Simplicial nerve of an A∞-category

Abstract

We introduce a functor called the simplicial nerve of an A∞-category defined on the category of A∞-categories with values in simplicial sets. We show that the nerve of an A∞-category is an (∞, 1)-category in the sense of J. Lurie [Lur1]. This construction generalizes the nerve construction for di_erential graded categories given in [Lur2]. We prove that if a di_erential graded category is pretriangulated in the sense of A.I. Bondal and M. Kapranov [Bo-Ka] then its nerve is a stable (∞, 1)-category in the sense of J. Lurie [Lur2].