Javier J. Gutiérrez
Radboud Universiteit Nijmegen
On autoequivalences of the (∞, 1)-category of ∞-operads
Abstract. Higher operad theory can be formalized by means of different approaches, including simplicial operads, ∞-operads (Lurie), dendroidal sets (Moerdijk–Weiss) and complete dendroidal Segal spaces (Cisinski–Moerdijk). Each of these theories is organized in a Quillen model category and these are connected by Quillen equivalences.
By using techniques introduced by Toën, Lurie and Barwick and Schommer-Pries, we show that the (∞, 1)-category of autoequivalences of the (∞, 1)-category of ∞-operads is a contractible ∞-groupoid. More precisely, we prove that the quasi-category of autoequivalences of Ω-spaces is a contractible Kan complex. This implies that if there is a way to compare two models for ∞-operads, then this can be done in an essentially unique way. Similarly, we show that the (∞, 1)-category of autoequivalences of the (∞, 1)-category of non-symmetric ∞-operads is the discrete category on the cyclic group Z/2Z of order two, the non-trivial element being the ``mirror autoequivalence". Our calculations are based on the model of complete dendroidal Segal spaces introduced by Cisinski–Moerdijk.
This is a joint work with Dimitri Ara and Moritz Groth.