Algebraic & Geometric Topology

Preorientations of the derived motivic multiplicative group

Jens Hornbostel

Full-text: Open access


We establish several new model structures and Quillen adjunctions both in the classical and in the motivic case for algebras over operads and for modules over strictly commutative ring spectra. As an application, we provide a proof in the language of model categories and symmetric spectra of Lurie’s Theorem that topological complex K–theory represents orientations of the derived multiplicative group. Then we generalize this result to the motivic situation.

Article information

Algebr. Geom. Topol., Volume 13, Number 5 (2013), 2667-2712.

Received: 22 March 2012
Revised: 2 April 2013
Accepted: 4 April 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra
Secondary: 18D50: Operads [See also 55P48] 19D99: None of the above, but in this section 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]

motivic homotopy theory motivic operads


Hornbostel, Jens. Preorientations of the derived motivic multiplicative group. Algebr. Geom. Topol. 13 (2013), no. 5, 2667--2712. doi:10.2140/agt.2013.13.2667.

Export citation


  • J Adámek, J Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189, Cambridge Univ. Press (1994)
  • J F Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press (1974)
  • M Ando, A Blumberg, D Gepner, M Hopkins, C Rezk, Units of ring spectra and Thom spectra
  • M G Barratt, P J Eccles, $\Gamma^{+}$–structures, I: A free group functor for stable homotopy theory, Topology 13 (1974) 25–45
  • C Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010) 245–320
  • M Behrens, Notes on the construction of tmf, preprint (2009) Available at \setbox0\makeatletter\@url {\unhbox0
  • C Berger, B Fresse, Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004) 135–174
  • C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805–831
  • J E Bergner, A survey of $(\infty,1)$–categories, from: “Towards higher categories”, (J C Baez, J P May, editors), IMA Vol. Math. Appl. 152, Springer, New York (2010) 69–83
  • A J Berrick, M Karoubi, P A Østvær, Hermitian $K$–theory and $2$–regularity for totally real number fields, Math. Ann. 349 (2011) 117–159
  • A J Berrick, M Karoubi, M Schlichting, P A Østvær, The homotopy fixed point theorem and the Quillen–Lichtenbaum conjecture in hermitian $K$–theory, preprint (2011) Available at \setbox0\makeatletter\@url {\unhbox0
  • B A Blander, Local projective model structures on simplicial presheaves, $K$–Theory 24 (2001) 283–301
  • J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer, Berlin (1973)
  • F Borceux, Handbook of categorical algebra, 2, Encyclopedia of Mathematics and its Applications 51, Cambridge Univ. Press (1994)
  • A K Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001) 2391–2426
  • A K Bousfield, E M Friedlander, Homotopy theory of $\Gamma$–spaces, spectra, and bisimplicial sets, from: “Geometric applications of homotopy theory”, Lecture Notes in Math. 658, Springer, Berlin (1978) 80–130
  • B I Dundas, O R öndigs, P A Østvær, Motivic functors, Doc. Math. 8 (2003) 489–525
  • W G Dwyer, D M Kan, Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math. 46 (1984) 147–153
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, American Mathematical Society (1997)
  • A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163–228
  • B Fresse, Modules over operads and functors, Lecture Notes in Mathematics 1967, Springer, Berlin (2009)
  • D Gepner, V Snaith, On the motivic spectra representing algebraic cobordism and algebraic $K$–theory, Doc. Math. 14 (2009) 359–396
  • P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: “Structured ring spectra”, (A Baker, B Richter, editors), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
  • S Gorchinskiy, V Guletski, Symmetric powers in stable homotopy categories
  • J E Harper, Homotopy theory of modules over operads in symmetric spectra, Algebr. Geom. Topol. 9 (2009) 1637–1680
  • J E Harper, Homotopy theory of modules over operads and non–$\Sigma$ operads in monoidal model categories, J. Pure Appl. Algebra 214 (2010) 1407–1434
  • A Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (1988) vi+78
  • P S Hirschhorn, Letter to Paul Coerss (1996)
  • P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)
  • J Hornbostel, Localizations in motivic homotopy theory, Math. Proc. Cambridge Philos. Soc. 140 (2006) 95–114
