with the dg category whose objects are triples $(C_1,C_2,f)$, where $C_i\in \mathrm {Ob}({\mathscr {C}}_i)$ for $i=1,2$ and $f\colon \mathsf {F}_1(C_1)\to \mathsf {F}_2(C_2)$ is a homotopy equivalence. A morphism of degree n from $(C_1,C_2,f)$ to $(C^{\prime }_1,C^{\prime }_2,f')$ in ${\mathscr {C}}_1\times ^h_{\mathscr {D}}{\mathscr {C}}_2$ is given by a triple $(a_1,a_2,h)$ with $a_i\in \mathrm {Hom}_{{\mathscr {C}}_i}(C_i,C^{\prime }_i)^n$ for $i=1,2$ and $h\in \mathrm {Hom}_{\mathscr {D}}(\mathsf {F}_1(C_1),\mathsf {F}_2(C^{\prime }_2)^{n-1}$. The differential is defined by

$$\begin{align*}d(a_1,a_2,h):=(d(a_1),d(a_2),d(h)+(-1)^n(f'\circ\mathsf{F}_1(a_1)-\mathsf{F}_2(a_2)\circ f)) \end{align*}$$

and the composition by

$$\begin{align*}(a^{\prime}_1,a^{\prime}_2,h')\circ(a_1,a_2,h):=(a^{\prime}_1\circ a_1,a^{\prime}_2\circ a_2,\mathsf{F}_2(a^{\prime}_2)\circ h+(-1)^nh'\circ \mathsf{F}_1(a_1)). \end{align*}$$

Remark 3.12. By the universal property of the pullback, if ${\mathscr {C}}$ is a dg category with dg functors $\mathsf {G}_i\colon {\mathscr {C}}\to {\mathscr {C}}_i$ for $i=1,2$ making the diagram

(3.7)

commutative in ${\mathbf {dgCat}}$, then there is a unique dg functor $\mathsf {F}\colon {\mathscr {C}}\to {\mathscr {C}}_1\times ^h_{\mathscr {D}} {\mathscr {C}}_2$ making the diagram

commutative in ${\mathbf {dgCat}}$. Given the explicit description of the homotopy pullback discussed above, the dg functor $\mathsf {F}$ is defined by

$$\begin{align*}\mathsf{F}(C):=\left(\mathsf{G}_1(C),\mathsf{G}_2(C),\mathrm{id}_{\mathsf{F}_1(\mathsf{G}_1(C))}\right) \end{align*}$$

on objects and

$$\begin{align*}\mathsf{F}(a):=\left(\mathsf{G}_1(a),\mathsf{G}_2(a),0\right) \end{align*}$$

on morphisms.

Remark 3.13. The homotopy pullback is a special instance of the concept of homotopy limit in a model category for which exhaustive presentations are available—for example, in [Reference Hirschhorn20, Section 19.1]. What is important for us is that homotopy limits are well-defined once we invert weak equivalences. Concretely, given a small category ${\mathscr {N}}$ and a functor $\mathsf {F}\colon {\mathscr {N}}\to {\mathbf {dgCat}}$, the homotopy limit $\operatorname {\mathrm {Holim}}(\mathsf {F})$ is a well-defined object of ${\mathbf {Hqe}}$ up to canonical isomorphism. Moreover, suppose we are given a diagram

