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Internal Homs via extensions of dg functors

We provide a simple proof of the existence of internal Homs in the localization of the category of dg categories with respect to all quasi-equivalences and of some of their main properties such as the so-called derived Morita theory. This was originally proved in a seminal paper by To\"{e}n.


Introduction
The problem of characterizing exact functors between triangulated categories is certainly one of the major open questions in the theory of triangulated categories. As soon as we deal with triangulated categories which are the bounded derived categories of coherent sheaves on smooth projective varieties, this challenge in the vague form above gets neater. More precisely, if X 1 and X 2 are smooth projective schemes and we denote by D b (X i ) the bounded derived category of coherent sheaves on X i , then one would expect that all exact functors F : D b (X 1 ) −→ D b (X 2 ) are of Fourier-Mukai type (see [2,15]). This means that there should exist E ∈ D b (X 1 × X 2 ) and an isomorphism of exact functors F ∼ = Φ E , where, denoting by p i : X 1 × X 2 → X i the natural projections, Φ E : D b (X 1 ) → D b (X 2 ) is the exact functor defined by Φ E := R(p 2 ) * (E ⊗ L p * 1 (−)). Unfortunately, only partial results confirm the expectation above (see [5] for a survey about this).
If one changes perspective slightly and moves to higher categorical structures, the situation becomes amazingly beautiful. More precisely, one looks at the localization Hqe of the category dgCat of (small) dg categories over a commutative ring k with respect to all quasi-equivalences. Then one can take dg enhancements D 1 and D 2 of D b (X 1 ) and D b (X 2 ) respectively. It turns out (see [2]) that all Fourier-Mukai functors at the triangulated level lift to morphisms between D 1 and D 2 in Hqe. More surprisingly, all morphisms in Hqe between these dg categories are of this type. This was observed in the seminal paper [18] and comes as a corollary of a very general and elegant result about the existence of internal Homs in Hqe. The statement can be formulated as follows. proving that the symmetric monoidal category Hqe is closed.
Here [−, −] denotes the set of morphisms in Hqe, while h-proj(−) rqr denotes right quasirepresentable h-projective dg modules, which will be precisely described in Section 3.1. In the original version [18] h-proj(−) rqr is replaced by the dg category of right quasi-representable cofibrant dg modules, which is actually quasi-equivalent to h-proj(−) rqr . Recall that the monoidal structure provided by the derived tensor product − ⊗ L − is said to be closed if, for every B, C ∈ Hqe, there exists RHom(B, C) ∈ Hqe, such that the functor A → [A ⊗ L B, C] is isomorphic to A → [A, RHom(B, C)].
Roughly speaking, Theorem 1.1 asserts that all dg (quasi-)functors are of Fourier-Mukai type. The reason is that, if one looks carefully at the proof of the bijection (1.1), one sees that all morphisms in Hqe are essentially provided by the tensorization by dg bimodules, mimicking the definition of Fourier-Mukai functors given above in the triangulated setting. It is worth pointing out that the first part of Theorem 1.1 comes in [18] as a corollary of a much more general result involving substantially the simplicial structure on dgCat, seen as a model category (see [18,Thm. 4.2]).
The purpose of this paper is to provide a simple proof of Theorem 1.1 essentially based on the circle of ideas emerging from [6] and [7].
Comparing Toën's approach and ours. Following [18], one uses the model category structure on dgCat (see [16]) in such a way that any morphism in Hqe can be represented by a 'canonical' roof by means of the cofibrant replacements. At this point, the description of the morphisms between two dg categories in Hqe can be essentially carried out assuming that we are working with actual dg functors. In this way, Toën proves that, for two dg categories A and B, there is a bijection between [A, B] and the isomorphism classes of the homotopy category of right quasirepresentable (fibrant and) cofibrant A • ⊗ B-dg modules (see Corollary 4.10 of [18]). This is essentially the first part of Theorem 1.1.
