## Geometry & Topology

### 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. 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 $ℂ$, 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 $ℂ$, 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.

#### Article information

Source
Geom. Topol., Volume 21, Number 5 (2017), 3093-3158.

Dates
Revised: 28 September 2016
Accepted: 11 November 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859284

Digital Object Identifier
doi:10.2140/gt.2017.21.3093

Mathematical Reviews number (MathSciNet)
MR3687115

Zentralblatt MATH identifier
06774941

#### Citation

Carchedi, David; Scherotzke, Sarah; Sibilla, Nicolò; Talpo, Mattia. Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes. Geom. Topol. 21 (2017), no. 5, 3093--3158. doi:10.2140/gt.2017.21.3093. https://projecteuclid.org/euclid.gt/1510859284

#### References

• D Abramovich, Q Chen, D Gillam, Y Huang, M Olsson, M Satriano, S Sun, Logarithmic geometry and moduli, from “Handbook of moduli, I” (G Farkas, I Morrison, editors), Adv. Lect. Math. 24, International Press, Somerville, MA (2013) 1–61
• M Artin, Algebraization of formal moduli, II: Existence of modifications, Ann. of Math. 91 (1970) 88–135
• M Artin, A Grothendieck, J L Verdier, Théorie des topos et cohomologie étale des schémas, Tome 1: Théorie des topos, Exposés I–IV (SGA $4\ssty\sb{\,\mathrm{1}}\kern-.1em$), Lecture Notes in Math. 269, Springer (1972)
• M Artin, A Grothendieck, J L Verdier, Théorie des topos et cohomologie étale des schémas, Tome 3: Exposés IX–XIX (SGA $4\ssty\sb{\,\mathrm{3}}\kern-.1em$), Lecture Notes in Math. 305, Springer (1973)
• M Artin, B Mazur, Etale homotopy, Lecture Notes in Math. 100, Springer (1969)
• I Barnea, Y Harpaz, G Horel, Pro-categories in homotopy theory, Algebr. Geom. Topol. 17 (2017) 567–643
• P Belmans, A J de Jong, et al., The Stacks project, electronic reference, Columbia University (2005–) \setbox0\makeatletter\@url http://stacks.math.columbia.edu {\unhbox0
• J Bochnak, M Coste, M-F Roy, Real algebraic geometry, Ergeb. Math. Grenzgeb. 36, Springer (1998)
• N Borne, A Vistoli, Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012) 1327–1363
• D Carchedi, Higher orbifolds and Deligne–Mumford stacks as structured infinity topoi, preprint (2013) To appear in Mem. Amer. Math. Soc.
• D Carchedi, On the étale homotopy type of higher stacks, preprint (2015)
• D Carchedi, On the homotopy type of higher orbifolds and Haefliger classifying spaces, Adv. Math. 294 (2016) 756–818
• T Coyne, B Noohi, Singular chains on topological stacks, I, Adv. Math. 303 (2016) 1190–1235
• D Dugger, D C Isaksen, Topological hypercovers and $\mathbb A^1$–realizations, Math. Z. 246 (2004) 667–689
• E M Friedlander, Fibrations in etale homotopy theory, Inst. Hautes Études Sci. Publ. Math. 42 (1973) 5–46
• E M Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies 104, Princeton Univ. Press (1982)
• M Groth, A short course on $\infty$–categories, preprint (2010)
• K Hagihara, Structure theorem of Kummer étale $K$–group, $K$–Theory 29 (2003) 75–99
• A Henriques, D Gepner, Homotopy theory of orbispaces, preprint (2007)
• K R Hofmann, Triangulation of locally semi-algebraic spaces, PhD thesis, University of Michigan (2009) \setbox0\makeatletter\@url http://search.proquest.com/docview/304931094 {\unhbox0
• M Hoyois, Higher Galois theory, preprint (2015)
• L Illusie, K Kato, C Nakayama, Quasi-unipotent logarithmic Riemann–Hilbert correspondences, J. Math. Sci. Univ. Tokyo 12 (2005) 1–66
• L Illusie, C Nakayama, T Tsuji, On log flat descent, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013) 1–5
• D C Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805–2841
• J F Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987) 35–87
• K Kato, Logarithmic structures of Fontaine–Illusie, from “Algebraic analysis, geometry, and number theory” (J-I Igusa, editor), Johns Hopkins Univ. Press, Baltimore, MD (1989) 191–224
• K Kato, C Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over ${\bf C}$, Kodai Math. J. 22 (1999) 161–186
• A Kock, I Moerdijk, Presentations of étendues, Cahiers Topologie Géom. Différentielle Catég. 32 (1991) 145–164
• S Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa 18 (1964) 449–474
• J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
• J Lurie, Derived algebraic geometry, XIII: Rational and $p$–adic homotopy theory, preprint (2011) \setbox0\makeatletter\@url http://tinyurl.com/lurie-dagXIII {\unhbox0
• A Mathew, V Stojanoska, Fibers of partial totalizations of a pointed cosimplicial space, Proc. Amer. Math. Soc. 144 (2016) 445–458
• J P McCammond, A general small cancellation theory, Internat. J. Algebra Comput. 10 (2000) 1–172
• W Nizio\l, $K$–theory of log-schemes, I, Doc. Math. 13 (2008) 505–551
• B Noohi, Foundations of topological stacks, I, preprint (2005)
• B Noohi, Homotopy types of topological stacks, Adv. Math. 230 (2012) 2014–2047
• A Ogus, Lectures on logarithmic algebraic geometry, lecture notes (2006) \setbox0\makeatletter\@url http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf {\unhbox0
• M C Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. 36 (2003) 747–791
• G Quick, Profinite homotopy theory, Doc. Math. 13 (2008) 585–612
• G Quick, Some remarks on profinite completion of spaces, from “Galois–Teichmüller theory and arithmetic geometry” (H Nakamura, F Pop, L Schneps, A Tamagawa, editors), Adv. Stud. Pure Math. 63, Math. Soc. Japan, Tokyo (2012) 413–448
• D G Quillen, Some remarks on etale homotopy theory and a conjecture of Adams, Topology 7 (1968) 111–116
• H Ruddat, N Sibilla, D Treumann, E Zaslow, Skeleta of affine hypersurfaces, Geom. Topol. 18 (2014) 1343–1395
• U Schreiber, Differential cohomology in a cohesive infinity-topos, preprint (2013)
• J-P Serre, Cohomologie galoisienne, Lecture Notes in Math. 5, Springer (1965)
• M Shiota, Whitney triangulations of semialgebraic sets, Ann. Polon. Math. 87 (2005) 237–246
• D Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. 100 (1974) 1–79
• M Talpo, A Vistoli, Infinite root stacks and quasi-coherent sheaves on logarithmic schemes, preprint (2014)
• B Toën, M Vaquié, Algébrisation des variétés analytiques complexes et catégories dérivées, Math. Ann. 342 (2008) 789–831