  • M Hovey, Monoidal model categories
  • M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999)
  • M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63–127
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • P Hu, I Kriz, K Ormsby, The homotopy limit problem for Hermitian $K$–theory, equivariant motivic homotopy theory and motivic real cobordism, Adv. Math. 228 (2011) 434–480
  • J F Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445–553
  • J F Jardine, Presheaves of spectra, Lectures at the Fields Institute (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • D Kobal, The Karoubi tower and $K$–theory invariants of Hermitian forms, $K$–Theory 17 (1999) 141–150
  • L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer, Berlin (1986)
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
  • J Lurie, A survey of elliptic cohomology, from: “Algebraic topology”, Abel Symp. 4, Springer, Berlin (2009) 219–277
  • J Lurie, Higher algebra (2012) Available at \setbox0\makeatletter\@url {\unhbox0
  • S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer, New York (1998)
  • M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
  • J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer, Berlin (1972)
  • F Morel, The stable $\mathbb{A}^1$–connectivity theorems, $K$–Th. 35 (2005) 1–68
  • F Morel, V Voevodsky, $\mathbb{A}^1$–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) 45–143
  • N Naumann, M Spitzweck, P A Østvær, Motivic Landweber exactness, Doc. Math. 14 (2009) 551–593
  • I Panin, K Pimenov, O R öndigs, A universality theorem for Voevodsky's algebraic cobordism spectrum, Homology, Homotopy Appl. 10 (2008) 211–226
  • I Panin, S Yagunov, Rigidity for orientable functors, J. Pure Appl. Algebra 172 (2002) 49–77
  • P Pelaez, Multiplicative properties of the slice filtration, Astérisque 335 (2011)
  • D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer, Berlin (1967)
  • C W Rezk, Spaces of algebra structures and cohomology of operads, PhD Thesis, Massachusetts Institute of Technology (1996)
  • M Robalo, Noncommutative motives I: From commutative to noncommutative motives
  • O R öndigs, M Spitzweck, P A Østvær, Motivic strict ring models for $K$–theory, Proc. Amer. Math. Soc. 138 (2010) 3509–3520
  • S Sagave, C Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012) 2116–2193
  • C Schlichtkrull, Thom spectra that are symmetric spectra, Doc. Math. 14 (2009) 699–748
  • S Schwede, Stable homotopical algebra and $\Gamma$–spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999) 329–356
  • S Schwede, An untitled book project about symmetric spectra, Preliminary version of a book (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
  • S Schwede, B E Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103–153
  • B E Shipley, A convenient model category for commutative ring spectra, from: “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$–theory”, (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc. (2004) 473–483
  • V Snaith, Localized stable homotopy of some classifying spaces, Math. Proc. Cambridge Philos. Soc. 89 (1981) 325–330
  • M Spitzweck, Operads, algebras and modules in general model categories
  • M Spitzweck, P A Østvær, The Bott inverted infinite projective space is homotopy algebraic $K$–theory, Bull. Lond. Math. Soc. 41 (2009) 281–292
  • B Toën, G Vezzosi, Homotopical algebraic geometry, II: Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008) x+224
  • V Voevodsky, $\mathbb A^1$–homotopy theory, from: “Proceedings of the International Congress of Mathematicians, Vol. I”, Extra Vol. I (1998) 579–604
  • V Voevodsky, Open problems in the motivic stable homotopy theory. I, from: “Motives, polylogarithms and Hodge theory, Part I”, (F Bogomolov, L Katzarkov, editors), Int. Press Lect. Ser. 3, Int. Press, Somerville, MA (2002) 3–34
  • C A Weibel, Homotopy algebraic $K$–theory, from: “Algebraic $K$–theory and algebraic number theory”, (M R Stein, R K Dennis, editors), Contemp. Math. 83, Amer. Math. Soc. (1989) 461–488
  • B Williams, Quadratic $K$–theory and geometric topology, from: “Handbook of $K$–theory, Vol. 1, 2”, (E M Friedlander, D R Grayson, editors), Springer, Berlin (2005) 611–651


  • Jens Hornbostel. Correction to the article Preorientations of the derived motivic multiplicative group. Algebr. Geom. Topol. 18 (2018), no. 2, 1257--1258.