A model for the homotopy theory of homotopy theory
HTML articles powered by AMS MathViewer
- by Charles Rezk
- Trans. Amer. Math. Soc. 353 (2001), 973-1007
- DOI: https://doi.org/10.1090/S0002-9947-00-02653-2
- Published electronically: June 20, 2000
- PDF | Request permission
Abstract:
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.References
- D. W. Anderson, Spectra and $\Gamma$-sets, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 23–30. MR 0367990
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
- W. G. Dwyer, P. Hirschhorn, and D. M. Kan, General abstract homotopy theory, in preperation.
- W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), no. 4, 427–440. MR 584566, DOI 10.1016/0040-9383(80)90025-7
- W. G. Dwyer and D. M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23 (1984), no. 2, 139–155. MR 744846, DOI 10.1016/0040-9383(84)90035-1
- W. G. Dwyer and D. M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23 (1984), no. 2, 139–155. MR 744846, DOI 10.1016/0040-9383(84)90035-1
- W. G. Dwyer, D. M. Kan, and C. R. Stover, An $E^2$ model category structure for pointed simplicial spaces, J. Pure Appl. Algebra 90 (1993), no. 2, 137–152. MR 1250765, DOI 10.1016/0022-4049(93)90126-E
- W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, DOI 10.1016/B978-044481779-2/50003-1
- P. G. Goerss and J. F. Jardine, Simplicial homotopy theory, Progress in Math., vol. 174, Birkhäuser, Basel, 1999.
- Alex Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (1988), no. 383, vi+78. MR 920963, DOI 10.1090/memo/0383
- P. Hirschhorn, Localization in model categories, http://www-math.mit.edu/$\sim$psh.
- J. F. Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87. MR 906403, DOI 10.1016/0022-4049(87)90100-9
- J. F. Jardine, Boolean localization, in practice, Doc. Math. 1 (1996), No. 13, 245–275. MR 1405671
- J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
- Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
- C. L. Reedy, Homotopy theory of model categories, unpublished manuscript.
- Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, DOI 10.1016/0040-9383(74)90022-6
- R. W. Thomason, Uniqueness of delooping machines, Duke Math. J. 46 (1979), no. 2, 217–252. MR 534053, DOI 10.1215/S0012-7094-79-04612-X
Bibliographic Information
- Charles Rezk
- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- MR Author ID: 638495
- ORCID: 0000-0003-4111-893X
- Email: rezk@math.nwu.edu
- Received by editor(s): November 4, 1998
- Published electronically: June 20, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 973-1007
- MSC (2000): Primary 55U35; Secondary 18G30
- DOI: https://doi.org/10.1090/S0002-9947-00-02653-2
- MathSciNet review: 1804411