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Keywords:
étale homotopy; higher categories; stacks; topos theory
étale homotopy; higher categories; stacks; topos theory
Summary:
A new approach to étale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the big étale site, and in particular to all algebraic stacks. This approach also produces a more refined invariant than the original construction of Artin-Mazur [2], namely we produce a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\Cal X$ is an arbitrary higher stack on the étale site of affine schemes of finite type over $\Bbb C$, then the étale homotopy type of $\Cal X$ agrees with the homotopy type of the underlying stack $\Cal X_{top}$ on the topological site, after profinite completion. In particular, if $\Cal X$ is an Artin stack locally of finite type over $\Bbb C$, our definition of the étale homotopy type of $\Cal X$ agrees up to profinite completion with the homotopy type of the underlying topological stack $\Cal X_{top}$ of $\Cal X$ in the sense of Noohi [35]. We also show this comparison is compatible in a suitable sense with the comparison theorem of Friedlander for simplicial schemes [17]. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of $\infty$-categories which we believe to be of independent interest.
References:
[1] Artin, M., Grothendieck, A., Verdier, J.L.: Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Lecture Notes in Mathematics, Vol. 269. Springer-Verlag, Berlin
[2] Artin, M., Mazur, B.: Etale homotopy, volume 100 of Lecture Notes in Mathematics. Springer-Verlag, Berlin. Reprint of the 1969 original
[3] Barnea, Ilan, Harpaz, Yonatan, Horel, Geoffroy: Pro-categories in homotopy theory. arXiv:1507.01564 http://arxiv.org/abs/1507.01564 MR 3604386
[4] Borsuk, Karol: Sur un espace compact localement contractile qui n’est pas un rétracte absolu de voisinage. Fundamenta Mathematicae, 35(1):175–180
[5] Bullejos, M., Faro, E., Garcia-Munoz, M. A.: Homotopy colimits and cohomology with local coefficients. Cah. Topol. Géom. Différ. Catég., 44(1):63–80 MR 1961526
[6] Carchedi, David: Higher orbifolds and deligne-mumford stacks as structured infinity topoi. Memoirs of the AMS,, 2020 MR 4075269
[7] Carchedi, David, Elmanto, Elden: Relative étale realizations of motivic spaces and Dwyer-Friedlander K-theory of noncommutative schemes. arXiv:1810.05544 https://arxiv.org/abs/1810.05544
[8] 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(5):3093–3158 MR 3687115
[9] Chough, C.-Y.: An equivalence of profinite completions. https://cgp.ibs.re.kr/~chough/chough_profinite.pdf
[10] Chough, C.-Y.: Topological types of algebraic stacks. International Mathematics Research Notices MR 4259160
[11] Dugger, Daniel, Isaksen, Daniel C.: Topological hypercovers and 𝔸¹-realizations. Math. Z., 246(4):667–689 MR 2045835
[12] Ebert, Johannes, Giansiracusa, Jeffrey: On the homotopy type of the Deligne-Mumford compactification. Algebr. Geom. Topol., 8(4):2049–2062 MR 2452916
[13] Edwards, David A., Hastings, Harold M.: Čech and Steenrod homotopy theories with applications to geometric topology. Lecture Notes in Mathematics, Vol. 542. Springer-Verlag, Berlin-New York
[14] Frediani, Paola, Neumann, Frank: Étale homotopy types of moduli stacks of polarised abelian schemes. Arxiv:1512.07544 http://arxiv.org/pdf/1512.07544 MR 3578997
[15] Fried, Michael D., Jarden, Moshe: Field arithmetic, volume 11 of. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, third edition. Revised by Jarden MR 2445111
[16] Friedlander, Eric M.: Fibrations in etale homotopy theory. Inst. Hautes Études Sci. Publ. Math., (42):5–46
[17] Friedlander, Eric M.: Étale homotopy of simplicial schemes, volume 104 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo
[18] Grothendieck, A., Dieudonné, J. A.: Eléments de géométrie algébrique. I, volume 166 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin
[19] Harpaz, Yonatan, Schlank, Tomer M.: Homotopy obstructions to rational points. In Torsors, étale homotopy and applications to rational points, volume 405 of London Math. Soc. Lecture Note Ser., pages 280–413. Cambridge Univ. Press, Cambridge MR 3077173
[20] Hoyois, Marc: Higher galois theory. Journal of Pure and Applied Algebra, Volume 222, Issue 7, July 2018 MR 3763287
[21] Isaksen, Daniel C.: A model structure on the category of pro-simplicial sets. Trans. Amer. Math. Soc., 353(7):2805–2841 MR 1828474
