# A question about the morphisms in the homotopy category of dg-Cat

Let $dg-Cat$ denote the category of (small) dg-categories and $Ho(dg-Cat)$ denote the localization of $dg-Cat$ at quasi-equivalence. Using the model structure on $dg-Cat$ we can describe the morphisms in $Ho(dg-Cat)$ as the homotopy classes of morphisms between cofibrant and fibrant objects.

In Toen's Lectures on dg-categories page 37 I found

(Let $T$ and $T^{\prime}$ be two dg-categories) Using the description of maps in $Ho(dg-Cat)$ as being homotopy classes of morphisms between cofibrant objects, we see that the morphism $\underline{h}$ induces an injective map $$[T,T^{\prime}]\hookrightarrow [T,Int((T^{\prime})^{op}-Mod)]$$ whose image consists of morphisms $T\to Int((T^{\prime})^{op}-Mod)$ factorizing in $Ho(dg-Cat)$ ) throught the quasi-essential image of $\underline{h}$.

Here $Int((T^{\prime})^{op}-Mod)$ denotes the full subcategory consisting of cofibrant and fibrant objects in the model category $(T^{\prime})^{op}-Mod$. Moreover $\underline{h}$ is the Yoneda embedding $\underline{h}: T^{\prime}\to Int((T^{\prime})^{op}-Mod)$.

I understand that $\underline{h}$ is quasi-fully faithful and the image of $\underline{h}$ consists of cofibrant and fibrant objects in $(T^{\prime})^{op}-Mod$. $\textbf{My question}$ is: how to prove that the induced map $$[T,T^{\prime}]\hookrightarrow [T,Int((T^{\prime})^{op}-Mod)]$$ is injective?

## 1 Answer

More generally one has the following statement: if $u : C \to D$ is a quasi-fully faithful functor of dg-categories, then the induced morphism of mapping spaces in the model category of dg-categories (with the Dwyer-Kan model structure) $$Map(E, C) \to Map(E, D)$$ induces an injection $\pi_0$ and an isomorphism on $\pi_{i > 0}$. (Note that the set of morphisms $[C,D]$ in $Ho(dg-cat)$ is the same as $\pi_0 Map(C, D)$.)

This follows directly from the characterization of quasi-fully faithful functors as the homotopy monomorphisms in the model category of dg-categories (see Lemma 2.4 in Derived Morita theory).