Mathematics – K-Theory and Homology
Scientific paper
2010-01-13
Mathematics
K-Theory and Homology
Added homotopy invariance section (homotopy K-theory, KV-theory), various minor revisions and improvements
Scientific paper
In this paper we establish a universal characterization of higher algebraic K-theory in the setting of stable infinity categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, i.e., the universal functor with values in a stable presentable infinity category which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. Similarly, we prove that non-connective algebraic K-theory is the universal localizing invariant, i.e., the universal functor that moreover satisfies the "Thomason-Trobaugh-Neeman" localization theorem. In addition, by adapting the standard cosimplicial affine space to the setting of stable categories, we generalize the classical notion of algebraic homotopy invariance and prove that Karoubi-Villamayor's algebraic K-theory is the universal additive homotopy invariant and that Weibel's homotopy K-theory is the universal localizing homotopy invariant. To prove these results, we construct and study various stable symmetric monoidal infinity categories of "non-commutative motives". In these infinity categories, Waldhausen's S. construction corresponds to the suspension functor and the various algebraic K-theory spectra becomes corepresentable by the unit object. Moreover, these infinity categories are enriched over Waldhausen's A-theory of a point and the homotopy K-theory of the sphere spectrum, respectively. We give several applications of our theory. We obtain a complete classification of all natural transformations from higher algebraic K-theory to THH and TC. Notably, we obtain a canonical construction and universal description of the cyclotomic trace map. We also exhibit a lax symmetric monoidal structure on the different algebraic K-theory functors, implying in particular that E_n ring spectra give rise to E_{n-1} K-theory spectra.
Blumberg Andrew J.
Gepner David
Tabuada Goncalo
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