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

Abstract

For a log scheme locally of finite type over $\mathbb{C}$, 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 $\mathbb{C}$, 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 $\mathbb{C}$, 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.

Final version. 57 pages

References in corpus (7)

• Higher Topos Theory
• Foundations of Topological Stacks I
• Homotopy Theory of Orbispaces
• Logarithmic Geometry and Moduli
• Pro-categories in homotopy theory
• Fibers of partial totalizations of a pointed cosimplicial space
• Singular chains on topological stacks

Cited by in corpus (6)

• Infinite root stacks and quasi-coherent sheaves on logarithmic schemes
• Holonomic and perverse logarithmic D-modules
• On a logarithmic version of the derived McKay correspondence
• Betti realization of varieties defined by formal Laurent series
• Gluing semi-orthogonal decompositions
• On the profinite homotopy type of log schemes