From homotopy operads to infinity-operads

Abstract

The goal of the present paper is to compare, in a precise way, two approaches of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk–Weiss in order to generalize the simplicial notion of infinity-category of Joyal–Lurie. This functor extends in two directions the simplicial nerve of Faonte–Lurie for A-infinity-categories and the homotopy coherent nerve of Moerdijk–Weiss for differential graded operads; it is also shown to be equivalent to a big nerve à la Lurie for differential graded operads. We prove that it satisfies some homotopy properties with respect to weak equivalences and fibrations; for instance, it is shown to be a right Quillen functor.