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In the framework of $\infty$-categories, I think it is not difficult to see that your claim for $\mathcal{A} = \Delta^1$ holds for all $\infty$-groupoids. Let $K$ be an $\infty$-groupoid (Kan complex …

See the recent paper Lee Cohn, Differential graded categories are k-linear stable infinity categories, arXiv:1308.2587 where a proof has been written down. The precise statement is that the under …

Toen proved that $RHom$ provides the internal hom in the homotopy category of dg-categories. For what you want to be true, you need something more than this: you need to know that $RHom$ is actually …

There are plenty of interesting dg-categories one can associate to a scheme. From the point of view of six functor yoga, these should be viewed as "categories of coefficients" for cohomology theories …

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 …

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) …

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 …