By S. Mardesic, J. Segal
Form idea: The Inverse procedure strategy
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Extra info for Shape theory : the inverse system approach
T O prove that ( 4 ) is commutative we need to show that for every μ G M ^ holds gf : X in = /μΡ*(μ) Sh. T h ; shape ( ) 5 morphism q^F : Χ ^ Υ μ is given by h = ( Υ ) , where g : Y —> ( Υ ) is given by 1 : Υ - > Υ . Consequently, μ μ μ Ii is the morphism of p r o - ^ given by f μ satisfies hp = / ρ μ ί Μ μ ) : Χ< Μ μ ) μ μ - > Υ . On the other hand, h μ , so that / ρ < ) as a shape morphism is also given by h μ Μμ and thus ( 5 ) holds. The uniqueness of F follows from the next lemma. C H . 3] A B S T R A C T SHAPE L E M M A 1.
1). , an infinite bouquet of circles as in Fig. 2. There is no rudimentary ^-expansion ρ : X — • ( Y ) . Indeed, any Fig. 2. Y 6 ^ has a finitely generated first homology group Η (Y). Therefore, if λ P Q X is the bouquet formed by the first r circles of X , and if r exceeds the rank of H i ( Y ) , then the obvious retraction h : X - * P cannot be factored homotopically through Y Consequently, property ( A E 1 ) fails to hold. E X A M P L E 3. It is a basic fact of shape theory that for ÏÏ and & as in Example 2 every metric compactum X admits an inverse sequence of compact A N R ' s X = (X ,p n+\) n n and a morphism ρ : X ^ X in pro-£T such that ρ is a ^-expansion of X (cf.
Let X , Y G J and let F:X—>Y morphism g G S T ( y , p ) , be a shape morphism. For every P G ^ , the shape morphism g F E S h ( X , P ) is actually a morphism / Ε 3~(X, P ) (3, Theorem 4). W e have thus defined for every Ρ Ε 9 a function F : 2Γ(Υ, P ) ^ 2T(X P ) , F (g) = f = gF. Notice P that for ν Ε 9(P\ y P P) vg' = g (1) «/' = / (2) implies because = vg'F = gF = f. If for two shape morphisms F, F ' : X — » Y we have F = F P P for every F, then F = F ' . This is because for every g G ? ( Y , F ) we have gF = gF' and applying Lemma 1 we get the desired conclusion.
Shape theory : the inverse system approach by S. Mardesic, J. Segal