We introduce a functor called the simplicial nerve of an $A_\infty$-category defined on the category of $A_\infty$-categories with values in simplicial sets. We show that the nerve of an $A_\infty$-category is an $(\infty,1)$-category in the sense of J. Lurie. This construction generalizes the nerve construction for differential graded categories given by Lurie. We prove that if a differential graded category is pretriangulated in the sense of A.I. Bondal and M. Kapranov then its nerve is a stable $(\infty,1)$-category in the sense of J. Lurie.
Keywords: $A_\infty$-categories, nerve, higher categories, pretriangulated dg-categories
2010 MSC: 18G30
Theory and Applications of Categories, Vol. 32, 2017, No. 2, pp 31-52.