Systems of diagram categories and K-theory. II. (English) Zbl 1074.19001
[For Part I see arxiv.org/abs/math/0401062 (2004).]
Summary: The additivity theorem for dérivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a \(K\)-theory space is associated. This \(K\)-theory is shown to satisfy the additivity, approximation and resolution theorems.
Summary: The additivity theorem for dérivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a \(K\)-theory space is associated. This \(K\)-theory is shown to satisfy the additivity, approximation and resolution theorems.
MSC:
| 19D99 | Higher algebraic \(K\)-theory |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 18E05 | Preadditive, additive categories |
References:
| [1] | Cisinski, D.-C.: Catégories dérivables. Preprint, 2002 |
| [2] | Cisinski, D.-C.: An e-interchange, January 2004 |
| [3] | Dwyer, W.G., Kan, D.M.: Simplicial localizations of categories. J. Pure Appl. Alg. 17(3), 267-284 (1980) · Zbl 0485.18012 · doi:10.1016/0022-4049(80)90049-3 |
| [4] | Dwyer, W.G., Kan, D.M.: Calculating simplicial localizations. J. Pure Appl. Alg. 18(1), 17-35 (1980) · Zbl 0485.18013 · doi:10.1016/0022-4049(80)90113-9 |
| [5] | Franke, J.: Uniqueness theorems for certain triangulated categories with an Adams spectral sequence. K-theory Preprint Archives 139, (1996) |
| [6] | Garkusha, G.: Systems of diagram categories and K-theory. I. Preprint, 2003 |
| [7] | Grothendieck, A.: Les Dérivateurs. Manuscript, 1990 |
| [8] | Heller, A.: Homotopy theories. Mem. Am. Math. Soc. 71(383), (1988) · Zbl 0643.55015 |
| [9] | Karoubi, M.: Foncteurs dérivés et K-théorie. In: Séminaire Heidelberg-Saarbrücken-Strasbourg sur la K-théorie 1967/68, Lecture Notes in Mathematics, No. 136, Springer-Verlag, 1970, pp. 107-186 |
| [10] | Keller, B.: Chain complexes and stable categories. Manus. Math. 67, 379-417 (1990) · Zbl 0753.18005 · doi:10.1007/BF02568439 |
| [11] | Keller, B.: Derived categories and their uses. In: Handbook of Algebra, vol. 1, North-Holland, Amsterdam, 1996, pp. 671-701 · Zbl 0862.18001 |
| [12] | Keller, B.: Le dérivateur triangulé associé à une catégorie exacte. Preprint, 2002 |
| [13] | Mac Lane, S.: Categories for the working mathematician. 2nd edn. Graduate Texts in Mathematics 5, Springer-Verlag, New-York, 1998 · Zbl 0906.18001 |
| [14] | Maltsiniotis, G.: La K-théorie d?un dérivateur triangulé. Preprint, 2002 |
| [15] | McCarthy, R.: On fundamental theorems of algebraic K-theory. Topology 32(2), 325-328 (1993) · Zbl 0818.55001 · doi:10.1016/0040-9383(93)90023-O |
| [16] | Pedersen, E.K., Weibel, C.A.: K-theory homology of spaces. In: Algebraic topology, Arcata/CA 1986, Lecture Notes in Mathematics, No. 1370, Springer-Verlag, 1989, pp. 346-361 |
| [17] | Schlichting, M.: Délaçage de la K-théorie des catégories exactes et K-groupes négatifs, Thése, Université Paris 7, 2000 |
| [18] | Schlichting, M.: A note on K-theory and triangulated categories. Inv. Math. 150, 111-116 (2002) · Zbl 1037.18007 · doi:10.1007/s00222-002-0231-1 |
| [19] | Schlichting, M.: Delooping the K-theory of exact categories. To appear Topology · Zbl 1059.18007 |
| [20] | Segal, G.: Categories and cohomology theories. Topology 13, 293-312 (1974) · Zbl 0284.55016 · doi:10.1016/0040-9383(74)90022-6 |
| [21] | Staffeldt, R.E.: On fundamental theorems of algebraic K-theory. K-theory 1, 511-532 (1989) · Zbl 0665.18010 · doi:10.1007/BF00533280 |
| [22] | Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift III, Progress in Mathematics 88, Birkhäuser, 1990, pp. 247-435 · Zbl 0731.14001 |
| [23] | Toën, B., Vezzosi, G.: Remark on K-theory and S-categories. Topology 43(4), 765-791 (2004) · Zbl 1054.55004 |
| [24] | Waldhausen, F.: Algebraic K-theory of generalized free products. Ann. Math. 108, 135-256 (1978) · Zbl 0397.18012 · doi:10.2307/1971165 |
| [25] | Waldhausen, F.: Algebraic K-theory of spaces. In: Algebraic and geometric topology. Proc. Conf., New Brunswick/USA 1983, Lecture Notes in Mathematics, No. 1126, Springer-Verlag, 1985, pp. 318-419 |
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