meaning ${\mathscr {N}}$ is a small category, $\mathsf {F},\mathsf {G}\colon {\mathscr {N}}\to {\mathbf {dgCat}}$ are functors, $\theta :\mathsf {F}\to \mathsf {G}$ is a natural transformation and $\pi \colon {\mathbf {dgCat}}\to {\mathbf {Hqe}}$ is the natural functor. Then $\operatorname {\mathrm {Holim}}(\theta )$ delivers a well-defined morphism in ${\mathbf {Hqe}}$ (up to canonical isomorphism), and if $\pi \circ \theta \colon \pi \colon \mathsf {F}\to \pi \colon \mathsf {G}$ is an isomorphism, then so is $\operatorname {\mathrm {Holim}} (\theta )$. Thus $\operatorname {\mathrm {Holim}}\colon \mathrm {Hom}\big ({\mathscr {N}},{\mathbf {dgCat}}\big )\to {\mathbf {Hqe}}$ takes morphisms in $\mathrm {Hom}\big ({\mathscr {N}},{\mathbf {dgCat}}\big )$ that induce isomorphisms in $\mathrm {Hom}\big ({\mathscr {N}},{\mathbf {Hqe}}\big )$ to isomorphisms in ${\mathbf {Hqe}}$.

What we will need in Section 7 is the following simple case. We start with a finite set of integers $N:=\{1,\dots ,n\}$ and form the category ${\mathscr {N}}$, whose objects are the subsets of N and whose morphisms are the inclusions. We then consider a functor ${\mathscr {N}}\to {\mathbf {dgCat}}$ and denote by ${\mathscr {C}}_I$ the dg category corresponding to $I\subseteq N$. We can then form the homotopy limit

$$\begin{align*}\operatorname{\mathrm{Holim}}_{\emptyset\ne I\subseteq N}{\mathscr{C}}_I. \end{align*}$$

The special case when $n=2$ is just a homotopy pullback, specifically the homotopy pullback of ${\mathscr {C}}_{\{1\}}\to {\mathscr {C}}_{\{1,2\}}\leftarrow {\mathscr {C}}_{\{2\}}$. In the discussion just preceding Remark 3.12, we spelled out an explicit, functorial construction for homotopy pullbacks, allowing us to enhance the homotopy pullback to a functor $\operatorname {\mathrm {Holim}}\colon \mathrm {Hom}\big (\{1\}\to \{1,2\}\leftarrow \{2\},{\mathbf {dgCat}}\big )\to {\mathbf {dgCat}}$. This will allow us to iterate the construction. We will end up reducing the computation of the homotopy limit, in the case where $n>2$, to iterated homotopy pullbacks.

3.4 Localisations in dg categories

Let us first recall a basic construction in the context of triangulated categories. Let ${\mathscr {T}}$ be such a category, and suppose further that ${\mathscr {T}}$ is closed under arbitrary coproducts. Let ${\mathscr {S}}\subset {\mathscr {T}}^c$ be a collection of compact objects closed under shifts. Then every object $T\in {\mathscr {T}}$ sits in a distinguished triangle

$$\begin{align*}S\longrightarrow T\longrightarrow T_S \end{align*}$$

with $S\in {\mathbf {Loc}}({\mathscr {S}})$, the smallest full triangulated subcategory of ${\mathscr {T}}$ containing ${\mathscr {S}}$ and closed under arbitrary coproducts, and $T_S$ in ${\mathscr {S}}^{\perp }:=\left \{T\in {\mathscr {T}}:\mathrm {Hom}_{{\mathscr {T}}}({\mathscr {S}},T)=0\right \}$.

The expert reader has certainly noticed that this observation follows from the existence of a semi-orthogonal decomposition for ${\mathscr {T}}$ with factors given by ${\mathscr {S}}$ and ${\mathscr {S}}^{\perp }\cong {\mathscr {T}}/{\mathscr {S}}$. But here, we want to stress that the construction of S and $T_S$ is explicit. Indeed, put $T_0=T$, and then inductively construct distinguished triangles

$$\begin{align*}\coprod_{C\to T_i}C\longrightarrow T_i\longrightarrow T_{i+1}, \end{align*}$$

where the first map is the coproduct of all morphisms $C\to T_i$ with $C\in \mathrm {Ob}({\mathscr {S}})$. Then we consider the map $T\longrightarrow \operatorname {\mathrm {Hocolim}} T_i$ and complete it to a distinguished triangle

$$\begin{align*}S\longrightarrow T\longrightarrow \operatorname{\mathrm{Hocolim}} T_i. \end{align*}$$

A direct computation shows that $S\in \mathrm {Ob}({\mathbf {Loc}}({\mathscr {S}}))$ and $T_S:=\operatorname {\mathrm {Hocolim}} T_i\in \mathrm {Ob}({\mathscr {S}}^{\perp })$ (see, for example, [Reference Neeman34, Lemma 1.7]).

