X 1, is applied, Ob f = Kan complexes, Obc = Ob. 11) to the cofibration category of simplicial sets. 12) one obtains the next example. 13) Closed model category (Bousfield (1975)). C = category of simplicial I Axioms and examples 18 sets, cof = injective maps, fib = M (cof, we) and we = maps f :K -+L which induce isomorphisms f*:k*(K) = k*(L), Ob f = k*-local Kan complexes, Obc = Ob.

9). (4) We call a a model functor if a preserves weak equivalences and if a is compatible with all push outs as in (3). Hence a model functor a carries homotopy cocartesian diagrams in C to homotopy cocartesian diagrams in K. 11 We will see that a based model functor is compatible with most of the constructions in a cofibration category described in this book. In general, we do not assume that a model functor carries a cofibration in C to a cofibration in K. 1) Definition. A fibration category is a category F with the structure (F, fib, we), subject to axioms (F1), (F2), (F3) and (F4).

Let (ChainR ), be the category of free chain complexes, which are bounded below, and of chain maps. 2). Clearly, the trivial chain complex V = 0 is the initial (and the final) object of Chain,. 10) Proposition. 2) yield the structure of an I-category on (Chain'' ),. 11) Corollary. 3)) chain complexes which are bounded below. Then f is a homotopy equivalence in the I-category (Chain' )c. 9). 10). 8). The push out axiom (12) is satisfied since for graded modules V, W we have (V (D W) QO I = V OO I O+ W QO I.