The existence of internal Homs follows from a characterization of the model category of dg functors between dg categories and a comparison between this and the presentation above ([18, Thm. 6.1]). As an application of the existence of internal Homs, Toën deduces in [18,Thm. 7.2] a restriction theorem asserting in particular that, given two dg categories A and B, the Yoneda embedding of A into the dg category Int(A) of cofibrant A-dg modules yields a quasi-equivalence between the continuous internal Hom RHom c (Int(A), Int(B)) and RHom(A, Int(B)) (see Sections 2.1 and 4 for the precise definitions), which goes under the name of derived Morita theory. This is a very interesting result in itself with various geometric applications as explained in [18,Sect. 8]. One should keep in mind that, for a dg category A, the dg category Int(A) is contained in the dg category h-proj(A) of h-projective A-dg modules and the inclusion is a quasi-equivalence (see, for example, [11,Lemma 2.6] and the discussion in Section 3.2 of [8] for this standard fact).
In a sense, our argument starts from this sort of ending point in [18]. Indeed, we prove directly a weaker form of this restriction result (see Proposition 3.10 and Corollary 3.13) which provides a nice description of some morphisms in Hqe in terms of isomorphism classes of special dg bimodules (see Proposition 3.12). The existence of internal Homs and the proof of Theorem 1.1 follow then from a simple and purely formal argument explained in Section 4. All of this is achieved using the notion of extension of dg functors which is already contained in [7]. This is carried out in Sections 3.1 and further developed to deal with morphisms in Hqe in Section 3.2. As we will see, this is a conceptually very simple application of the notion of tensor product of dg modules (see [6] and [7]). In particular, it should be noted that the core and the really non-trivial part of this paper is the content of Section 3.
This slight change of perspective makes our proof easier also because we can forget about the model category structure and content ourselves with the fact that the category of dg categories is a category of fibrant objects. This has certainly been known for a long time and is summarized in Section 2.2 (after a short introduction to dg categories in Section 2.1).
Once Theorem 1.1 is settled, some important properties of internal Homs which are proved in [18] can be deduced in a straightforward way. This is the case of Corollaries 4.1 and 4.2. The last one covers the dg Morita theory mentioned above.
It is probably worth pointing out here that Toën's result gave an input to further generalizations at the level of ∞-categories (see, for example, [1]). But for this one really needs the model and simplicial structures on dgCat. This is out of the scope of our paper.
The reader should be also aware that a different approach to the existence of internal Homs was proposed by Tabuada in [17]. In particular, he constructs a new model category structure for the homotopy category of dg categories, where the localization takes place with respect to the so-called Morita equivalences and not just the quasi-equivalences. Clearly, this means that [17] is not in the same generality as [18]. Moreover, [17] goes in a transversal direction with respect to the present work. Nevertheless, one can observe that in Tabuada's approach the internal Homs can be naturally interpreted as derived functors.
We conclude this summary going back to the triangulated setting presented at the beginning. It is important to observe that there is no hope that the beauty of Theorem 1.1 can appear in the triangulated context as well. Indeed, it has been shown in [4] that the object E ∈ D b (X 1 × X 2 ) realizing a Fourier-Mukai functor is by no means unique (up to isomorphism).
Notation and general assumptions. We denote by k a commutative ring. By a k-linear category we mean a category whose Hom spaces are k-modules and such that the compositions are k-bilinear, not assuming that finite direct sums exist. For a category A, we denote by Iso(A) the set of isomorphism classes of objects in A.
Throughout the paper, we assume that a universe U containing an infinite set is fixed. Anticipating some definitions that will be explained in Section 2, let us spend some words to clarify the context. We will consider U-small dg categories, meaning dg categories D such that Hom D (D 1 , D 2 ) is a complex of k-modules which are isomorphic to objects of U and such that the collection of objects of D is isomorphic to an object of U as well. Analogously one can speak about U-small sets. We will then define the dg categories of D-dg modules and of h-projective D-dg modules. Again, with this we tacitly mean the dg categories of U-small D-dg modules and h-projective D-dg modules. This simply refers to the dg modules that take values in the dg category of complexes of U-small k-modules. These are no longer U-small dg categories. Nevertheless, due to the results in [10, Appendix A], they are dg equivalent to V-small dg categories, for some universe U ∈ V. By its definition and main properties (see Section 4), the dg category h-proj(−) rqr mentioned above is essentially U-small. This means that the isomorphism classes of its objects form a U-small set. After these warnings and to simplify the notation, we will not mention explicitly the universe where we are working any longer in the paper, as it should be clear from the context. We refer to [10, Appendix A] for all possible subtle logical issues.