The following result will be used later and provides an enhancement of the above discussion to the setting of dg categories. Actually, the content of the proposition is more elaborate and deals with two distinct (and orthogonal) localisations.

Proposition 3.14. Let ${\mathscr {C}}$ be a dg category, and suppose that ${\mathscr {D}}_1$ and ${\mathscr {D}}_2$ are full dg subcategories of ${\mathscr {C}}$ such that

1. (i) ${\mathscr {D}}_1$ and ${\mathscr {D}}_2$ are closed under shifts.

2. (ii) $\mathrm {Hom}_{H^0({\mathscr {C}})}({\mathscr {D}}_i,{\mathscr {D}}_j)=0$ for $i\neq j\in \{1,2\}$.

Then for any $C\in \mathrm {Ob}(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$, there exists a diagram in $Z^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$

(3.8)

that is commutative in $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$ and such that

1. (1) its rows and columns yield distinguished triangles in $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$;

2. (2) $D_1\in \mathrm {Ob}(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}_1))$ and $D_2\in \mathrm {Ob}(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}_2))$;

3. (3) the complexes

   $$ \begin{align*}\mathrm{Hom}_{\mathrm{h}\text{-}\mathrm{proj}({\mathscr{C}})}\left(\mathrm{h}\text{-}\mathrm{proj}({\mathscr{D}}_i),C_{D_i}\right)\qquad\mathrm{Hom}_{\mathrm{h}\text{-}\mathrm{proj}({\mathscr{C}})}\left(\mathrm{h}\text{-}\mathrm{proj}({\mathscr{D}}_i),C_{D_1,D_2}\right)\end{align*} $$
   are acyclic for $i=1,2$.

Proof. To obtain the objects $D_i$ and $C_{D_i}$ and the triangles in the second row and the second column in equation (3.8), we proceed by enhancing the construction at the beginning of this section to the context of dg categories. Since the situation is symmetric, we denote by ${\mathscr {D}}$ either of the two dg subcategories ${\mathscr {D}}_1$ and ${\mathscr {D}}_2$. Given $C\in \mathrm {Ob}(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$, we set $C_0=C$ and then inductively define $C_{i+1}$ to be the cone of the map

$$\begin{align*}\coprod_{P\to C_i}P\longrightarrow C_i, \end{align*}$$

where the coproduct is over all objects $P\in \mathrm {Ob}({\mathscr {D}})$ and a set of morphisms in $\mathrm {Hom}_{Z^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}))}(P,C_i)$ representing all morphisms in $\mathrm {Hom}_{H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}))}(P,C_i)$. As above, the natural map $C\to \operatorname {\mathrm {Hocolim}} C_i$ in $Z^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$ sits in the triangle

$$\begin{align*}D\longrightarrow C\longrightarrow\operatorname{\mathrm{Hocolim}} C_i, \end{align*}$$

which is distinguished in $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$. The argument above now shows that D is an object of $\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}})$ and $C_D:=\operatorname {\mathrm {Hocolim}} C_i$ is such that $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}(Q,C_D)$ is acyclic for all $Q\in \mathrm {Ob}(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}))$.

Since $\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})$ is pretriangulated, the second row and second column we just constructed can be completed to the diagram of equation (3.8), where the rows and columns are cofibre sequences, in particular yielding distinguished triangles in $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$.

