Monadicity of the Bousfield–Kuhn functor
HTML articles powered by AMS MathViewer
- by Rosona Eldred, Gijs Heuts, Akhil Mathew and Lennart Meier PDF
- Proc. Amer. Math. Soc. 147 (2019), 1789-1796 Request permission
Abstract:
Let $\mathscr {M}_n^f$ be the localization of the $\infty$-category of spaces at the $v_n$-periodic equivalences, the case $n=0$ being rational homotopy theory. We prove that $\mathscr {M}_n^f$ is for $n\geq 1$ equivalent to algebras over a certain monad on the $\infty$-category of $T(n)$-local spectra. This monad is built from the Bousfield–Kuhn functor.References
- A. K. Bousfield and E. M. Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. MR 513569
- A. K. Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391–2426. MR 1814075, DOI 10.1090/S0002-9947-00-02649-0
- Mark Behrens and Charles Rezk, Spectral algebra models of unstable $v_n$-periodic homotopy theory, preprint, arXiv:1703.02186, 2017.
- Mark Behrens and Charles Rezk, The Bousfield-Kuhn functor and topological André-Quillen cohomology, preprint, arXiv:1712.03045, 2017.
- Donald M. Davis, Computing $v_1$-periodic homotopy groups of spheres and some compact Lie groups, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 993–1048. MR 1361905, DOI 10.1016/B978-044481779-2/50021-3
- Gijs Heuts, Lie algebras and $v_n$-periodic spaces, preprint, arXiv:1803.06325, 2018.
- Michael J. Hopkins and Jeffrey H. Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. (2) 148 (1998), no. 1, 1–49. MR 1652975, DOI 10.2307/120991
- Nicholas J. Kuhn, A guide to telescopic functors, Homology Homotopy Appl. 10 (2008), no. 3, 291–319. MR 2475626
- Jacob Lurie, Higher algebra, http://www.math.harvard.edu/~lurie/papers/HA.pdf, May 2016.
- Mark Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65–112. MR 662118, DOI 10.2307/2007048
- Charles Rezk, When are homotopy colimits compatible with homotopy base change? preprint, http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf.
- Emily Riehl and Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads, Adv. Math. 286 (2016), 802–888. MR 3415698, DOI 10.1016/j.aim.2015.09.011
- Robert D. Thompson, The $v_1$-periodic homotopy groups of an unstable sphere at odd primes, Trans. Amer. Math. Soc. 319 (1990), no. 2, 535–559. MR 1010890, DOI 10.1090/S0002-9947-1990-1010890-8
Additional Information
- Rosona Eldred
- Affiliation: Max-Planck-Institut, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1015629
- Gijs Heuts
- Affiliation: Mathematical Institute, Utrecht University, 3584CD Utrecht, The Netherlands
- MR Author ID: 1082480
- Akhil Mathew
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1514
- MR Author ID: 891016
- Lennart Meier
- Affiliation: Mathematical Institute, Utrecht University, 3584CD Utrecht, The Netherlands
- MR Author ID: 955940
- Received by editor(s): March 29, 2018
- Received by editor(s) in revised form: July 10, 2018
- Published electronically: January 8, 2019
- Additional Notes: This work was begun through a Junior Trimester Program at the Hausdorff Institute of Mathematics, and we thank the HIM for its hospitality.
The second author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 682922).
The third author was supported by the NSF Graduate Fellowship under grant DGE-114415 and was a Clay Research Fellow when this work was finished.
The fourth author was supported by DFG SPP 1786. - Communicated by: Mark Behrens
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1789-1796
- MSC (2010): Primary 55Q51
- DOI: https://doi.org/10.1090/proc/14331
- MathSciNet review: 3910443