Abstract
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.
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Notes
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.
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.
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Acknowledgements
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.
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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
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DOI: https://doi.org/10.1007/s00209-019-02394-7