4 I. AN OVERVIEW (ii) strengthening his factorization axiom by requiring the therein mentioned factorizations to be functorial, and we will therefore use the term model category for a closed model category which satisfies these stronger versions of Quillen's limit and factorization axioms. 2.2. Categories. We will use the term locally small category for what one usually calls a category, i.e. a category with small horn-sets, and reserve the term category for a more general notion which ensures that (i) for every two categories C and D , the functors C D and the natural transformations between them also form a category (which is usually denoted by D ), and (ii) for every category C and subcategory W C C, there exists a localiza- tion of C with respect to W, i.e. a category C[T^_1] together with a localization functor 7: C —* C[W~ ] which is 1-1 and onto on objects and sends the maps of W to isomorphisms in C[W _ 1 ] , with the univer- sal property that, for every category B and functor b: C B which sends the maps of W to isomorphisms in B, there is a unique functor b1: C[W~l) - B such that b^ = b. It is also convenient to consider, for use in this chapter only, the notion of 2.3. Categories with weak equivalences. By a category with weak equivalences or we-category we will mean a category C with a single distin- guished class W of maps (called weak equivalences) such that (i) W contains all the isomorphisms, and (ii) W has Quillen's two out of three property that, for every pair of maps f,g £ C such that gf exists, if two of / , g and gf are in W, then so is the third which readily implies that (iii) W is a subcategory of C. For such a we-category C one then can (iv) define the homotopy category HoC of C as the category obtained from C by "formally inverting" the weak equivalences, i.e. the category with the same objects as C in which, for every pair of objects X,Y G C, the horn-set Ho C(X, Y) consists of the equivalence classes of zigzags in C from X to Y in which the backward maps are weak equivalences, by the weakest equivalence relation which puts two zigzags in the same class when one can be obtained from the other by (a) omitting an identity map, (b) replacing two adjacent maps which go in the same direction by their composition, or (c) omitting two adjacent maps when they are the same but go in op- posite directions, note that this homotopy category Ho C, together with the functor 7: C -* Ho C which is the identity on the objects and which sends a map c: X —» Y C to the class containing the zigzag which consists of the map c only, is a localization of C with respect to its subcategory of weak equivalences (2.2(ii)) and call C saturated whenever a map in C is a weak equivalence
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