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Keywords:
Rational homotopy theory; motive; motivic Galois action; Tannakian formalism
Rational homotopy theory; motive; motivic Galois action; Tannakian formalism
Summary:
We propose a motivic generalization of rational homotopy types. The algebraic invariants we study are defined as algebra objects in the category of mixed motives. This invariant plays a role of Sullivan’s polynomial de Rham algebras. Another main notion is that of cotangent motives. Our main objective is to investigate the topological realization of these invariants and study their structures. Applying these machineries and the Tannakian theory, we construct actions of a derived motivic Galois group on rational homotopy types. Thanks to this, we deduce actions of the motivic Galois group of pro-unipotent completions of homotopy groups.
References:
[1] Ahlfors, L., Sario, L.: Riemann Surfaces. Princeton Univ. Press Zbl 0196.33801
[2] Ancona, G., Enright-Ward, S., Huber, A.: On the motives of a commutative algebraic group. Documenta Math. 20, 807–858
[3] André, Y.: Une introduction aux motifs (motifs purs, motifs mixtes, periods). Panoramas et Synthèses (Panoramas and Synthèses), vol.17, Société Mathématique de France, Paris
[4] André, Y., Kahn, B.: Nilpotence, radicaux et structures monoidales. Rend. Sem. Math. Univ. Padova, 108,, 107–291
[5] Berger, C., Fresse, B.: Combinatorial operad operations of cochains. Math. Proc. Cambridge Phil. Soc. 137, 135–174 DOI 10.1017/S0305004103007138
[6] Bergner, J.: A survey of (∞,1)-categories. Towards higher categories, IMA Volumes in Mathematics and Its Applications 152, Springer 69–83
[7] Ben-Zvi, D., Nadler, D.: Loop space and connections. J. Topol. 5 377–430 DOI 10.1112/jtopol/jts007
[8] Borel, A.: Linear Algebraic Groups. (the second edition) Springer-Verlag 1991
[9] Bott, R., Tu, L.W.: Differential forms in Algebraic Topology. Graduate texts in mathematics vol. 82 Springer
[10] Carlson, J., Clemens, H., Morgan, J.: On the mixed Hodge structure associated to π₃ of a simply connected projective manifold. Ann. Sci. E.N.S. 14, 323–338
[11] Cisinski, D.-C., Déglise, F.: Local and stable homological algebra in Grothendieck abelian categories. Homotopy Homology Applications, 11 (1),, 219–260
[12] Cisinski, D.-C., Déglise, F.: Mixed Weil cohomologies. Adv. Math. 230,, 55–130 DOI 10.1016/j.aim.2011.10.021
[13] Cisinski, D.-C., Déglise, F.: Triangulated Categories of Mixed Motives. Springer Monographs in Mathematics
[14] Deligne, P., Goncharov, A. B.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. Ecole Norm. Sup., Serie 4 : Vol. 38, 1–56
[15] Dugger, D., Isaksen, D.C.: Hypercovers in topology. preprint available at arXiv:math/0111287
[16] Félix, Y., Halperin, S., Thomas, J.-C.: Rational homotopy theory. Graduate Texts in Math. Springer
[17] Fresse, B.: Differential Graded Commutative Algebras and Cosimplicial Algebras. preprint version in, Chapater II.6 of “Homotopy of operads and Grothendieck-Teichmullar groups”
[18] Fukuyama, H., Iwanari, I.: Monoidal infinity category of complexes from tannakian viewpoint. Math. Ann. 356, 519–553 DOI 10.1007/s00208-012-0843-8
[19] Grothendieck, A.: Récoltes et Semailles. Gendai-Sugakusha, Japanese translation by Y. Tsuji
[20] Hain, R.: The de Rham Homotopy Theory of Complex Algebraic Variety I. K-theory 1, 271–324 DOI 10.1007/BF00533825
[21] Hess, K.: Rational Homotopy Theory: A Brief Introduction. Interactions between homotopy theory and algebra, 175–202, Comtemp. Math., 436 Amer. Math. Soc
[22] Hinich, V.: Homological algebra of homotopy algebras. Comm. in Algebra, 25, 3291–3323 DOI 10.1080/00927879708826055
