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For a log scheme locally of finite type over $\mathbb{C}$, a naturalcandidate for its profinite homotopy type is the profinite completion of itsKato-Nakayama space. Alternatively, one may consider the profinite homotopytype of the underlying topological stack of its infinite root stack. Finally,for a log scheme not necessarily over $\mathbb{C}$, another natural candidateis the profinite \'etale homotopy type of its infinite root stack. We provethat, for a fine saturated log scheme locally of finite type over $\mathbb{C}$,these three notions agree. In particular, we construct a comparison map fromthe Kato-Nakayama space to the underlying topological stack of the infiniteroot stack, and prove that it induces an equivalence on profinite completions.In light of these results, we define the profinite homotopy type of a generalfine saturated log scheme as the profinite \'etale homotopy type of itsinfinite root stack.

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