## Geometry & Topology

### A universal characterization of higher algebraic $K\mkern-4mu$–theory

#### Abstract

In this paper we establish a universal characterization of higher algebraic $K$–theory in the setting of small stable $∞$–categories. Specifically, we prove that connective algebraic $K$–theory is the universal additive invariant, ie the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits and satisfies Waldhausen’s additivity theorem. Similarly, we prove that nonconnective algebraic $K$–theory is the universal localizing invariant, ie the universal functor that moreover satisfies the Thomason–Trobaugh–Neeman Localization Theorem.

To prove these results, we construct and study two stable $∞$–categories of “noncommutative motives”; one associated to additivity and another to localization. In these stable $∞$–categories, Waldhausen’s $S∙$–construction corresponds to the suspension functor and connective and nonconnective 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 the case of a connective ring spectrum $R$, we prove moreover that its negative $K$–groups are isomorphic to the negative $K$–groups of the ordinary ring $π0R$.

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 $∞$–categories. We also explain in detail the comparison between the $∞$–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.

#### Article information

Source
Geom. Topol., Volume 17, Number 2 (2013), 733-838.

Dates
Revised: 2 October 2012
Accepted: 4 November 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732567

Digital Object Identifier
doi:10.2140/gt.2013.17.733

Mathematical Reviews number (MathSciNet)
MR3070515

Zentralblatt MATH identifier
1267.19001

#### Citation

Blumberg, Andrew J; Gepner, David; Tabuada, Gonçalo. A universal characterization of higher algebraic $K\mkern-4mu$–theory. Geom. Topol. 17 (2013), no. 2, 733--838. doi:10.2140/gt.2013.17.733. https://projecteuclid.org/euclid.gt/1513732567

