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Yes. This is well-known and can be deduced, for example, from the general statements in the last section of this paper of Barwick. Take $V$ to be the symmetric monoidal model category of simplicial …

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 …

Let $sSet$ be the category of simplicial sets with the Quillen model structure. Define a homotopy monomorphism in $sSet$ to be a morphism whose homotopy fibres are empty or weakly contractible. In a …

This answer is an elaboration on Dylan's comments. 1) Let us define a homotopy theory to be a pair $(C, W)$, where $C$ is a category and $W$ is some class of morphisms called weak equivalences. (Let' …

The model structure on the category of dg-categories presents an $(\infty,1)$-category DGCat. This structure is essentially provided by the existence of mapping spaces (or mapping $\infty$-groupoids) …

I am not sure if this will answer your question, but it may at least point you in the right direction (or at least some direction). Let me start with some classical background. Let $C$ be a category …

Let $\Delta^1_k$ be the $k$-linear dg-category with two objects $0$ and $1$, mapping complexes $$ Map(0,0) = [k], $$ $$ Map(0,1) = [k], $$ $$ Map(1,0) = [0], $$ $$ Map(1,1) = [k] $$ where $[k] …

In some sense, the "universal" version of this fact was proved by Blumberg-Gepner-Tabuada as Proposition 3.3 in this paper. That is, they proved the analogue for stable $\infty$-categories, which is t …