Volume 17, issue 2 (2013)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 27
Issue 2, 417–821
Issue 1, 1–415

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
A universal characterization of higher algebraic $K\mkern-4mu$–theory

Andrew J Blumberg, David Gepner and Gonçalo Tabuada

Geometry & Topology 17 (2013) 733–838
1 J Adámek, J Rosický, Locally presentable and accessible categories, London Math. Soc. Lecture Note Ser. 189, Cambridge Univ. Press (1994) MR1294136
2 C Barwick, On the algebraic $K$–theory of higher categories, I. The universal property of Waldhausen ${K}$–theory arXiv:1204.3607
3 C Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010) 245 MR2771591
4 C Barwick, D M Kan, Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. 23 (2012) 42 MR2877401
5 H Bass, Algebraic $K$–theory, W A Benjamin (1968) MR0249491
6 D Ben-Zvi, J Francis, D Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909 MR2669705
7 J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043 MR2276611
8 J E Bergner, A survey of $(\infty,1)$–categories, from: "Towards higher categories" (editors J C Baez, J P May), IMA Vol. Math. Appl. 152, Springer (2010) 69 MR2664620
9 A J Blumberg, D Gepner, G Tabuada, Uniqueness of the multiplicative cyclotomic trace arXiv:1103.3923
10 A J Blumberg, M A Mandell, Localization for ${THH}(ku)$ and the topological Hochschild and cyclic homology of Waldhausen categories arXiv:1111.4003
11 A J Blumberg, M A Mandell, The localization sequence for the algebraic $K$–theory of topological $K$–theory, Acta Math. 200 (2008) 155 MR2413133
12 A J Blumberg, M A Mandell, Algebraic $K$–theory and abstract homotopy theory, Adv. Math. 226 (2011) 3760
13 A J Blumberg, M A Mandell, Derived Koszul duality and involutions in the algebraic $K$–theory of spaces, J. Topol. 4 (2011) 327 MR2805994
14 A J Blumberg, M A Mandell, Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol. 16 (2012) 1053 MR2928988
15 J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer (1973) MR0420609
16 M Bökstedt, Topological Hochschild homology, preprint
17 M Bökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic $K$–theory of spaces, Invent. Math. 111 (1993) 465 MR1202133
18 A K Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133 MR0380779
19 D C Cisinski, Invariance de la $K$–théorie par équivalences dérivées, J. $K$–Theory 6 (2010) 505 MR2746284
20 D C Cisinski, G Tabuada, Non-connective $K$–theory via universal invariants, Compos. Math. 147 (2011) 1281 MR2822869
21 D C Cisinski, G Tabuada, Symmetric monoidal structure on non-commutative motives, J. $K$–Theory 9 (2012) 201 MR2922389
22 G Cortiñas, A Thom, Bivariant algebraic $K$–theory, J. Reine Angew. Math. 610 (2007) 71 MR2359851
23 V Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004) 643 MR2028075
24 D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177 MR1870516
25 D Dugger, D I Spivak, Mapping spaces in quasi-categories, Algebr. Geom. Topol. 11 (2011) 263
26 B I Dundas, Relative $K$–theory and topological cyclic homology, Acta Math. 179 (1997) 223 MR1607556
27 W G Dwyer, D M Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980) 17 MR578563
28 W G Dwyer, D M Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267 MR579087
29 W G Dwyer, D M Kan, Equivalences between homotopy theories of diagrams, from: "Algebraic topology and algebraic $K$–theory" (editor W Browder), Ann. of Math. Stud. 113, Princeton Univ. Press (1987) 180 MR921478
30 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) MR1417719
31 Z Fiedorowicz, R Schwänzl, R Steiner, R Vogt, Nonconnective delooping of $K$–theory of an $A_\infty$ ring space, Math. Z. 203 (1990) 43 MR1030707
32 T Geisser, L Hesselholt, Topological cyclic homology of schemes, from: "Algebraic $K$–theory" (editors W Raskind, C Weibel), Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 41 MR1743237
33 T Geisser, L Hesselholt, On relative and bi-relative algebraic $K$–theory of rings of finite characteristic, J. Amer. Math. Soc. 24 (2011) 29
34 P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser (1999) MR1711612
35 T G Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985) 187 MR793184
36 L Hesselholt, Witt vectors of non-commutative rings and topological cyclic homology, Acta Math. 178 (1997) 109 MR1448712
37 L Hesselholt, I Madsen, On the $K$–theory of local fields, Ann. of Math. 158 (2003) 1 MR1998478
38 N Higson, A characterization of $KK$–theory, Pacific J. Math. 126 (1987) 253 MR869779
39 P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monographs 99, Amer. Math. Soc. (2003) MR1944041
