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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

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.

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
Received: 25 April 2016
Revised: 28 September 2016
Accepted: 11 November 2016
Published: 15 August 2017
Proposed: Dan Abramovich
Seconded: Paul Goerss, Frances Kirwan
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
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