Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-19T07:00:32.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopy colimits in the category of small categories

Published online by Cambridge University Press:  24 October 2008

R. W. Thomason
R. W. Thomason
Affiliation:
Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Extract

In (13), Quillen defines a higher algebraic K-theory by taking homotopy groups of the classifying spaces of certain categories. Certain questions in K-theory then become questions such as when do functors induce a homotopy equivalence of classifying spaces, or when is a square of categories homotopy cartesian? Quillen has given some techniques for answering such questions. F. Waldhausen has extended these ideas in (19), and broadened the range of applications to include geometric topology (20).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 1 , January 1979 , pp. 91 - 109
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Anderson, D. W. Simplicial K-theory and generalized homology theories. (Preprint.)Google Scholar
(2)Bousfield, A. K. and Kan, D. M. Homotopy limits, completions, and localizations (Springer Lecture Notes, no. 304).Google Scholar
(3)Charney, R. (TO appear.)Google Scholar
(4)Gray, J. The categorical comprehension scheme. In Category theory, homology theory, and their applications III, pp. 242–312 (Springer Lecture Notes, no. 99).Google Scholar
(5)Gray, J. Fibred and cofibred categories. Proc. Conf. Categorical Algebra, La Jolla (1965), pp. 21–83. Springer.Google Scholar
(6)Gray, J. Formal category theory: adjointness for 2-categories (Springer Lecture Notes, no. 391).Google Scholar
(7)Grayson, D. Higher algebraic K-theory II (after Quillen). In Algebraic K-theory: Evanston 1976, pp. 217–240. (Springer Lecture Notes, no. 551).CrossRefGoogle Scholar
(8)Kelly, G. M. Coherence theorems for lax algebras and for distributive laws. In Category Seminar: Sydney 1972/1973, pp. 281–375. (Springer Lecture Notes, no. 420).Google Scholar
(9)Lewis, G. Coherence for a closed functor. In Coherence in categories, pp. 148–195 (Springer Lect. Notes, no. 281).Google Scholar
(10)MacLane, S.Categories for the working mathematician (Graduate Texts in Math., Springer).Google Scholar
(11)May, J. P.E ∞ spaces, group completions, and permutative catgories. New developments in topology, pp. 61–93 (Lond. Math. Soc. Lecture Notes).Google Scholar
(12)May, J. P. The spectra associated to a permutative category. (To appear.)Google Scholar
(13)Quillen, D. Higher algebraic K-theory I. In Algebraic K-theory I (Springer Lecture Notes, 341).Google Scholar
(14)Segal, G.Categories and cohomology theories. Topology 13 (1974), 293–312.CrossRefGoogle Scholar
(15)Segal, G.Classifying spaces and spectral sequences. Inst. Hautes Études Sci. Publ. Math. 34, 105–112.CrossRefGoogle Scholar
(16)Street, R. Two constructions on lax functors. Cahiers de Topologie et Geometrie Differentielle, xii-3, pp. 217–264.Google Scholar
(17)Thomason, R. Thesis, Princeton University, 1977. Available from University Microfilms.Google Scholar
(18)Vogt, R.Homotopy limits and colimits. Math. Z. 134 (1973), 11–52.CrossRefGoogle Scholar
(19)Waldhausen, F.Algebraic K-theory of generalized free products. Ann. of Math, (in the Press).Google Scholar
(20)Waldhausen, F.Algebraic K-theory of topological spaces. Proc. 1976 AMS Summer Institute (in the Press).Google Scholar