40 A Hirschowitz, C Simpson, Descente pour les $n$–champs arXiv:math/9807049
41 M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149 MR1695653
42 A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207 MR1935979
43 A Joyal, M Tierney, Quasi-categories vs Segal spaces, from: "Categories in algebra, geometry and mathematical physics" (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 277 MR2342834
44 B Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999) 1 MR1667558
45 H Krause, On Neeman's well generated triangulated categories, Doc. Math. 6 (2001) 121 MR1836046
46 H Krause, Localization theory for triangulated categories, from: "Triangulated categories" (editors T Holm, P Jørgensen, R Rouquier), London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press (2010) 161 MR2681709
47 J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009) MR2522659
48 J Lurie, Higher algebra, preprint (2012)
49 M Makkai, R Paré, Accessible categories: the foundations of categorical model theory, Contemporary Mathematics 104, Amer. Math. Soc. (1989) MR1031717
50 J P May, $A_{\infty }$ ring spaces and algebraic $K$–theory, from: "Geometric applications of homotopy theory, II" (editors M G Barratt, M E Mahowald), Lecture Notes in Math. 658, Springer (1978) 240 MR513580
51 R McCarthy, On fundamental theorems of algebraic $K$–theory, Topology 32 (1993) 325 MR1217072
52 R McCarthy, The cyclic homology of an exact category, J. Pure Appl. Algebra 93 (1994) 251 MR1275967
53 R McCarthy, Relative algebraic $K$–theory and topological cyclic homology, Acta Math. 179 (1997) 197 MR1607555
54 R Meyer, R Nest, The Baum–Connes conjecture via localisation of categories, Topology 45 (2006) 209 MR2193334
55 F Morel, V Voevodsky, $\mathbf{A}^1$–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) 45 MR1813224
56 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 MR1191736
57 A Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton Univ. Press (2001) MR1812507
58 E K Pedersen, C A Weibel, $K$–theory homology of spaces, from: "Algebraic topology" (editors G Carlsson, R L Cohen, H R Miller, D C Ravenel), Lecture Notes in Math. 1370, Springer (1989) 346 MR1000388
59 D Quillen, Higher algebraic $K$–theory. I, from: "Algebraic $K$–theory, I: Higher $K$–theories" (editor H Bass), Lecture Notes in Math. 341, Springer (1973) 85 MR0338129
60 C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973 MR1804411
61 C Rezk, S Schwede, B Shipley, Simplicial structures on model categories and functors, Amer. J. Math. 123 (2001) 551 MR1833153
62 J Rognes, A spectrum level rank filtration in algebraic $K$–theory, Topology 31 (1992) 813 MR1191383
63 J Rognes, Two-primary algebraic $K$–theory of spaces and related spaces of symmetries of manifolds, from: "Algebraic $K$–theory" (editors W Raskind, C Weibel), Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 213 MR1743242
64 M Schlichting, Negative $K$–theory of derived categories, Math. Z. 253 (2006) 97 MR2206639
65 S Schwede, B Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103 MR1928647
66 C Simpson, A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$–pretopoi arXiv:math/9903167
67 R J Steiner, Infinite loop structures on the algebraic $K$–theory of spaces, Math. Proc. Cambridge Philos. Soc. 90 (1981) 85 MR611287
68 G Tabuada, Higher $K$–theory via universal invariants, Duke Math. J. 145 (2008) 121 MR2451292
69 G Tabuada, Homotopy theory of spectral categories, Adv. Math. 221 (2009) 1122 MR2518634
70 G Tabuada, Matrix invariants of spectral categories, Int. Math. Res. Not. 2010 (2010) 2459 MR2669656
71 Z Tamsamani, On non-strict notions of $n$–category and $n$–groupoid via multisimplicial sets arXiv:alg-geom/9512006
72 R W Thomason, T Trobaugh, Higher algebraic $K$–theory of schemes and of derived categories, from: "The Grothendieck Festschrift, Vol. 3" (editors P Cartier, L Illusie, N M Katz, G Laumon, K A Ribet), Progr. Math. 88, Birkhäuser (1990) 247 MR1106918
73 B Toën, The homotopy theory of $dg$–categories and derived Morita theory, Invent. Math. 167 (2007) 615 MR2276263
74 B Toën, G Vezzosi, A remark on $K$–theory and $S$–categories, Topology 43 (2004) 765 MR2061207
75 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 MR1764202
76 J B Wagoner, Delooping classifying spaces in algebraic $K$–theory, Topology 11 (1972) 349 MR0354816
77 F Waldhausen, Algebraic $K$–theory of topological spaces. II, from: "Algebraic topology" (editors J L Dupont, I H Madsen), Lecture Notes in Math. 763, Springer (1979) 356 MR561230
78 F Waldhausen, Algebraic $K$–theory of spaces, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 318 MR802796