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dg: is for differential graded

Suppose that $F: C\rightarrow D$ is a dg-functor between small dg-categories such that:

  1. F: Objects of $C$ $\rightarrow$ Objects of $D$ is injective.

  2. $Hom_{C}(a,b)\rightarrow Hom_{D}(F(a),F(b))$ induces an isomorphism in homology for any $a, b \in C$

Let $\hat{C}$ be the category of dg-$C$-modules. Is it true that the induced dg-functor $$\hat{F}:\hat{C}\rightarrow \hat{D}$$ is faithfull in the sense that

for any $x,y\in \hat{C} $, $Hom_{\hat{C}}(a,b)\rightarrow Hom_{\hat{D}}(\hat{F}(a),\hat{F}(b))$ induces an isomorphism in homology.

Edit: I'm also interested in the particular case when $F:C\rightarrow D$ is an embedding of dg-categories e.g. when $C$ is a full dg-subcategory of $D$.

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  • $\begingroup$ The example given in this question may be useful: mathoverflow.net/questions/244699/… (it is for sets instead of chain complexes, but can probably be adapted). $\endgroup$ Jun 21 '17 at 10:48
  • $\begingroup$ Which induced dg functor? There are two, and the most natural one goes in the other direction. $\endgroup$ Jun 25 '17 at 6:17
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Let me start with an answer for your last claim, namely, that a fully faithful dg-functor $F \colon \mathcal C \to \mathcal D$ induces a fully faithful dg-functor $\hat{F} \colon \hat{\mathcal C} \to \hat{\mathcal D}$ between the dg-categories of right dg-modules. This is true, and the reason lies in general facts about Kan extensions. In any case, a sketch of proof would be as follows.

First, $\hat{F}$ is usually defined as the "induction" dg-functor $\operatorname{Ind}_F$, and in particular $$ \operatorname{Ind}_F(M)(D) = M \otimes_{\mathcal C} \mathcal D(D,F(-)) = \int^C M(C) \otimes\mathcal D(D,F(C)), $$ if you also like coends. To show that $\operatorname{Ind}_F$ is fully faithful, we first recall that it is by definition the left adjoint of the restriction dg-functor $\operatorname{Res}_F \colon \hat{\mathcal D} \to \hat{\mathcal C}$, which maps $N$ to $N \circ F$. There is a unit morphism $$ M \to \operatorname{Res}_F \operatorname{Ind}_F(M), $$ and if we check that this is an isomorphism, then $\operatorname{Ind}_F$ will be fully faithful. But now we can compute: \begin{align*} \operatorname{Res}_F \operatorname{Ind}_F(M)(C) &= M \otimes_{\mathcal C} \mathcal D(F(C),F(-)) \\ &\cong M \otimes_{\mathcal C} \mathcal C(C,-)\\ & \cong M(C), \end{align*} where I used fully faithfulness and for the last step the "co-Yoneda lemma" or "density theorem".

Now, your main claim that a quasi-fully faithful dg-functor induces something which is again quasi-fully faithful between the dg-category of dg-modules is a little more nuanced. Working with this "cohomological" assumptions, you'll want derived stuff all around. So, instead of working with dg-modules, it's best to work with h-projective/cofibrant/semi-free dg-modules, namely your favourite version of the "derived dg-category". Let us work with the dg-category of h-projective dg-modules, just to fix ideas. What is true is that if you have a quasi-equivalence $G \colon \mathcal A \to \mathcal B$ between dg-categories, then $\operatorname{Ind}_G$ will induce a quasi-equivalence $$ \operatorname{Ind}_G \colon \operatorname{h-proj}(\mathcal A) \to \operatorname{h-proj}(\mathcal B). $$ This is quite a well-known fact: if you want to check it yourself, you can work instead with semi-free dg-modules and prove that the unit and counit maps are quasi-isomorphism "by hand", or you can use the fact that you have compact generators: a reference is Drinfeld's famous article (Remark 4.3).

Now, that was the hard part. If you have just a quasi-fully faithful dg-functor $F \colon \mathcal C \to \mathcal D$ as in your post (you don't even need injectivity on objects, just that you have quasi-isomorphisms on hom-complexes), then clearly you can factor it as a quasi-equivalence and a (strictly) fully faithful dg-functor. So, in the end, I think you'll end up with a quasi-fully faithful dg-functor $\operatorname{h-proj}(\mathcal C) \to \operatorname{h-proj}(\mathcal D)$.

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