A multiplicative comparison of Segal and Waldhausen K-Theory


In this paper, we establish a multiplicative equivalence between two multiplicative algebraic K-theory constructions, Elmendorf and Mandell’s version of Segal’s K-theory and Blumberg and Mandell’s version of Waldhausen’s \(S_\bullet \) construction. This equivalence implies that the ring spectra, algebra spectra, and module spectra constructed via these two classical algebraic K-theory functors are equivalent as ring, algebra or module spectra, respectively. It also allows for comparisons of spectrally enriched categories constructed via these definitions of K-theory. As both the Elmendorf–Mandell and Blumberg–Mandell multiplicative versions of K-theory encode their multiplicativity in the language of multicategories, our main theorem is that there is multinatural transformation relating these two symmetric multifunctors that lifts the classical functor from Segal’s to Waldhausen’s construction. Along the way, we provide a slight generalization of the Elmendorf–Mandell construction to symmetric monoidal categories.

This is a preview of subscription content, log in to check access.


  1. 1.

    This definition varies slightly from that of Elmendorf and Mandell in that they do not require \(\rho \) to be invertible here. However, there is a natural adjunction between the category where the \(\rho \)’s are isomorphisms and the category where the \(\rho \)’s are merely morphisms, and so upon geometric realization we obtain weakly equivalent K-theory spaces.

  2. 2.

    Using the same idea as in the case of simplicial sets, one can also enrich \(\mathbf {Spec}({{ s}}\mathcal {C} at )\) in \({{ s}}\mathcal {C} at \). We will not need this second enrichment.


  1. 1.

    Barwick, C.: Multiplicative structures on algebraic \(K\)-theory. Doc. Math. 20, 859–878 (2015)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Barwick, C.: On the algebraic \(K\)-theory of higher categories. J. Topol. 9(1), 245–347 (2016)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Blumberg, A.J., Gepner, D., Tabuada, G.: \(K\)-theory of endomorphisms via noncommutative motives. Trans. Am. Math. Soc. 368(2), 1435–1465 (2016)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Blumberg, A.J., Mandell, M.A.: Derived Koszul duality and involutions in the algebraic \(K\)-theory of spaces. J. Topol. 4(2), 327–342 (2011)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bohmann, A.M., Osorno, A.: Equivariant spectra via Waldhausen Mackey functors. In preparation.

  6. 6.

    Bohmann, A.M., Osorno, A.: Constructing equivariant spectra via categorical Mackey functors. Algebraic Geom. Topol. 15(1), 537–563 (2015)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Elmendorf, A.D., Mandell, M.A.: Rings, modules, and algebras in infinite loop space theory. Adv. Math. 205(1), 163–228 (2006)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Elmendorf, A.D., Mandell, M.A.: Permutative categories, multicategories and algebraic \(K\)-theory. Algebraic Geom. Topol. 9(4), 2391–2441 (2009)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Gepner, D., Groth, M., Nikolaus, T.: Universality of multiplicative infinite loop space machines. Algebraic Geom. Topol. 15(6), 3107–3153 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Geisser, T., Hesselholt, L.: On the \(K\)-theory of complete regular local \({\mathbb{F}}_p\)-algebras. Topology 45(3), 475–493 (2006)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149–208 (2000)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    May, J.P.: Pairings of categories and spectra. J. Pure Appl. Algebra 19, 299–346 (1980)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    May, J.P.: Multiplicative infinite loop space theory. J. Pure Appl. Algebra 26(1), 1–69 (1982)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    May, J.P.: The construction of \(E_\infty \) ring spaces from bipermutative categories. In: New Topological Contexts for Galois Theory and Algebraic Geometry (BIRS 2008), Volume 16 of Geom. Topol. Monogr., pp. 283–330. Geom. Topol. Publ., Coventry, (2009)

  15. 15.

    May, J.P., Merling, M., Osorno, A.M.: Equivariant infinite loop space theory, I. The space level story. ArXiv e-prints. 1704.03413

  16. 16.

    Mandell, M.A., May, J.P., Schwede, S., Shipley, B.: Model categories of diagram spectra. Proc. Lond. Math. Soc. 82(02), 441–512 (2001)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    May, J.P., Thomason, R.: The uniqueness of infinite loop space machines. Topology 17(3), 205–224 (1978)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Schwede, S.: A untitled book project about symmetric spectra. Available on the author’s webpage., (2007)

  19. 19.

    Segal, G.: Categories and cohomology theories. Topology 13, 293–312 (1974)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Waldhausen, F.: Algebraic \(K\)-theory of spaces. In: Algebraic and Geometric Topology (New Brunswick, N.J., 1983), Volume 1126 of Lecture Notes in Math. pp. 318–419. Springer, Berlin (1985)

  21. 21.

    Yau, D.: Colored operads. Graduate Studies in Mathematics, vol. 170. American Mathematical Society, Providence (2016)

    Google Scholar 

  22. 22.

    Zakharevich, I.: The category of Waldhausen categories is a closed multicategory. In: New Directions in Homotopy Theory, Volume 707 of Contemp. Math., pp. 175–194. American Mathematical Society, Providence, RI (2018)

Download references


Many thanks to Clark Barwick, Andrew Blumberg, Tony Elmendorf, Lars Hesselholt, Mona Merling and Inna Zakharevich for interesting and helpful conversations. The authors also thank the anonymous referee for their very careful reading of this paper and for several clarifying suggestions. It is a pleasure to acknowledge the support of several institutions that helped make this research possible. The first author was partially supported by NSF DMS-1710534. The second author was partially supported by the Simons Foundation Grant No. 359449, the Woodrow Wilson Career Enhancement Fellowship, and NSF grant DMS-1709302. The first author also thanks the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program “Homotopy Harnessing Higher Structures” when some of the work on this paper was undertaken. This work was supported by EPSRC grant numbers EP/K032208/1 and EP/R014604/1.

Author information



Corresponding author

Correspondence to Angélica M. Osorno.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bohmann, A.M., Osorno, A.M. A multiplicative comparison of Segal and Waldhausen K-Theory. Math. Z. 295, 1205–1243 (2020). https://doi.org/10.1007/s00209-019-02394-7

Download citation


  • K-theory
  • Ring spectra
  • Waldhausen categories
  • Multiplicative structure