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
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Trans. Amer. Math. Soc. 353 (2001), 973-1007 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