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

TL;DR

This paper establishes the equivalence of different notions of profinite homotopy types for fine saturated log schemes over complex numbers, connecting Kato-Nakayama spaces, infinite root stacks, and étale homotopy types.

Contribution

It proves the equivalence of three natural profinite homotopy type candidates for log schemes over complex numbers and defines the profinite homotopy type for general cases.

Findings

The three notions of profinite homotopy type agree for fine saturated log schemes over .

Constructs a comparison map inducing an equivalence on profinite completions.

Defines the profinite homotopy type for general log schemes via étale homotopy of infinite root stacks.

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 \'etale 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 \'etale homotopy type of its infinite root stack.