ABSTRACT

Some properties of weak factorization systems . . . . . . . . . . . . . . . . . . 106 Orthogonal factorization systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Tierney Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.4 The homotopy category of a model category . . . . . . . . . . . . . . . . . . . . 117 Homotopy categories via localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Homotopy categories via cylinder and path objects . . . . . . . . . . . . . 119

5.5 Monoidal model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1 Introduction The definition of a model category was introduced by Daniel Quillen in

1967 [Qui67] to formalize some notions and constructions from homotopy theory in the more general context of category theory. Quillen’s definition was motivated by the category of topological spaces and the category of simplicial sets, which play an important role in algebraic topology, and by the category of (bounded) chain complexes, in a more algebraic setting.