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Faithfulness of a functor of Quillen. (English) Zbl 1227.18014

Summary: There exists a canonical functor from the category of fibrant objects of a model category modulo cylinder homotopy to its homotopy category. We show that this functor is faithful under certain conditions, but not in general.

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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References:

[1] K S Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1974) 419 · Zbl 0245.55007
[2] D Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Math. Society Lecture Note Ser. 119, Cambridge Univ. Press (1988) · Zbl 0635.16017
[3] P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003) · Zbl 1017.55001
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[5] D G Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer (1967) · Zbl 0168.20903
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