It remains to prove that the object $C_{D_1,D_2}$ that we constructed is such that the complex $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}\left (\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}_i),C_{D_1,D_2}\right )$ is acyclic for $i=1,2$. To this end, we apply the functor $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}^{}\left ({\mathscr {D}}_i,-\right )$ to the triangle

$$\begin{align*}D_j\longrightarrow C_{D_i}\longrightarrow C_{D_1,D_2}, \end{align*}$$

where $i\neq j\in \{1,2\}$. The complex $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}\left ({\mathscr {D}}_i,C_{D_i}\right )$ is clearly acyclic from the first part of the construction. By (ii), we know that $\mathrm {Hom}_{\mathscr {C}}({\mathscr {D}}_i,{\mathscr {D}}_j)$ is acyclic, and thus $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}\left ({\mathscr {D}}_i,\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}_j)\right )$ is acyclic as well, because the objects in ${\mathscr {D}}_i$ are in ${\mathscr {C}}$ and thus are compact in $H^0(\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}}))$. Since $D_j$ belongs to $\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}_j)$, we obtain that $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}\left ({\mathscr {D}}_i,D_j\right )$ is acyclic. From this, we deduce that $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}\left ({\mathscr {D}}_i,C_{D_1,D_2}\right )$ is acyclic, and this implies that $\mathrm {Hom}_{\mathrm {h}\text {-}\mathrm {proj}({\mathscr {C}})}\left (\mathrm {h}\text {-}\mathrm {proj}({\mathscr {D}}_i),C_{D_1,D_2}\right )$ is acyclic as well.

3.5 Dg enhancements and their uniqueness

In this section, we will be careful about the universe where the triangulated and dg categories live—mostly to show that such care is superfluous.

By abuse of notation, one says that ${\mathscr {C}}$ is a $\mathbb {V}$-enhancement of ${\mathscr {T}}$ if $\mathsf {E}$ is clear from the context.

Example 3.16. If ${\mathscr {A}}$ is a $\mathbb {U}$-small additive category, then ${\mathbf {C}}_{{\mathbf {dg}}}^?({\mathscr {A}})$ is in a natural way a $\mathbb {U}$-enhancement of ${\mathbf {K}}^?({\mathscr {A}})$ (see Example 3.1). If ${\mathscr {A}}$ is abelian, then, since ${\mathbf {D}}^?({\mathscr {A}})={\mathbf {K}}^?({\mathscr {A}})/{\mathbf {K}}^?_{\mathrm {acy}}({\mathscr {A}})$, the discussion in Section 3.2 allows us to conclude that the Drinfeld quotient

$$\begin{align*}{\mathbf{D}}^?_{\mathbf{dg}}({\mathscr{A}}):={\mathbf{C}}_{{\mathbf{dg}}}^?({\mathscr{A}})/{\mathbf{Acy}}^?({\mathscr{A}}) \end{align*}$$

is a $\mathbb {U}$-enhancement of ${\mathbf {D}}^?({\mathscr {A}})$. Here ${\mathbf {Acy}}^?({\mathscr {A}})$ denotes the full dg subcategory of ${\mathbf {C}}_{{\mathbf {dg}}}^?({\mathscr {A}})$ whose objects correspond to those of ${\mathbf {K}}^?_{\mathrm {acy}}({\mathscr {A}})$.

The core of this paper is to study when dg enhancements are unique in the following sense.

Let us now prove the following result.

Given this, in the rest of the paper, it will rarely be necessary to specify in which universe we are working. For this reason, we will freely talk about dg enhancements and their uniqueness without mentioning any universe.

An equivalent way to state Definition 3.17 is by saying that there is an isomorphism $f\in \mathrm {Hom}_{\mathbf {Hqe}}({\mathscr {C}}_1,{\mathscr {C}}_2)$. With this in mind, we can state the following stronger definition.

4.1 An enhancement for ${\mathbf {B}}^?({\mathscr {A}})$

Remembering Example 3.16, it is clear that for $?=b,+,-,\emptyset $, the full dg subcategory ${\mathscr {V}}={\mathscr {V}}^?({\mathscr {A}})$ of ${\mathbf {C}}_{{\mathbf {dg}}}^?({\mathscr {A}})$ whose objects are complexes with trivial differential is in a natural way an enhancement of ${\mathbf {V}}^?({\mathscr {A}})$.

