Profinite completions of topological operads
Thomas Blom, Ieke Moerdijk
TL;DR
The work develops a robust homotopical framework for profinite completions of topological operads by constructing a model category of profinite up-to-homotopy operads and a left Quillen functor from dendroidal spaces to profinite dendroidal spaces. It provides a Dwyer–Kan-style characterization of weak equivalences, identifies the underlying ∞-category as a pro-category of π-finite ∞-operads, and proves a comparison with the naive levelwise profinite completion in a large class of operads. The results extend the earlier notion of profinite ∞-operads and tie the profinite completion to derived automorphism spaces relevant to the Grothendieck–Teichmüller program. The framework unifies profinite completion notions and confirms their coherence with Boavida–Horel–Robertson’s constructions in key cases, offering a versatile toolkit for studying profinite structures in ∞-operads and their homotopy theories.
Abstract
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 $\infty$-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 $\infty$-operads", we obtain analogous results in that setting.