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

Abstract

For a log scheme locally of finite type over C, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space [26]. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack [48]. Finally, for a log scheme not necessarily over C, another natural candidate is the profiniteétale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over 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 profiniteétale homotopy type of its infinite root stack.2010 Mathematics Subject Classification. MSC Primary: 14F35, 55P60; Secondary: 55U35. Key words and phrases. log scheme, Kato-Nakayama space, root stack, profinite spaces, infinity category, etale homotopy type, topological stack. 1 1.2. The comparison map and the equivalence of profinite completions. Our main result states:Theorem (see Theorem 6.4). Let X be a fine saturated log scheme locally of finite type over C. Then there is a canonical map of pro-topological stacksthat induces an equivalence upon profinite completion

The Kato–Nakayama Space as a Transcendental Root Stack

“…Let us show how the functorial interpretation of Theorem 3.2 gives a globally defined morphism of topological stacks X log → n √ X top for every n, and these assemble into a morphism of pro-topological stacks X log → ∞ √ X top . We will check later (see Proposition 4.6) that this morphism coincides with the one constructed in [CSST,Proposition 4.1].…”

“…Building on this idea, in the paper [CSST] by Carchedi, Scherotzke, Sibilla and the first author it is shown that there is a canonical morphism Φ X : X log → ∞ √ X top from the Kato-Nakayama space to the topological realization of the infinite root stack of X, that is moreover a "profinite equivalence". In that paper the morphism is constructed locally on X, in presence of a Kato chart for the log structure, and then globalized by gluing [CSST,Section 4].…”

“…As a consequence, we obtain, in Section 3.4, a global description of the canonical morphism Φ X : X log → ∞ √ X top constructed in [CSST,Section 4]: for every n there is a factorization…”

Parabolic sheaves with real weights as sheaves on the Kato–Nakayama space

“…Therefore for every i we have j a ij b j = 0 in B. Now using the description of B given by (6), let us write…”

“…The use of the Kato-Nakayama space X log is heuristically justified by the fact that the infinite root stack is a sort of "profinite algebraic incarnation" of the former: there is a morphism X log → ∞ √ X top to the topological realization of ∞ √ X, which is a "profinite equivalence" [6,Theorem 6.4]. The fiber of X log → X over a point x can be identified with a real torus (S 1 ) r , and the fiber of ∞ √ X top → X with B Z r , where r is the "rank of the log structure" at x.…”