Assume from now on that ${\mathscr {A}}$ is abelian, and fix an enhancement $({\mathscr {C}},\mathsf {E})$ of ${\mathbf {D}}^?({\mathscr {A}})$. We are going to define a dg category ${\mathscr {B}}^?_{{\mathscr {C}},\mathsf {E}}({\mathscr {A}})$, which will turn out to be an enhancement of ${\mathbf {B}}^?({\mathscr {A}})$. Recall that the latter category is the full subcategory of ${\mathbf {D}}^?({\mathscr {A}})$ consisting of complexes with trivial differential. The notation for ${\mathscr {B}}^?_{{\mathscr {C}},\mathsf {E}}({\mathscr {A}})$ alludes to the fact that its definition depends on the pair $({\mathscr {C}},\mathsf {E})$. But in the sequel, when there is no risk of confusion, we will use the shorthand ${\mathscr {B}}:={\mathscr {B}}^?_{{\mathscr {C}},\mathsf {E}}({\mathscr {A}})$.

An object $B=(B^-,B^+,B^i,\alpha ^i,\beta ^i)_{i\in \mathbb {Z}}$ of ${\mathscr {B}}$ is given by objects $B^-$, $B^+$ and $B^i$ of ${\mathscr {C}}$ together with morphisms ${B^i}[-i]\xrightarrow {\alpha ^i}B^?\xrightarrow {\beta ^i}{B^i}[-i]$ of $Z^0({\mathscr {C}})$ (where $?=+$ if $i>0$ and $?=-$ if $i\le 0$) such that the following conditions are satisfied:

1. (B.1) If $i,j$ are both $>0$ or both $\le 0$, then $\beta ^j\circ \alpha ^i$ is $\mathrm {id}_{{B^i}[-i]}$ for $i=j$ and is $0$ for $i\ne j$.

2. (B.2) $\mathsf {E}(B^i)\in \mathrm {Ob}({\mathscr {A}})$ for every $i\in \mathbb {Z}$.

3. (B.3) The morphisms $\alpha ^i$ with $i\le 0$ and $\beta ^i$ with $i>0$ induce isomorphisms in $H^0({\mathscr {C}})\cong {\mathbf {D}}^?({\mathscr {A}})$

   $$\begin{align*}\coprod_{i\le0}{B^i}[-i]\xrightarrow{\sim} B^-,\qquad B^+\xrightarrow{\sim}\prod_{i>0}{B^i}[-i]. \end{align*}$$

Given objects $B_k=(B_k^-,B_k^+,B_k^i,\alpha _k^i,\beta _k^i)_{i\in \mathbb {Z}}$, for $k=1,2$, we define

$$\begin{align*}\mathrm{Hom}_{{\mathscr{B}}}(B_1,B_2):=\mathrm{Hom}_{{\mathscr{C}}}(B_1^-\oplus B_1^+,B_2^-\oplus B_2^+). \end{align*}$$

It is straightforward to check that ${\mathscr {B}}$ (with the obvious composition of morphisms) is a dg category and that the map defined on objects by $(B^-,B^+,B^i,\alpha ^i,\beta ^i)_{i\in \mathbb {Z}}\mapsto B^-\oplus B^+$ extends to a fully faithful dg functor $\mathsf {B}\colon {\mathscr {B}}\to {\mathscr {C}}$.

One can consider the following variant, which will be used in the proof of Theorem B. Let ${\mathscr {A}}$ be an abelian category with a Serre subcategory ${\mathscr {E}}\subseteq {\mathscr {A}}$. Consider then the full triangulated subcategory ${\mathbf {D}}^?_{\mathscr {E}}({\mathscr {A}})$ of ${\mathbf {D}}^?({\mathscr {A}})$ consisting of all complexes with cohomology in ${\mathscr {E}}$. There is a natural exact functor

$$\begin{align*}\pi\colon{\mathbf{D}}^?({\mathscr{E}})\longrightarrow{\mathbf{D}}^?_{\mathscr{E}}({\mathscr{A}}), \end{align*}$$

which in general is not an equivalence.

