Profinite completions of topological operads

We show that the particular profinite completion used by Boavida-Horel-Robertson in their study of the Grothendieck-Teichmüller group fits in the framework of profinite completion as a left Quillen functor. More precisely, we construct a model category of profinite up-to-homotopy operads based on dendroidal objects in Quick’s model category of profinite spaces and show that the construction of Boavida-Horel-Robertson extends to a left Quillen functor into this model category. We also characterize the underlying $\mathrm{\infty }$-category of this model category and obtain a Dwyer-Kan style characterization of the weak equivalences between such profinite up-to-homotopy operads. Since this model category of profinite up-to-homotopy operads is Quillen equivalent to the one considered in our earlier paper “Profinite $\mathrm{\infty }$-operads”, we obtain analogous results in that setting.