Basic properties of dg categories
This section collects some very well-known facts concerning dg categories. The emphasis is on the properties of morphisms in the localization of the category of dg categories by quasi-equivalences.  (ii) Every dg algebra A over k defines a dg category with one object and A as its space of endomorphisms. Notice that an ordinary k-algebra (in particular, k itself) can be regarded as a dg algebra in degree 0, hence as a dg category with one object.
(iii) We denote by C dg (k) the dg category whose objects are complexes of k-modules. We refer to [8,Sect. 2.2] for the precise definition.
(iv) Given two dg categories A and B, one can construct the dg categories Hom(A, B) and A ⊗ B (see [8,Sect. 2.3] for the precise definitions). The objects of Hom(A, B) are dg functors from A to B and morphisms are given by (dg) natural transformations. On the other hand, the objects of A ⊗ B are pairs (A, B) with A ∈ A and B ∈ B, while the morphisms are defined by for all (A i , B i ) ∈ A ⊗ B and i = 1, 2. Notice that the tensor product defines a symmetric monoidal structure on dgCat. Namely, up to isomorphism, the tensor product is associative, commutative and k acts as the identity. It is also easy to see that two dg functors F : A → B and G : C → D naturally induce a dg functor F ⊗ G : A ⊗ C → B ⊗ D.
(v) If A is a dg category, A • denotes the opposite dg category. The objects of A • are the same as those of A, while A • (A, B) := A(B, A) and the compositions in A • are defined as in A, up to a sign (see [8,Sect. 2.2] for details).
If A is a dg category, we denote by Z 0 (A) (respectively H 0 (A)) the (k-linear) category with the same objects as A and whose morphisms from A to B are given by Z 0 (A(A, B)) (respectively H 0 (A(A, B))). The category H 0 (A) is called the homotopy category of A, and it has a natural structure of triangulated category if A is pretriangulated (see [8,Sect. 4.5] for the precise definition). A morphism of Z 0 (A) is a dg isomorphism (respectively a homotopy equivalence) if it is an isomorphism (respectively if its image in H 0 (A) is an isomorphism). Accordingly, two objects A and B of A are dg isomorphic (respectively homotopy equivalent) if A ∼ = B in Z 0 (A) (respectively in H 0 (A)). If B is another dg category, two dg functors from A to B will be said to be dg isomorphic (respectively homotopy equivalent) if they are dg isomorphic (respectively homotopy equivalent) in Hom (A, B).
For all dg categories A, B and C, there is a natural isomorphism in dgCat In particular, there is a natural bijection between the objects of the two dg categories above, which shows that the functor − ⊗ B : dgCat → dgCat is left adjoint to Hom(B, −) : dgCat → dgCat.
Hence the symmetric monoidal structure on dgCat discussed in Example 2.1 (iv) is closed. At the same time we denote by Perf(A) the full dg subcategory of h-proj(A) consisting of perfect A-dg modules, i.e. the compact objects in the triangulated category H 0 (h-proj(A)). Recall that an object C in a triangulated category D is compact if, given {D i } i∈I ⊂ D such that I is a set and The Yoneda embedding of A is the fully faithful (and injective on objects) dg functor and H 0 (F) is an equivalence of categories. One can consider the localization Hqe of dgCat with respect to quasi-equivalences (which is denoted by Ho(dgCat) in [18] Remark 2.4. First of all, one observes that the natural transformation in Definition 2.3 induces a natural transformation θ ′ between H 0 (G 1 ) and H 0 (G 2 ) and θ is a termwise homotopy equivalence if and only if θ ′ is an isomorphism. If A and B are pretriangulated dg categories, the functors H 0 (G 1 ) and H 0 (G 2 ) are exact and the check that θ is a termwise homotopy equivalence is just a question involving standard techniques in the theory of triangulated categories.
