zbMATH — the first resource for mathematics

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”.

55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
18E20 Categorical embedding theorems
18C99 Categories and theories
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.