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Simplicial approximation. (English) Zbl 1062.55019
D. M. Kan defined a homotopy theory for simplicial sets using Kan complexes and Kan fibrations. The homotopy category obtained in this way and the homotopy category of topological spaces are equivalent. Later, D. Quillen gave a model category structure for the category of simplicial sets. In this paper, a new approach to the construction of the homotopy theory of simplicial sets is given. The equivalence with the homotopy theory of topological spaces is proved using simplicial approximation techniques. The author initially uses cofibrations and weak equivalences of simplicial sets to define fibrations as those maps which have the right lifting property with respect to all trivial cofibrations. In this way, a model structure for simplicial sets is easily obtained. Fibrations are Kan fibrations. Proving that all Kan fibrations are fibrations and deriving the equivalence of homotopy categories is the subject of the rest of the paper.

##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 55U35 Abstract and axiomatic homotopy theory in algebraic topology
##### Keywords:
simplicial sets; simplicial approximation; model category
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