# “Sameness” of dg and A-infinity categories

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).

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.

• Any $A_\infty$-algebra is quasiequivalent to a dg-algebra through the cobar-bar construction. – Aaron Bergman Jan 24 at 1:38
• Ah, so perhaps the equivalence of categories I'm looking for sends an $A_{\infty}$ algebra $A$ to the dg algebra $\Omega B A$. I need to think a bit more, but I think this answers the question at the level of algebras. – user142700 Jan 24 at 5:45