Let $k$ be a field. 


A folklore theorem states that dg-categories (over $k$), $A\_{\infty}$-categories (over $k$) and stable ($k$-linear) $(\infty, 1)$-categories are "the same" (see for example
[Stable infinity categories vs dg-categories](https://mathoverflow.net/questions/114251/stable-infinity-categories-vs-dg-categories)).


I would like to understand in what sense (and why) dg- and $A\_{\infty}$-categories are "the same". 


There is a canonical inclusion functor from the category of dg categories to the category of $A\_{\infty}$ categories. However, this functor is not full, so it does not define an equivalence of categories.



"Sameness" of dg and A-infinity categories
By Corollary 9.2.1 in the paper <https://arxiv.org/abs/1410.5675>
the model category of small A\_∞-categories
is Quillen equivalent to the model category of small categories
(with a fixed set of objects for simplicity, but see
also Proposition 9.2.3 for the general case),
where both types of categories are enriched over a fixed monoidal tractable model category,
such as chain complexes over a commutative ring,
simplicial sets,
simplicial modules over a commutative ring,
symmetric spectra, etc.


The proof there also gives explicit formulas for the rectification operation.


