Profinite completions of topological operads
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908628434288640 |
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| author | Blom, Thomas Moerdijk, Ieke |
| author_facet | Blom, Thomas Moerdijk, Ieke |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_12567 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Profinite completions of topological operads Blom, Thomas Moerdijk, Ieke Algebraic Topology Category Theory 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. |
| title | Profinite completions of topological operads |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2312.12567 |