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2 GEORGE W . WHITEHEA D right formulatio n i s no t ye t clear . On e suc h theore m ha s bee n foun d b y D . S . Kah n i n his thesi s (unpublished) . Fo r a genera l discussio n o f thes e problems , th e reade r i s re - ferred t o Adams ' Seattl e lecture s [7] . 1. Preliminarie s In wha t follows , al l space s wil l b e assume d t o b e provide d wit h base-points , an d all map s an d homotopie s t o preserv e th e base-point th e base-poin t o f an y spac e wil l be denote d indiscriminatel y b y a n asterisk . Moreover , base-point s o f CT-complexe s are assume d to b e vertices . Categorie s o f space s an d map s withou t base-poin t (her e termed "fre e s p a c e s " an d "fre e maps" ) wil l b e assume d t o b e imbedde d i n th e corre - sponding base d categorie s b y th e devic e o f adjoinin g a n externa l base=point . Let S Q b e th e categor y o f compactl y generate d Hausdorf f space s [68 ] havin g th e homotopy typ e o f a CJF-complex . I f X, A ar e object s o f S Q wit h A a close d subspac e of X, th e pai r {X, A) i s sai d t o b e admissible i f an d onl y i f th e inclusio n ma p o f A into X is in th e categor y S Q (i.e. , X an d A hav e th e sam e base-point ) an d i s a cofibration. I t i s easil y see n tha t (1.1) / / A is the base-point of X, then (X, A) is admissible. (1.2) If (Z , A) is an admissible pair in S Q , then there is a CW-pair {K 7 L) such that {X, A) and {K, L) have the same homotopy type. Let {Xj A) b e a n admissibl e pai r i n S Q . The n th e quotien t spac e X/A i s th e identification spac e obtaine d fro m X b y collapsin g A t o a point , th e base-poin t o f X/A X/A belong s t o § 0 . Recall tha t th e categor y S ha s (finite ) product s an d sums i f X, Y € S Q , the n their categorica l produc t X x Y i s th e cartesia n produc t suitabl y retopologized . More - over thei r su m X V Y = X x {* ! \J {* ! x Y i s a subspac e o f X x Y , an d th e pai r (X x Y , X V Y ) i s admissible . Henc e thei r reduced join X A Y = X x Y/X V Y i s de - fined an d belong s t o S Q . Moreover , th e reduce d joi n functor i s defined map s /: X -+X' 9 g: Y -+Y' induc e a m a p / A g : X A X ' _ Y A Y ' wit h th e usua l properties. On e o f th e advantage s o f workin g i n th e categor y S i s (1.3) The reduced join functor is commutative and associative, up to natural homeomorphism. Let X, Y , Z € S 0 , an d le t i^ X - * X V Y , i^ Y - X V Y b e th e natura l inclu - sions. The n i x A 1 : X A Z - » (X V Y ) A Z an d % 2 A 1 : Y A Z — {X V Y ) A Z induc e a ma p h-. U A Z ) V ( y A Z ) - , ( i v y ) A Z and i t i s easil y verifie d tha t h i s a homeomorphism . Henc e (1.4) TA e operation of reduced join is distributive over that of addition, up to