4 GEORG E W . WHITEHEA D in it s second . I f / : X' — * X, g: Y — Y ', le t f# = [/, i] : [x , y]-*[*' , Y], g# = [i,gi= L* , y ] - t f, y' L so tha t [/ , g\ = f* ° g | = g # ° / # . Note tha t (1.11) The sets [SX, Y] and [X, Q,Y] are in natural one-to-one correspondence. (1.12) The sets [X V Y, Z ] and [X, Z ] x [Y , Z ] are in natural one-to-one correspondence. (1.13) The sets [X, Y x Z] and [X, Y ] x [A' , Z ] are in natural one-to-one correspondence. Define a ma p \jj: I — » / x O | J l x / b y tfr(*)-= (2f, 0 ) (tV 2 ) ( l , 2 i - l ) U H ) . The identificatio n ma p o f / int o S induce s a ma p p: / x 0 {J 1 x / — S V 5 an d pif/{0) - pif/{l) = * . Henc e i/ r induce s a ma p y : S — S V 5 . Let I , y 6 S 0 , an d conside r th e compositio n [SAT, Y ] x [SX, Y] — [S X V SX , y ] _ + [( S V s) A X , Y ] [S A X , Y ] in whic h th e firs t tw o map s ar e induce d b y th e natura l equivalence s o f (1.12 ) an d (1=4). (1.14) The above operation defines a group structure on [SX, Y]. (1.15) If f: X' — X and g: Y — * Y are maps, then {Sf)*: [SX, Y]-+[SX', Y] g # : [SX, Y]-^[SX, Y'] are homomorphisms . The one-to-on e correspondenc e o f [SX, Y] wit h [X, QY ] o f ( U l ) induce s a group structur e o n th e latte r set , an d w e hav e (1 c 16) If f: X* ~* X and g: Y —• Y' are maps, then / # : [X,QY]-+[X',aY] {Qg)^[X,QY]^[X,QY']

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