A map which is both a fibration and a weak equivalence is called a trivial fihration The initial object of C shall be denoted objects exist by Axiom M0). natural map cofibration. , trivial cofibration). 4) object Definition. X ~ [0,i] Let X E C. (i0,il) pl 0 = pi I = id X. X~ i for product of i 0 + iI '--x| consists of an [o,i] P~x is a cofibratlon, the map We shall frequently write il(X ). X X and a commutative diagram x_[ix where the map A cylinder object for Caution: with an object not depend functorially upon in general, [0,i]; X | X | p is a weak equivalence, end 0 for [0,i] i0(X) and is not the "tensor" in fact, in general, X | [0,i] need X.
Čech and Steenrod Homotopy Theories with Applications to Geometric Topology by David A. Edwards, Harold M. Hastings (auth.)