There are two definitions of $A_\infty$-category in common use: linear $A_\infty$-categories and general $A_\infty$-categories.

Regarding the linear definition, a functor $N\colon\mathrm{Cat}_{A_\infty}\longrightarrow\mathrm{Cat}_\infty$ is constructed in a paper \[1\]  by Faonte. This gives a way to _get_ an $\infty$-category by starting with an $A_\infty$-category.

Going the other way, **is it possible to _define_ a linear $A_\infty$-category as a special kind of an $\infty$-category?**

**What about general $A_\infty$-categories (i.e. categories over the $A_\infty$-operad)?**

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\[1\] Simplicial nerve of an A-infinity category (Giovanni Faonte, [arXiv:1312.2127)][2], suggested by DamienC in an answer to [MO152370][1]. 
  [1]: https://mathoverflow.net/questions/152370/are-infty-1-categories-a-infty-categories
  [2]: https://arxiv.org/abs/1312.2127