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Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes

David, Carchedi, Sarah, Scherotzke, Nicolò, Sibilla, Mattia, Talpo (2017) Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes. Geometry & Topology, 21 (5). pp. 3093-3158. ISSN 1364-0380. (doi:10.2140/gt.2017.21.3093) (KAR id:61900)

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http://dx.doi.org/10.2140/gt.2017.21.3093

Abstract

For a log scheme locally of finite type over C, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over C, another natural candidate is the profinite \'etale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over C, these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite \'etale homotopy type of its infinite root stack.

Item Type: Article
DOI/Identification number: 10.2140/gt.2017.21.3093
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Nicolo Sibilla
Date Deposited: 31 May 2017 13:44 UTC
Last Modified: 06 May 2020 03:16 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/61900 (The current URI for this page, for reference purposes)
Nicolò, Sibilla: https://orcid.org/0000-0003-4438-7369
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