Simplicial nerve of an A-infinity category

Abstract

In this paper we introduce a functor, called the simplicial nerve of an A-infinity category, defined on the category of (small) A-infinity categories with values in simplicial sets. We prove that the simplicial nerve of any A-infinity category is an infinity category. This construction extends functorially the nerve construction for differential graded categories proposed by J.Lurie in Higher Algebra. We prove that if a differential graded category is pretriangulated in the sense of A.I.Bondal-M.Kapranov, then its nerve is a stable infinity category in the sense of J.Lurie.