Volume 21, issue 5 (2017)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 22
Issue 5, 2511–3144
Issue 4, 1893–2510
Issue 3, 1267–1891
Issue 2, 645–1266
Issue 1, 1–644

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 , 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.

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
References
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