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Zbl 1030.55015
Rosický, J.; Tholen, W.
Left-determined model categories and universal homotopy theories. (English)
[J] Trans. Am. Math. Soc. 355, No. 9, 3611-3623 (2003). ISSN 0002-9947; ISSN 1088-6850/e

A {\it left-determined model category} $\cal K$ has the property that the class of weak equivalences $\cal W$ is determined by the class of cofibrations. More precisely: Let $\tilde{\cal W}$ be the smallest class of morphisms satisfying: 1) ${\cal C}^{\square} \subseteq \tilde{\cal W}$. 2) $\tilde {\cal W}$ is closed under retracts and satisfies the 2-out-of 3 property. 3) $\cal C \cap \tilde{\cal W}$ is stable under pushouts and closed under transfinite composition. Here ${\cal L}^{\square}$ denotes the class of all morphisms having the right lifting property with respect to $\cal L$. A model category is left-determined, whenever $\tilde{\cal W} = \cal W$.\par The main results of the present paper are: 1) The category of simplicial sets is not left-determined (which is probably not too surprising, because the cofibrations are all inclusions). 2) The category SSimp of symmetric simplicial sets is left determined, as well as the category Simp of simplicial complexes. 3) The authors show that the category SSimp is a universal model category over the one-morphism category in the sense of D. Dugger. For an erratum to this paper see Trans. Am. Math. Soc. 360, No. 11, 6179--6179 (2008; Zbl 1225.55010).
[Friedrich Wilhelm Bauer (Frankfurt / Main)]

MSC 2000:: *55U35 Abstract homotopy theory
55U10 Semisimplicial complexes

Keywords: left-determined model category; weak equivalences; simplicial sets; cofibrations; universal model category

Citations: Zbl 1225.55010

Cited in: Zbl 1225.55010

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