Given a dg enhancement $({\mathscr {C}},\mathsf {E})$ of ${\mathbf {D}}^?_{\mathscr {E}}({\mathscr {A}})$, we define a dg category $\widehat {\mathscr {B}}^?_{{\mathscr {C}},\mathsf {E}}({\mathscr {A}},{\mathscr {E}})$, for which we use the shorthand $\widehat {\mathscr {B}}$, in a fashion that is very similar to ${\mathscr {B}}$.

In particular, an object $B=(B^-,B^+,B^i,\alpha ^i,\beta ^i)_{i\in \mathbb {Z}}$ of $\widehat {\mathscr {B}}$ is given by objects $B^-$, $B^+$ and $B^i$ of ${\mathscr {C}}$ together with morphisms ${B^i}[-i]\xrightarrow {\alpha ^i}B^?\xrightarrow {\beta ^i}{B^i}[-i]$ of $Z^0({\mathscr {C}})$ (where $?=+$ if $i>0$ and $?=-$ if $i\le 0$) such that (B.1) is satisfied, while (B.2) and (B.3) are replaced, respectively, by the following:

1. (B.2’) $\mathsf {E}(B^i)\cong \pi (Q_i)$ for some $Q_i\in \mathrm {Ob}({\mathscr {E}})$ and for every $i\in \mathbb {Z}$.

2. (B.3’) The morphisms $\alpha ^i$ with $i\le 0$ and $\beta ^i$ with $i>0$ induce isomorphisms in ${\mathbf {D}}^?_{\mathscr {E}}({\mathscr {A}})$

   $$\begin{align*}\pi\left(\coprod_{i\le0}{Q^i}[-i]\right)\xrightarrow{\sim}\mathsf{E}(B^-),\qquad \mathsf{E}(B^+)\xrightarrow{\sim}\pi\left(\prod_{i>0}{Q^i}[-i]\right). \end{align*}$$

As for the Hom-spaces, given objects $B_k=(B_k^-,B_k^+,B_k^i,\alpha _k^i,\beta _k^i)_{i\in \mathbb {Z}}$, for $k=1,2$, we keep the same definition and set

$$\begin{align*}\mathrm{Hom}_{\widehat{\mathscr{B}}}(B_1,B_2):=\mathrm{Hom}_{{\mathscr{C}}}(B_1^-\oplus B_1^+,B_2^-\oplus B_2^+). \end{align*}$$

Recall the h-flat resolution ${\mathsf {I}}^{\mathrm {hf}}_{\mathscr {C}}\colon {{\mathscr {C}}}^{\mathrm {hf}}\to {\mathscr {C}}$ of Proposition 3.10.

Proposition 4.1. If $({\mathscr {C}},\mathsf {E})$ is a dg enhancement of ${\mathbf {D}}^?_{\mathscr {E}}({\mathscr {A}})$, then ${\mathsf {I}}^{\mathrm {hf}}_{\mathscr {C}}$ induces a quasi-equivalence

$$\begin{align*}{\widehat{\mathsf{I}}}^{\mathrm{hf}}_{\mathscr{C}}\colon\widehat{\mathscr{B}}^?_{{{\mathscr{C}}}^{\mathrm{hf}},\mathsf{E}\circ H^0({\mathsf{I}}^{\mathrm{hf}}_{\mathscr{C}})}({\mathscr{A}},{\mathscr{E}})\longrightarrow\widehat{\mathscr{B}}^?_{{\mathscr{C}},\mathsf{E}}({\mathscr{A}},{\mathscr{E}}), \end{align*}$$

which is also surjective on objects.