Dendroidal sets and simplicial operads

Abstract

Abstract. We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for ∞-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by A. Joyal and J. Lurie. We also explain how this theory applies to the usual notion of operad (i.e. with a single colour) in the category of spaces.

An Additivity Theorem for the interchange of En structures

“…Recent work of Cisinski and Moerdijk [11][12][13] and Heuts, Hinich and Moerdijk [17] relates the dendroidal sets world with that of Lurie. We hope that the methods presented in our paper will be of help in understanding the precise relation of their tensor products with the interchange of structures.…”

Dendroidal Segal spaces and ∞-operads

“…Moreover, they are all Quillen equivalent to the model category of simplicial categories discovered by Bergner [2], thus providing a strictification or rigidification result for each of these notions of category-up-to-homotopy. The goal of this paper and its sequel [8] is to develop analogous theories of Segal operads (rather than categories) and complete dendroidal (rather than simplicial) Segal spaces, to relate these to each other and to dendroidal sets via Quillen equivalences, and to prove a strictification result for each of them by relating them to simplicial operads. By a simple slicing procedure like in (I), the earlier results just mentioned for categories-up-to-homotopy can all be recovered from our results, which can in this sense be said to be more general.…”

“…We believe these results are of interest in themselves, and because they generalize important classical results from the simplicial-categorical context to the dendroidal-operadic one. In addition, they will all be used in our proof of the strictification theorem for ∞-operads presented in [8].…”

Relative left properness of colored operads

“…Our goal in this paper is to further the study of the Quillen model category structure of colored operads initiated in [Rob11,CM13,Cav14]. Specifically we are interested in understanding if the category of colored, symmetric operads is left proper; ie we wish to know if weak equivalences between all colored, symmetric operads are closed under cobase change along cofibrations.…”

“…Other examples of colored operads 1 encode complicated algebraic structures such as operadic modules, enriched categories, and even categories of operads themselves. The study of model category structures on categories of colored operads has found many recent applications including the rectification of diagrams of operads [BM07] and the construction of simplicial models for ∞-operads [CM13].…”