[22] Isaksen, Daniel C.: Etale realization on the 𝔸¹-homotopy theory of schemes. Adv. Math., 184(1):37–63 MR 2047848
[23] Jardine, J. F.: Simplicial presheaves. J. Pure Appl. Algebra, 47(1):35–87
[24] Johnstone, Peter T.: Sketches of an elephant: a topos theory compendium. Vol. 2, volume 44 of Oxford Logic Guides. The Clarendon Press, Oxford University Press, Oxford MR 2063092
[25] Kato, Kazuya, Nakayama, Chikara: Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C. Kodai Math. J., 22(2):161–186
[26] Lojasiewicz, S.: Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa (3), 18:449–474
[27] Lurie, Jacob: Spectral algebraic geometry. (In Preparation)
[28] Lurie., Jacob: Derived algebraic geometry V: Structured spaces. Derived algebraic geometry v: Structured spaces. Arxiv:0905.0459 http://arxiv.org/pdf/0905.0459 MR 2717174
[29] Lurie., Jacob: Derived algebraic geometry XIII: Rational and p-adic homotopy theory. http://www.math.harvard.edu/ lurie/papers/DAG-XIII.pdf
[30] Lurie., Jacob: Higher algebra. http://www.math.harvard.edu/ lurie/papers/higheralgebra.pdf
[31] Lurie, Jacob: Higher topos theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ MR 2522659
[32] Lurie., Jacob: Derived algebraic geometry VII: Spectral schemes. http://www.math.harvard.edu/ lurie/papers/DAG-VII.pdf
[33] Milne, J. S.: Some estimates from étale cohomology. J. Reine Angew. Math., 328:208–220
[34] Nikolaus, Thomas, Schreiber, Urs, Stevension, Danny: Principal ∞-bundles: general theory. Journal of Homotopy and Related Structures MR 3423073
[35] Noohi, Behrang: Foundations of topological stacks I. Arxiv:0503247 http://arxiv.org/pdf/0503247
[36] Noohi, Behrang: Homotopy types of topological stacks. Adv. Math., 230(4-6):2014–2047 MR 2927363
[37] Noohi, Behrang, Coyne, Thomas: Singular chains on topological stacks. Adv. Math., Volume 303(Issue 5):Pages 1190–1235, November 2016 DOI 10.1016/j.aim.2016.08.037 | MR 3552548
[38] Oda, Takayuki: Etale homotopy type of the moduli spaces of algebraic curves. In Geometric Galois actions, 1, volume 242 of London Math. Soc. Lecture Note Ser., pages 85–95. Cambridge Univ. Press, Cambridge
[39] Pál, Ambrus: Étale homotopy equivalence of rational points on algebraic varieties. Algebra Number Theory, 9(4):815–873 MR 3352821
[40] Quick, Gereon: Profinite homotopy theory. Doc. Math., 13:585–612 MR 2466189
[41] Quick, Gereon: Some remarks on profinite completion of spaces. In Galois-Teichmüller theory and arithmetic geometry, volume 63 of Adv. Stud. Pure Math., pages 413–448. Math. Soc. Japan, Tokyo MR 3051250
[42] Quillen, Daniel G.: Some remarks on etale homotopy theory and a conjecture of Adams. Topology, 7:111–116
[43] Robinson, C. A.: Moore-Postnikov systems for non-simple fibrations. Illinois J. Math., 16:234–242
[44] Schmidt., Alexander: On the étale homotopy type of Morel-Voevodsky spaces. https://www.mathi.uni-heidelberg.de/ schmidt/papers/homotyp.pdf
[45] Serre, Jean-Pierre: Cohomologie galoisienne, volume 1965 of With a contribution by Jean-Louis Verdier. Lecture Notes in Mathematics, No. 5. Troisième édition. Springer-Verlag, Berlin-New York
[46] Skorobogatov, Alexei N., editor, : Torsors, étale homotopy and applications to rational points, volume 405 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2013. Papers from the Workshop “Torsors: Theory and Applications” held in Edinburgh, January 10–14 MR 1845760
[47] Sullivan, Dennis: Genetics of homotopy theory and the Adams conjecture. Ann. of Math. (2), 100:1–79
[48] Talpo, Mattia, Vistoli, Angelo: Infinite root stacks and quasi-coherent sheaves on logarithmic chemes. Proceedings of the London Mathematical Society, 10 2014 MR 3805055
[49] Toën, Bertrand, Vaquié, Michel: Algébrisation des variétés analytiques complexes et catégories dérivées. Math. Ann., 342(4):789–831 MR 2443764
[50] Toen, Bertrand, Vezzosi, Gabriele: Segal topoi and stacks over segal categories. in Proceedings of the Program Stacks, Intersection theory and Non-abelian Hodge Theory , MSRI, Berkeley, January-May 2002
[51] Toën, Bertrand, Vezzosi, Gabriele: Homotopical algebraic geometry. II. Geometric stacks and applications. Mem. Amer. Math. Soc., 193(902):x+224 MR 2394633
[52] Vakil, Ravi, Wickelgren, Kirsten: Universal covering spaces and fundamental groups in algebraic geometry as schemes. J. Théor. Nombres Bordeaux, 23(2):489–526 MR 2817942
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