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* linear $A_\infty$-categories as special $\infty$-categories?

## References

[1] Simplicial nerve of an A-infinity category (Giovanni Faonte, arXiv:1312.2127), suggested by DamienC in an answer to MO152370.