A universal characterization of higher algebraic K-theory

By Andrew Justin Blumberg, D. Gepner, Gonçalo Tabuada. 2010

347 citations recorded by the source corpus.

In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, i.e., the universal functor with values in spectra 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. To prove these results, we construct and study two stable infinity categories of "noncommutative motives"; one associated to additivity and another to localization. In these stable infinity categories, Waldhausen's S. construction corresponds to the suspension functor and connective and non-connective algebraic K-theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K-theory of every scheme, stack, and ring spectrum can be recovered from these categories of noncommutative motives. In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable infinity categories. We also explain in detail the comparison between the infinity categorical version of Waldhausen K-theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b08ecf3ba70f425c2a8712fc6b9b90fb2e110ac8

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Citations in the corpus

The Galois group of a stable homotopy theory (2014, background): “Compare also the treatment of stable ∞-categories in [8], which shows (cf.”
Algebraic K-theory and descent for blow-ups (2016, methodology): “ean more precisely an (∞,1)-category. For the purpose of this paper, the choice of model of ∞-categories does not play any essential role. However, to bedeﬁnite, and in accordance with main reference [BGT13] for the K-theory of stable ∞-categories, we use the language of quasi-categories as develope”
Differential Graded Categories are k-linear Stable Infinity Categories (2013, methodology): “el structure. We exploit this equivalence between dg categories and categories enriched in Hk-module spectra in our comparison theorem by using machinery that was developed for spectral categories in [BGT1] and [BGT2]. Moreover, one can conclude that: Proposition 3.13. A dg category T is pretriangul”
Uniqueness of the multiplicative cyclotomic trace (2011, methodology): “: D ( A ) → D ( B ) which restricts to The category Cat M allows us to correct a small error in [11].”
On exact $\infty$-categories and the Theorem of the Heart (2012, background): “On the other hand, there are other proposed versions of algebraic K -theory for ∞ - categories (most notably that of Blumberg, Gepner, and Tabuada [6]) that restrict attention to stable ∞ -categories or the like.”
Brauer groups and étale cohomology in derived algebraic geometry (2012, background): “The theory of smooth and proper algebras is also fundamental in the theory of noncommutative motives, and has been studied in that setting by Cisinski and Tabuada [18] and Blumberg, Gepner and Tabuada [12].”
Higher traces, noncommutative motives, and the categorified Chern character (2015, methodology): “We are grateful to Clark Barwick, Kai Behrend, David Ben-Zvi, Andrew Blumberg, David Carchedi, Rune Haugseng, Claudia Scheimbauer, Chris Schommer-Pries, Jay Shah, Gonçalo Tabuada, Bertrand Toën, and Jesse Wolfson for useful conversations and for their interest in this project.”
Universality of multiplicative infinite loop space machines (2013, methodology): “Nevertheless, in many cases of interest, eg that of a connective ring spectrum R, the algebraic K–theory of R, defined in terms of Waldhausen’s S construction applied to the stable 1–category of R–modules (which agrees with the K–theory of any suitable model category of R–modules; see [6] for detail”