From homotopy operads to infinity-operads

TL;DR

This paper constructs a functor linking strict unital homotopy colored operads to infinity-operads, clarifying their relationship and extending known nerve constructions, with key homotopy properties established.

Contribution

It introduces a functor from strict unital homotopy colored operads to infinity-operads, generalizing and connecting existing nerve constructions in operad theory.

Findings

The functor extends the simplicial nerve for A-infinity-categories.

It is shown to be a right Quillen functor.

The functor is equivalent to a big nerve for differential graded operads.

Abstract

The goal of the present paper is to compare, in a precise way, two notions 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 \`a 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.