Braided injections and double loop spaces
HTML articles powered by AMS MathViewer
- by Christian Schlichtkrull and Mirjam Solberg PDF
- Trans. Amer. Math. Soc. 368 (2016), 7305-7338 Request permission
Abstract:
We consider a framework for representing double loop spaces (and more generally $E_2$ spaces) as commutative monoids. There are analogous commutative rectifications of braided monoidal structures and we use this framework to define iterated double deloopings. We also consider commutative rectifications of $E_{\infty }$ spaces and symmetric monoidal categories and we relate this to the category of symmetric spectra.References
- Andrew J. Blumberg, Ralph L. Cohen, and Christian Schlichtkrull, Topological Hochschild homology of Thom spectra and the free loop space, Geom. Topol. 14 (2010), no. 2, 1165–1242. MR 2651551, DOI 10.2140/gt.2010.14.1165
- Clemens Berger, Double loop spaces, braided monoidal categories and algebraic $3$-type of space, Higher homotopy structures in topology and mathematical physics (Poughkeepsie, NY, 1996) Contemp. Math., vol. 227, Amer. Math. Soc., Providence, RI, 1999, pp. 49–66. MR 1665460, DOI 10.1090/conm/227/03252
- Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
- F. R. Cohen, J. P. May, and L. R. Taylor, Splitting of certain spaces $CX$, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 465–496. MR 503007, DOI 10.1017/S0305004100055298
- Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 1–38. MR 0272852
- A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR 1417719, DOI 10.1090/surv/047
- A. D. Elmendorf and M. A. Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006), no. 1, 163–228. MR 2254311, DOI 10.1016/j.aim.2005.07.007
- Z. Fiedorowicz, The symmetric bar construction, available on the authors homepage.
- Z. Fiedorowicz, M. Stelzer, and R. M. Vogt, Rectification of weak product algebras over an operad in $Cat$ and $Top$ and applications, arXiv:1311.2817.
- Z. Fiedorowicz, M. Stelzer, and R. M. Vogt, Homotopy colimits of algebras over $\scr {Cat}$-operads and iterated loop spaces, Adv. Math. 248 (2013), 1089–1155. MR 3107537, DOI 10.1016/j.aim.2013.07.016
- Z. Fiedorowicz and R. Vogt, Simplicial $n$-fold monoidal categories model all loop spaces, Cah. Topol. Géom. Différ. Catég. 44 (2003), no. 2, 105–148 (English, with French summary). MR 1985834
- Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR 1711612, DOI 10.1007/978-3-0348-8707-6
- Daniel Grayson, Higher algebraic $K$-theory. II (after Daniel Quillen), Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 217–240. MR 0574096
- Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041, DOI 10.1090/surv/099
- Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
- Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR 1695653, DOI 10.1090/S0894-0347-99-00320-3
- J. Hollender and R. M. Vogt, Modules of topological spaces, applications to homotopy limits and $E_\infty$ structures, Arch. Math. (Basel) 59 (1992), no. 2, 115–129. MR 1170635, DOI 10.1007/BF01190675
- André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, DOI 10.1006/aima.1993.1055
- J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 0420610
- J. P. May, $E_{\infty }$ spaces, group completions, and permutative categories, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) London Math. Soc. Lecture Note Ser., No. 11, Cambridge Univ. Press, London, 1974, pp. 61–93. MR 0339152
- M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512. MR 1806878, DOI 10.1112/S0024611501012692
- Christian Schlichtkrull, The homotopy infinite symmetric product represents stable homotopy, Algebr. Geom. Topol. 7 (2007), 1963–1977. MR 2366183, DOI 10.2140/agt.2007.7.1963
- Christian Schlichtkrull, Thom spectra that are symmetric spectra, Doc. Math. 14 (2009), 699–748. MR 2578805
- Brooke Shipley, A convenient model category for commutative ring spectra, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 473–483. MR 2066511, DOI 10.1090/conm/346/06300
- M. Solberg, Injective braids, braided operads and double loop spaces, Master thesis, available on BORA: bora.uib.no/handle/1956/5766, 2011.
- Stefan Schwede and Brooke E. Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491–511. MR 1734325, DOI 10.1112/S002461150001220X
- Steffen Sagave and Christian Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012), no. 3-4, 2116–2193. MR 2964635, DOI 10.1016/j.aim.2012.07.013
- R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91–109. MR 510404, DOI 10.1017/S0305004100055535
- R. W. Thomason, Cat as a closed model category, Cahiers Topologie Géom. Différentielle 21 (1980), no. 3, 305–324. MR 591388
Additional Information
- Christian Schlichtkrull
- Affiliation: Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway
- MR Author ID: 640767
- Email: christian.schlichtkrull@math.uib.no
- Mirjam Solberg
- Affiliation: Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen, Norway
- Email: mirjam.solberg@math.uib.no
- Received by editor(s): March 5, 2014
- Received by editor(s) in revised form: September 29, 2014
- Published electronically: November 16, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7305-7338
- MSC (2010): Primary 18D10, 18D50, 55P48; Secondary 55P43
- DOI: https://doi.org/10.1090/tran/6614
- MathSciNet review: 3471092