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Universal homotopy theories. (English) Zbl 1009.55011
Author’s summary: “Begin with a small category \({\mathcal C}\). The goal of this short note is to point out that there is such a thing as a ‘universal model category built from \({\mathcal C}\)’. We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, sheaf theory, and the homotopy theory of schemes”.

MSC:
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
18E20 Categorical embedding theorems
18C99 Categories and theories
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[1] M. Artin, Grothendieck topologies, seminar notes, Harvard University, Department of Mathematics, 1962. · Zbl 0208.48701
[2] Artin, M.; Mazur, B., Étale homotopy, Lecture notes in mathematics, 100, (1969), Springer-Verlag Berlin · Zbl 0182.26001
[3] Adámek, J.; Rosicky, J., Locally presentable and accessible categories, (1994), Cambridge University Press Cambridge · Zbl 0795.18007
[4] Beke, T., Sheafifiable homotopy model categories, Math. proc. Cambridge philos soc., 129, 447-475, (2000) · Zbl 0964.55018
[5] Borceux, F., Handbook of categorical algebra. II. categories and structures, (1994), Cambridge University Press Cambridge · Zbl 0843.18001
[6] Bousfield, A.K., The localization of spaces with respect to homology, Topology, 14, 133-150, (1975) · Zbl 0309.55013
[7] Brown, K.; Gersten, S., Algebraic K-theory and generalized sheaf cohomology, Trans. amer. math. soc., 186, 419-458, (1973)
[8] Bousfield, A.K.; Friedlander, E.M., Homotopy theory of γ-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory, II, Springer lecture notes in mathematics, 658, (1978), Springer-Verlag New York/Berlin, p. 80-130 · Zbl 0405.55021
[9] Bousfield, A.K.; Kan, D.M., Homotopy limits, completions, and localizations, Springer lecture notes in mathematics, 304, (1972), Springer-Verlag New York · Zbl 0259.55004
[10] W. Chachólski, and, J. Scherer, Homotopy meaningful constructions: Homotopy colimits, preprint.
[11] Dugger, D., Replacing model categories by simplicial ones, Trans. amer. math. soc., 353, 5003-5027, (2001) · Zbl 0974.55011
[12] Dugger, D., Combinatorial model categories have presentations, Adv. math., 164, 177-201, (2001) · Zbl 1001.18001
[13] D. Dugger, S. Hollander, and, D. Isaksen, Hypercovers and simplicial presheaves, preprint. · Zbl 1045.55007
[14] W. G. Dwyer, P. S. Hirschhorn, and, D. M. Kan, Model categories and more general abstract nonsense, in preparation.
[15] Dwyer, W.G.; Kan, D.M., Function complexes in homotopical algebra, Topology, 19, 427-440, (1980) · Zbl 0438.55011
[16] Heller, A., Homotopy theories, Memoirs American mathematical society, 71, (1988), American Mathematical Society Providence
[17] P. S. Hirschhorn, Localization of model categories, preprint, 1998. [Available at http://www-math.mit.edu/∼psh].
[18] Hovey, M., Model categories, Mathematical surveys and monographs, 63, (1999), American Mathematical Society Providence
[19] Jardine, J.F., Simplicial objects in a Grothendieck topos, Contemp. math., 55, 193-239, (1986) · Zbl 0606.18006
[20] Jardine, J.F., Simplicial presheaves, J. pure appl. algebra, 47, 35-87, (1987) · Zbl 0624.18007
[21] A. Joyal, unpublished letter to A. Grothendieck.
[22] Makkai, M.; Paré, R., Accessible categories: the foundations of categorical model theory, Contemporary mathematics, 104, (1989), American Mathematical Society Providence · Zbl 0703.03042
[23] Morel, F.; Voevodsky, V., \(A\)^1-homotopy theory of schemes, Inst. hautes études sci. publ. math., 90, 45-143, (2001) · Zbl 0983.14007
[24] Quillen, D., Homotopical algebra, Springer lecture notes in mathematics, 43, (1969), Springer-Verlag Berlin
[25] Schwede, S., Stable homotopical algebra and γ-spaces, Math. proc. Cambridge philos. soc., 126, 329-356, (1999) · Zbl 0920.55011
[26] J. Smith, Combinatorial model categories, in preparation.
[27] Thomason, R., Algebraic K-theory and étale cohomology, Ann. sci. école norm. sup. (4), 18, 437-552, (1985) · Zbl 0596.14012
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