#### References

• J Adámek, J Rosický, Locally presentable and accessible categories, London Math. Soc. Lecture Note Ser. 189, Cambridge Univ. Press (1994)
• C Barwick, On the algebraic $K$–theory of higher categories, I. The universal property of Waldhausen ${K}$–theory
• C Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010) 245–320
• C Barwick, D M Kan, Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. 23 (2012) 42–68
• H Bass, Algebraic $K$–theory, W A Benjamin, New York (1968)
• D Ben-Zvi, J Francis, D Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909–966
• J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043–2058
• J E Bergner, A survey of $(\infty,1)$–categories, from: “Towards higher categories”, (J C Baez, J P May, editors), IMA Vol. Math. Appl. 152, Springer, New York (2010) 69–83
• A J Blumberg, D Gepner, G Tabuada, Uniqueness of the multiplicative cyclotomic trace
• A J Blumberg, M A Mandell, Localization for ${THH}(ku)$ and the topological Hochschild and cyclic homology of Waldhausen categories
• A J Blumberg, M A Mandell, The localization sequence for the algebraic $K$–theory of topological $K$–theory, Acta Math. 200 (2008) 155–179
• A J Blumberg, M A Mandell, Algebraic $K$–theory and abstract homotopy theory, Adv. Math. 226 (2011) 3760–3812
• A J Blumberg, M A Mandell, Derived Koszul duality and involutions in the algebraic $K$–theory of spaces, J. Topol. 4 (2011) 327–342
• A J Blumberg, M A Mandell, Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol. 16 (2012) 1053–1120
• J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer, Berlin (1973)
• M Bökstedt, Topological Hochschild homology, preprint
• M B ökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic $K$–theory of spaces, Invent. Math. 111 (1993) 465–539
• A K Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133–150
• D-C Cisinski, Invariance de la $K$–théorie par équivalences dérivées, J. $K$–Theory 6 (2010) 505–546
• D-C Cisinski, G Tabuada, Non-connective $K$–theory via universal invariants, Compos. Math. 147 (2011) 1281–1320
• D-C Cisinski, G Tabuada, Symmetric monoidal structure on non-commutative motives, J. $K$–Theory 9 (2012) 201–268
• G Cortiñas, A Thom, Bivariant algebraic $K$–theory, J. Reine Angew. Math. 610 (2007) 71–123
• V Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004) 643–691
• D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177–201
• D Dugger, D I Spivak, Mapping spaces in quasi-categories, Algebr. Geom. Topol. 11 (2011) 263–325
• B I Dundas, Relative $K$–theory and topological cyclic homology, Acta Math. 179 (1997) 223–242
• W G Dwyer, D M Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980) 17–35
• W G Dwyer, D M Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267–284
• W G Dwyer, D M Kan, Equivalences between homotopy theories of diagrams, from: “Algebraic topology and algebraic $K$–theory”, (W Browder, editor), Ann. of Math. Stud. 113, Princeton Univ. Press (1987) 180–205
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monographs 47, Amer. Math. Soc. (1997)
• Z Fiedorowicz, R Schwänzl, R Steiner, R Vogt, Nonconnective delooping of $K$–theory of an $A\sb \infty$ ring space, Math. Z. 203 (1990) 43–57
• T Geisser, L Hesselholt, Topological cyclic homology of schemes, from: “Algebraic $K$–theory”, (W Raskind, C Weibel, editors), Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 41–87
• T Geisser, L Hesselholt, On relative and bi-relative algebraic $K$–theory of rings of finite characteristic, J. Amer. Math. Soc. 24 (2011) 29–49
• P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
• T G Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985) 187–215
• L Hesselholt, Witt vectors of non-commutative rings and topological cyclic homology, Acta Math. 178 (1997) 109–141
• L Hesselholt, I Madsen, On the $K$–theory of local fields, Ann. of Math. 158 (2003) 1–113
• N Higson, A characterization of $KK$–theory, Pacific J. Math. 126 (1987) 253–276
• P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monographs 99, Amer. Math. Soc. (2003)
• A Hirschowitz, C Simpson, Descente pour les $n$–champs
• M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
• A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207–222
• A Joyal, M Tierney, Quasi-categories vs Segal spaces, from: “Categories in algebra, geometry and mathematical physics”, (A Davydov, M Batanin, M Johnson, S Lack, A Neeman, editors), Contemp. Math. 431, Amer. Math. Soc. (2007) 277–326
• B Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999) 1–56
• H Krause, On Neeman's well generated triangulated categories, Doc. Math. 6 (2001) 121–126
• H Krause, Localization theory for triangulated categories, from: “Triangulated categories”, (T Holm, P Jørgensen, R Rouquier, editors), London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press (2010) 161–235
• J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
• J Lurie, Higher algebra, preprint (2012) Available at \setbox0\makeatletter\@url http://tinyurl.com/cdeqd7z {\unhbox0
• M Makkai, R Paré, Accessible categories: the foundations of categorical model theory, Contemporary Mathematics 104, Amer. Math. Soc. (1989)
• J P May, $A\sb{\infty }$ ring spaces and algebraic $K$–theory, from: “Geometric applications of homotopy theory, II”, (M G Barratt, M E Mahowald, editors), Lecture Notes in Math. 658, Springer, Berlin (1978) 240–315
• R McCarthy, On fundamental theorems of algebraic $K$–theory, Topology 32 (1993) 325–328
• R McCarthy, The cyclic homology of an exact category, J. Pure Appl. Algebra 93 (1994) 251–296
• R McCarthy, Relative algebraic $K$–theory and topological cyclic homology, Acta Math. 179 (1997) 197–222
• R Meyer, R Nest, The Baum–Connes conjecture via localisation of categories, Topology 45 (2006) 209–259
• F Morel, V Voevodsky, ${\bf A}\sp 1$–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) 45–143
• A Neeman, The connection between the $K$–theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. 25 (1992) 547–566
• A Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton Univ. Press (2001)
• E K Pedersen, C A Weibel, $K$–theory homology of spaces, from: “Algebraic topology”, (G Carlsson, R L Cohen, H R Miller, D C Ravenel, editors), Lecture Notes in Math. 1370, Springer, Berlin (1989) 346–361
• D Quillen, Higher algebraic $K$–theory. I, from: “Algebraic $K$–theory, I: Higher $K$–theories”, (H Bass, editor), Lecture Notes in Math. 341, Springer, Berlin (1973) 85–147
• C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973–1007
• C Rezk, S Schwede, B Shipley, Simplicial structures on model categories and functors, Amer. J. Math. 123 (2001) 551–575
• J Rognes, A spectrum level rank filtration in algebraic $K$–theory, Topology 31 (1992) 813–845
• J Rognes, Two-primary algebraic $K$–theory of spaces and related spaces of symmetries of manifolds, from: “Algebraic $K$–theory”, (W Raskind, C Weibel, editors), Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 213–229
• M Schlichting, Negative $K$–theory of derived categories, Math. Z. 253 (2006) 97–134
• S Schwede, B Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103–153
• C Simpson, A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$–pretopoi
• R J Steiner, Infinite loop structures on the algebraic $K$–theory of spaces, Math. Proc. Cambridge Philos. Soc. 90 (1981) 85–111
• G Tabuada, Higher $K$–theory via universal invariants, Duke Math. J. 145 (2008) 121–206
• G Tabuada, Homotopy theory of spectral categories, Adv. Math. 221 (2009) 1122–1143
• G Tabuada, Matrix invariants of spectral categories, Int. Math. Res. Not. 2010 (2010) 2459–2511
• Z Tamsamani, On non-strict notions of $n$–category and $n$–groupoid via multisimplicial sets
• R W Thomason, T Trobaugh, Higher algebraic $K$–theory of schemes and of derived categories, from: “The Grothendieck Festschrift, Vol. 3”, (P Cartier, L Illusie, N M Katz, G Laumon, K A Ribet, editors), Progr. Math. 88, Birkhäuser, Boston, MA (1990) 247–435
• B Toën, The homotopy theory of $dg$–categories and derived Morita theory, Invent. Math. 167 (2007) 615–667
• B Toën, G Vezzosi, A remark on $K$–theory and $S$–categories, Topology 43 (2004) 765–791
• V Voevodsky, Triangulated categories of motives over a field, from: “Cycles, transfers, and motivic homology theories”, Ann. of Math. Stud. 143, Princeton Univ. Press (2000) 188–238
• J B Wagoner, Delooping classifying spaces in algebraic $K$–theory, Topology 11 (1972) 349–370
• F Waldhausen, Algebraic $K$–theory of topological spaces. II, from: “Algebraic topology”, (J L Dupont, I H Madsen, editors), Lecture Notes in Math. 763, Springer, Berlin (1979) 356–394
• F Waldhausen, Algebraic $K$–theory of spaces, from: “Algebraic and geometric topology”, (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 318–419