Dendroidal sets and simplicial operads

Denis-Charles Cisinski (1) , Ieke Moerdijk (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IMAPP - Institute for Mathematics, Astrophysics and Particle Physics

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 infinity-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.

Domaines: Topologie algébrique [math.AT]; Catégories et ensembles [math.CT]

Denis-Charles Cisinski, Ieke Moerdijk. Dendroidal sets and simplicial operads. Journal of topology, 2013, 6 (3), pp.705-756. ⟨10.1112/jtopol/jtt006⟩. ⟨hal-00619248⟩