A comparison between pretriangulated $\mbox{A}_{\infty}$-categories and $\infty$-Stable categories

Abstract: In this paper we will prove that the $\mbox{A}{\infty}$-nerve of two quasi-equivalent $\mbox{A}{\infty}$-categories are weak-equivalent in the Joyal model structure, a consequence of this fact is that the $\mbox{A}{\infty}$-nerve of a pretriangulated $\mbox{A}{\infty}$-category is $\infty$-stable. Moreover we give a comparison between the notions of pretriangulated $\mbox{A}_{\infty}$-categories, pretriangulated dg-categories and $\infty$-stable categories.