More precisely, there is indeed a general principle that will be applied later on. Namely, assume that α is a natural transformation between two exact continuous functors F 1 , F 2 : D → D ′ , where D and D ′ are triangulated categories with arbitrary direct sums. Let D 1 be a full triangulated subcategory of D consisting of compact generators of D. Suppose further that, for all D ∈ D 1 , we have that α(D) is an isomorphism. Then one observes that the full subcategory D 2 of D consisting of all objects D such that α(D) is an isomorphism is obviously triangulated and contains D 1 . On the other hand, it is easy to see that α is automatically compatible with arbitrary direct sums, since F 1 and F 2 are continuous. Hence D 2 is closed under arbitrary direct sums and so D 2 = D (see [14]), which proves that α is an isomorphism.
Tensor product of dg modules. Following [6], if A is a dg category, the tensor product of M ∈ dgMod(A) and N ∈ dgMod(A • ) is defined as In this version, M ⊗ A N is a complex of k-modules, hence an object of C dg (k) ∼ = dgMod(k). Clearly, one can repeat the same definition taking M ∈ dgMod(A ⊗ B) and N ∈ dgMod(B • ⊗ C).
In this case, M ⊗ B N is an object in dgMod(A ⊗ C). (ii) Given M ∈ dgMod(A) and N ∈ dgMod(B), we can think of them as objects in dgMod(A⊗k) and N ∈ dgMod(k • ⊗ B). Now take the object M ⊗ k N ∈ dgMod(A ⊗ B). We claim that, for where the tensor product on the right-hand side is the usual tensor product of complexes of k-modules. Indeed, as k consists of only one object and every morphism is a scalar multiple of the identity, the map Ξ in (2.3) is trivial in this case, and thus we get the result from (2.2). In the rest of the paper, we write M ⊗ N for M ⊗ k N .
Remark 2.6. The tensor product of dg modules over a dg category looks very much like the ordinary tensor product of modules over a (not necessarily commutative) ring (see, for example, [12,Sect. VI.7] for the latter case). In particular, they share many properties, such as the associativity. Of course they can be proved directly using the definition. The easy computations are left to the reader, who, on the other hand, can have a look at [13,Sect. 6] for the proof of associativity in the case of tensor product of modules over a Z-linear category.
For a dg category A, an A-dg module M is h-flat if, for any N ∈ dgMod(A • ) which is acyclic, the tensor product M ⊗ A N is acyclic. One can check that any h-projective dg module is h-flat and that a dg module which is homotopy equivalent to an h-flat dg module is h-flat itself (see [9,Sect. 3.5], for some more details).
Derived tensor product of dg categories. We say that a dg category projective as well and all dg categories are h-projective if k is a field. It is also easy to see that if A and B are h-projective, then A ⊗ B is h-projective, too. We denote by hp-dgCat the full subcategory of dgCat consisting of h-projective dg categories.
Remark 2.7. Using [6, Sect. 13.5] (or the fact that dgCat is a model category [16]), for any dg category A, one can construct an explicit h-projective dg category A hp with a quasi-equivalence Q A : A hp → A, which will be fixed once and for all. In particular, if A is h-projective we assume that Q A = id A . If not, following [6], A hp can be constructed as a semi-free resolution of A, namely a semi-free dg category B with a quasi-equivalence B → A. Although it is not needed in the rest of the paper, let us briefly recall that, following [6, Sect. 13.4], a dg category A is semi-free if it can be represented as the union of an increasing sequence of dg subcategories A i , where i ∈ N, such that A 0 is a discrete dg category and, for i > 0, each A i is freely generated over A i−1 , as a graded category, by a family of homogeneous morphisms f j whose differentials d(f j ) are morphisms in A i−1 . It is then a simple calculation to see that A hp (A, B) is h-projective for all A, B ∈ A hp (one can also combine [6, Lemma 13.6], [18,Prop. 2.3 (3)] and the fact that a cofibrant complex of k-modules is h-projective).
