Geometry & Topology

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

David Carchedi, Sarah Scherotzke, Nicolò Sibilla, and Mattia Talpo

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For a log scheme locally of finite type over , 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 , another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over , 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 étale homotopy type of its infinite root stack.

Article information

Geom. Topol., Volume 21, Number 5 (2017), 3093-3158.

Received: 25 April 2016
Revised: 28 September 2016
Accepted: 11 November 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 55U35: Abstract and axiomatic homotopy theory
Secondary: 55P60: Localization and completion

log scheme Kato–Nakayama space root stack profinite spaces infinity category étale homotopy type topological stack


Carchedi, David; Scherotzke, Sarah; Sibilla, Nicolò; Talpo, Mattia. Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes. Geom. Topol. 21 (2017), no. 5, 3093--3158. doi:10.2140/gt.2017.21.3093.

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