Uniqueness of the multiplicative cyclotomic trace

Abstract

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.