Denoting by hp-Hqe the localization of hp-dgCat by all quasi-equivalences, it is then easy to verify that the natural functor hp-Hqe → Hqe is an equivalence (one can use [6,Lemma 13.5] for the faithfulness).
Hence, given two dg categories A and B, we define the derived tensor product as Remark 2.8. Given an h-projective dg category A and a quasi-equivalence F : B → B ′ , the induced dg functor G := id A ⊗ F : A ⊗ B → A ⊗ B ′ is again a quasi-equivalence. Indeed, the complex A(A 1 , A 2 ) is h-flat (being h-projective), for all A 1 , A 2 ∈ A, and thus the maps (G((A 1 , B 1 )), G((A 2 , B 2 ))) are quasi-isomorphisms, for all A 1 , A 2 ∈ A and all B 1 , B 2 ∈ B, since the maps B(B 1 , B 2 ) → B ′ (F(B 1 ), F(B 2 )) are quasi-isomorphisms. Hence it remains to show that H 0 (id A ⊗ F) is essentially surjective. This is clear from the definition of tensor product of dg categories, being H 0 (F) essentially surjective.
It follows from the universal property of the localization of dgCat by quasi-equivalences that the functor A ⊗ − : dgCat → dgCat naturally induces a functor A ⊗ − : Hqe → Hqe.
Similarly, given a dg category B and a quasi-equivalence G : Putting Remarks 2.7 and 2.8 together, we get a well-defined functor − ⊗ L − : Hqe × Hqe −→ Hqe which endows Hqe with a symmetric monoidal structure.
2.2. Some properties of morphisms in Hqe. The content of this section is probably well known to experts. Moreover, most of the properties of dgCat and Hqe mentioned here have trivial proofs when we regard dgCat as a model category (see [16]). Nonetheless, to achieve a proof of Theorem 1.1 much less is needed and we sketch in this section the minimal amount of information which is required. Trying to keep the paper as much self-contained as possible, we outline the proofs of the results of this section.
Recall from [3, Sect. 1] the notion of category of fibrant objects. Let C be a category with finite products and assume that C has two distinguished classes of morphisms, called weak equivalences and fibrations. A morphism which is both a weak equivalence and a fibration will be called a trivial fibration. A path object for C ∈ C is an object C I of C together with a weak equivalence C → C I and a fibration C I → C × C whose composition is the diagonal C → C × C. We say that C, together with its weak equivalences and fibrations, is a category of fibrant objects if the following axioms are satisfied.
(A) Let f and g be morphisms of C such that g • f is defined. If two of the morphisms f , g and g • f are weak equivalences, then so is the third. Any isomorphism is a weak equivalence. Remark 2.9. If g : B → C is a weak equivalence and f 0 , f 1 : C → D are morphisms such that f 0 • g and f 1 • g are homotopic, it follows from [3, Prop. 1] that for any path object D I for D there exists a weak equivalence g ′ : B ′ → C such that f 0 • g ′ and f 1 • g ′ are homotopic by a homotopy B ′ → D I . Now we want to prove that dgCat is a category of fibrant objects, if one takes as weak equivalences the quasi-equivalences and as fibrations the full dg functors whose H 0 is an isofibration (a functor F : C → D is an isofibration if for every C ∈ C and every isomorphism f : To this purpose, we need to fix some notation. As in [6, Sect. 2.9], for every dg category A we denote by Mor(A) the dg category whose objects are triples (A, B, f ) with f ∈ Z 0 (A(A, B)), and whose morphisms are given by . Actually we will be interested in the full dg subcategory P (A) of Mor(A) with objects the triples (A, B, f ) such that f is a homotopy equivalence (see [17,Sect. 3

]).
Notice that there is a natural dg functor I A : A → P (A), defined on objects by A → (A, A, id A ) and on morphisms by f → (f, f, 0). Similarly, there are obvious dg functors S A , T A : P (A) → A ("source" and "target") defined both on objects and morphisms as the projection respectively on the first and on the second component. Proof. The result depends on some elementary (but tedious) checks which are left to the reader. We simply outline the main ingredients in the proof.
