Volume 21, issue 5 (2017)

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

David Carchedi, Sarah Scherotzke, Nicolò Sibilla and Mattia Talpo

Geometry & Topology 21 (2017) 3093–3158
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 étale 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 étale homotopy type of its infinite root stack.

Keywords
log scheme, Kato–Nakayama space, root stack, profinite spaces, infinity category, étale homotopy type, topological stack
Mathematical Subject Classification 2010
Primary: 14F35, 55U35
Secondary: 55P60
Publication
Received: 25 April 2016
Revised: 28 September 2016
Accepted: 11 November 2016
Published: 15 August 2017
Proposed: Dan Abramovich
Seconded: Paul Goerss, Frances Kirwan
Authors
 David Carchedi Department of Mathematical Sciences George Mason University Fairfax, VA 22030 United States Sarah Scherotzke Mathematisches Institut University of Bonn Endenicher Allee 60 D-53115 Bonn Germany Nicolò Sibilla School of Mathematics Statistics and Actuarial Sciences University of Kent Canterbury, CT2 7FS United Kingdom Mattia Talpo Department of Mathematics Simon Fraser University Burnaby, BC V5A 1S6 Canada