idc, such that for all objects X pi,:X - IX -* X is the identity of X for r = 0 and e = 1. 1). (12) Push out axiom: For a cofibration i:B-+A and a map f there exists the push out 3 Categories with a natural cylinder 19 A ------ ),AUX lB X B f where 1 is also a cofibration. Moreover, the functor I carries the push out diagram into a push out diagram, that is I(AUBX)=IAUIBIX. Moreover, 10 = 0. (I3) Cofibration axiom: Each isomorphism is a cofibration and for each object X the map 0 -> X is a cofibration.

CX=limX" of cofibrations in Top is given with the following properties: X° is the disjoint union of A with a discrete set Z° and X" is obtained by a push out diagram (n>1)inTop U D"cZ _+ X" U U USn-1 Z. 4 , Xn-1 fn where 0 denotes the disjoint union. The space X" is the relative n-skeleton of (X, A) and f" is the attaching map of n-cells, cn is the characteristic map of ncells. If A is empty we call X a CW-complex. A CW-space is a topological space which is homotopy equivalent in Top to a CW-complex.

7). 3) Theorem. Let (C, cof, I, 4)) be an I-category. Then C is a cofibration category with the following structure. Cofbbrations are those of C, weak equivalences are the homotopy equivalences and all objects are fibrant and cofibrant in C. Remark. Assume the initial object 0 = * is also a final object. Then we obtain the suspension functor E:C-+C by the push out diagram IA ) EA J push A+A in C. 10). We leave the proof of this useful fact as an exercise, compare Chapter II. 3) we derive some useful facts from axioms (I 1), ...