First of all observe that finite products exist in dgCat, and they are given by the corresponding products both on objects and on morphisms (with differentials and compositions defined componentwise). In particular, a terminal object is the dg category with one object and 0 as the space of morphisms.
Axioms (A), (B) and (E) are straightforward to check from the definitions. As for axiom (C), note that for every dg functors F : A → C and G : B → C the fibre product D := A × C B along F and G exists in dgCat, and it is given by the full subcategory of A × B with objects those (A, B) ∈ A × B such that F(A) = G(B) and morphisms those morphisms (f, g) of A × B such that F(f ) = G(g). It is easy to show that, if G is a fibration (resp. a trivial fibration), then the projection dg functor D → A is a fibration (resp. a trivial fibration), too.
Passing to axiom (D), one shows that the dg functors A (2) Given two dg functors F : A → C and G : B → C with G a quasi-equivalence, there exist dg functors F ′ : D → B and G ′ : (1)). If moreover F is a quasi-equivalence, then F ′ is a quasi-equivalence, too.  (2) is then an easy consequence of axiom (A), using the fact that a morphism homotopic to a weak equivalence is also a weak equivalence (to see this, one uses again axiom (A)). For n = 1, (3) follows from part (i) of [3, Thm. 1]. In general, one can easily reduce by induction to n = 2. Then, by the case n = 1, for i = 1, 2 there exist dg functors On the other hand, by part (2), there exist two quasi-equivalences J i :

Extensions of morphisms in Hqe and bimodules
In this section we develop the key ingredients in our proof of Theorem 1.1. As it turns out, they rely on some natural properties of extension of dg functors. Finally, we provide an interpretation of the morphisms in Hqe in terms of dg modules over tensor dg categories. A and B, by (2.1) there is an isomorphism of dg categories dgMod(A • ⊗ B) ∼ = Hom(A, dgMod(B)), so in particular an object E ∈ dgMod(A • ⊗ B) corresponds to a dg functor Φ E : A → dgMod(B). Conversely, for every dg functor F : Let F : A → dgMod(B) be a dg functor corresponding to E ∈ dgMod(A • ⊗ B). Following [7, Sect. 6.1], we define the extension of F to be the dg functor

Extensions of dg functors. Given two dg categories
Notice that the definition of F is a reformulation of the usual notion of Kan extension in the context of dg functors. There is also a natural dg functor for every M ∈ dgMod(B).
Remark 3.1. Notice that, in [7], the dg functors F and F above are denoted by T E and H E respectively.  (1) F is left adjoint to F (hence Ind G is left adjoint to Res G ).
h-proj(B)). Proof. All the statements above are probably well known (see [7]). Thus we simply sketch the main ingredients in the proofs. The proof of (1) uses exactly the same argument as in [6,Sect. 14.9] for the adjunction between Ind G and Res G . The first part of (2) follows from Eq. (14.2) in [6], and H 0 ( F) is continuous because it is left adjoint to H 0 ( F) by (1).
The non-trivial implication in (3) is a consequence of (2) and of F(A) ⊆ h-proj(B). Indeed, we use here that the objects of A form a set of compact generators for h-proj(A) (see [7,Sect. 4.2]) and that H 0 (F) is continuous by (2), arguing exactly as at the end of Remark 2.4.
A similar argument applies to the first part of (4). For the fact that H 0 (Res G ) is continuous, we use that Res G has a right adjoint (see, for example, [10, Sect. 1]). Finally, (5) is observed in [6,Remark 4.3].
is a quasi-isomorphism in h-proj(B), and hence a homotopy equivalence, for every A ∈ A. In other words, Φ α is a termwise homotopy equivalence.  Proof. The first part of the statement is a simple consequence of the fact that, by definition, the tensor product of dg modules is functorial (see Remark 2.5 (i)). For the second part, observe that, by definition, Φ E (A) ⊆ h-proj(B), when E ∈ dgMod hp (A • ⊗ B). Hence we apply part (3)   Moreover, if F ′ : A → B and G ′ : B → C are two other dg functors, then there are also dg isomor- by the associativity of the tensor product. Using part (2) of Proposition 3.2, it follows that which proves the first part. The last statement then follows taking F = Y B • F ′ or G = Y C • G ′ and using again part (2) of Proposition 3.2.