[23] Hovey, M.: Model categories. Math. Survey and Monograph Vol. 83
[24] Iwanari, I.: Tannakization in derived algebraic geometry. J. K-Theory, 14, 642–700 DOI 10.1017/is014008019jkt278
[25] Iwanari, I.: Bar constructions and Tannakization. Publ. Res. Int. Math. Sci., 50, 515–568 DOI 10.4171/prims/143
[26] Iwanari, I.: Tannaka duality and stable infinity-categories. J. Topol. 11, 469–526 DOI 10.1112/topo.12057
[27] Iwanari, I.: On the structure of Galois groups of mixed motives. preprint, available at the author’s webpage https://sites.google.com/site/isamuiwanarishomepage/
[28] Kondo, S., Yasuda, S.: Product structures in motivic cohomology and higher Chow groups. J. Pure Appl. Algebra 215, 511–522 DOI 10.1016/j.jpaa.2010.06.003
[29] Künnemann, K.: On the Chow motive of an abelian scheme. in: Motives (Seattle, WA 1991) 189–205, Proc. Symposia in Pure Math. Vol. 55 1994
[30] Levine, M.: Tate motives and the vanishing conjectures for algebraic K-theory. (English summary) in: Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991), 167-188, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht
[31] Loday, J.-L.: Cyclic Homology. Springer-Verlag
[32] Lurie, J.: Higher topos theory. Ann. Math. Studies, 170 Princeton Univ. Press
[33] Lurie, J.: Higher algebra. preprint, the version of September 2017, available at the author’s webpage
[34] Lurie, J.: Derived algebraic geometry series. preprint available at the author’s webpage
[35] Mandell, M. A.: Cochains and homotopy type. Publ. I.H.E.S., 103, 213–246
[36] Mazel-Gee, A.: Quillen adjucntions induce adjunction of quasicategories. preprint available at Arxiv:1501.03146 http://arxiv.org/pdf/1501.03146
[37] Mazza, C., Voevodsky, V., Weibel, C.: Lecture Notes on Motivic cohomology. Clay Math. Monographs Vol. 2
[38] McClure, J., Smith, J.: Multivariable cochain operations and little n-cubes. J. Amer. Math. Soc., 16, 681–704 DOI 10.1090/S0894-0347-03-00419-3
[39] Morgan, J.: The algebraic topology of smooth algebraic varieties. Publ. I.H.E.S., 48, 137–204
[40] Murre, J.P., Nagel, J., Peters, C.: Lectures on the Theory of Pure Motives. AMS University Lecture Series, Vol. 61
[41] Olsson, M.: The bar construction and affine stacks. Comm. in Algebra, 44, 3088–3121 DOI 10.1080/00927872.2015.1082578
[42] Orgogozo, F.: Isomotifs de dimension inférieure ou égale à un. manuscripta math. 115,, 339–360
[43] Quillen, D.: Rational homotopy theory. Ann. Math. 90 205–295 DOI 10.2307/1970725
[44] Röndigs, O., Ostvaer, P. A.: Modules over motivic cohomology. Adv. Math. 219,, 689–727 DOI 10.1016/j.aim.2008.05.013
[45] Scholl, A. J.: Classical motives. in: Motives (Seattle, WA 1991) 163–187, Proc. Symposia in Pure Math. Vol. 55
[46] Schwede, S., Shipley, B.: Equivalences of monoidal model categories. Algebraic and Geometric Topology Vol.3, 287–334
[47] Spitzweck, M.: Derived fundamental groups for Tate motives. preprint available at Arxiv:1005.2670 http://arxiv.org/pdf/1005.2670
[48] Sullivan, D.: Infinitesimal computations in topology. Publ. Math. I.H.E.S., 47, 269–331 DOI 10.1007/BF02684341
[49] Toën, B.: Champs affine. Selecta Math. (N.S.) 12, 39–135 DOI 10.1007/s00029-006-0019-z
[50] Voevodsky, V.: Triangulated categories of motives over a field. Cycles, Transfers and motivichomology theories, 188–238, Ann. Math. Stud., 143 Princeton Univ. Press
[51] Wojtkowiak, Z.: Cosimplicial objects in algebraic geometry. in: Algebraic K-theory and algebraic topology, 287–327, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407
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