A model for the homotopy theory of homotopy theory

Rezk, Charles

doi:10.1090/S0002-9947-00-02653-2

A model for the homotopy theory of homotopy theory

Author: Charles Rezk
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
Published electronically: June 20, 2000
MathSciNet review: 1804411
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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.

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

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 https://doi.org/10.1016/0040-9383%2880%2990025-7

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 https://doi.org/10.1016/0040-9383%2884%2990035-1

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 https://doi.org/10.1016/0022-4049%2893%2990126-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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0022-4049%2887%2990100-9

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

Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI https://doi.org/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 https://doi.org/10.1016/0040-9383%2874%2990022-6

R. W. Thomason, Uniqueness of delooping machines, Duke Math. J. 46 (1979), no. 2, 217–252. MR 534053