Proof. The commutativity of the diagram follows directly from the fact that Φ E i (A) = E i ((A, −)) and Remark 2.5 (ii). The second part amounts to showing that, for all M ∈ dgMod( This is an easy exercise using the definition (2.2). (1) The dg functor F induces a natural map Iso(H 0 (h-proj(A ′ • ⊗B) rqr )) → Iso(H 0 (h-proj(A • ⊗ B) rqr )); if moreover F is a quasi-equivalence, then this map is bijective and which is a quasi-equivalence if G is such.
Proof. As for (1), notice that, setting ). On the other hand, one can also define a natural map Iso It is not difficult to show that, if F (hence Ind F 1 , by Remark 2.8 and part (5) of Proposition 3.2) is a quasi-equivalence, then these two maps are bijective and inverse to each other (see, for example, [6, Sect. 14.12]). Therefore it is enough to prove that Clearly, if F is a quasi-equivalence, the same argument shows that As for (2), we just need to show that, setting rqr (because then the second part of the statement can be proved with an argument which is completely similar to the one used in (1)). Given E ∈ h-proj(A • ⊗ B) and setting where the last isomorphism is due to [6,Sect. 14.6]. It follows from the associativity of the tensor product that (2) Proof. As [F 1 ] = [F 2 ], by part (1) of Proposition 2.11 there exists a quasi-equivalence I : C → A such that G i := F i • I for i = 1, 2 sit in the commutative diagram for some dg functor H : C → P (h-proj(B)). Thus, by Lemma 3.6, we have where the first and the last isomorphisms are again due to Lemma 3.6. On the other hand, given three dg categories A, B and C with a fully faithful dg functor J : B → C, we have another natural map of sets Proposition 3.11. The natural map of sets Σ A,J is injective.
Proof. Set C ′ to be the full dg subcategory of C consisting of all objects in the essential image of H 0 (J) and denote by J 1 : B → C ′ the natural quasi-equivalence. Let J 2 : C ′ → C be the natural inclusion inducing a natural dg functor J 3 : P (C ′ ) → P (C) such that J 3 ((C 1 , C 2 , f )) = (J 2 (C 1 ), J 2 (C 2 ), J 2 (f )) for every (C 1 , C 2 , f ) ∈ P (C ′ ). It is easy to verify that J 3 is fully faithful since J 2 is.
Given It is easy to see that H factors through J 3 . This means that there exists a dg functor H ′ :  is an h-projective resolution of ∆ A . It is easy to see, using Lemma 3.7, that , and so (by Lemma 3.9) Denoting as before by D an h-projective resolution of ∆ A , again by Lemma 3.7 we have As D → ∆ A is a quasi-isomorphism between dg modules which are h-flat over A, also the induced Corollary 3.13. If A and B are dg categories, the map of Proposition 3.10 is bijective.
Proof. It remains to prove that the map is injective. To this end, we can clearly assume that B is h-projective. Then it is very easy to see that, for every dg category C, the map Λ C,B can be identified with the map By Proposition 3.12 this implies that every morphism C → h-proj(B) in Hqe can be represented by a dg functor. In particular, this is true when C = h-proj(A). Therefore, given Let A, B and C be dg categories. In view of the basic properties of the derived tensor product, of Remark 2.8 and of Proposition 3.8, we can assume in the proof of Theorem 1.1, without loss of generality, that these three dg-categories are h-projective. In this way the tensor product does not need to be derived. Putting the results in the previous section together, we get the following maps:  1). We can prove the following, which is Theorem 7.2 in [18] and is usually referred to as derived Morita theory. For the second part, we argue as at the beginning of the proof of Theorem 7.2 of [18]. So we have to show that, for any dg category C, the Yoneda embedding Y A induces a bijection