$k$-linear Morita theory

In this paper, we prove the standard comparison used by mathematicians between the idempotent complete pretriangulated dg-categories, over a unitary and commutative ring $k$, and the idempotent complete $k$-linear stable $\infty$-categories. Our approach is completely included in the $\infty$-categorical theory. To achieve the target we will reinterpret the Morita theory for dg-categories and we set the Morita theory for $k$-linear stable $\infty$-category.