Uniqueness of the multiplicative cyclotomic trace

By Andrew Justin Blumberg, D. Gepner, Gonçalo Tabuada. 2011

64 citations recorded by the source corpus.

Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology (TC). Moreover, we prove that the space of all multiplicative structures on algebraic K-theory is contractible. We also show that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, which in particular implies that E_n ring spectra give rise to E_{n-1} ring algebraic K-theory spectra. Along the way, we develop a "multiplicative Morita theory", establishing a symmetric monoidal equivalence between the infinity category of small idempotent-complete stable infinity categories and the Morita localization of the infinity category of spectral categories.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08e90427a4b93af8bb513396f1be90e16b4ee92f

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Citations in the corpus

The Local Structure of Algebraic K-Theory (2012, unknown): “A further reason for not rewriting the presentation is that with results as the one suggested in the preprint [21] of Blumberg, Gepner and Tabuada, concrete constructions may be needed only for technical considerations.”
Differential Graded Categories are k-linear Stable Infinity Categories (2013, methodology): “y structures). This is one reason why our comparison between dg categories and k-linear stable ∞-categories is not completely trivial. To resolve this issue, we use ideas inspired by [To1], [CT], and [BGT2] to derive the tensor product of dg categories via “ﬂat” dg categories. Following [Tab1], we u”
Universality of multiplicative infinite loop space machines (2013, methodology): “In [7] it is shown that the algebraic K–theory KW Cat 1 !Sp of small idempotent-complete stable 1–categories is a lax symmetric monoidal functor, as is the nonconnective version; the methods employed to do so are similar to the ones used in the present paper, in that K is shown to be the tensor unit”
On the algebraic K‐theory of higher categories (2012, methodology): “(For fully general solutions to the problem of finding O structures on K-theory spectra using machinery of the kind developed here, see either Blumberg–Gepner–Tabuada [13] or [2].)”