Abstract
Consider pairs ( X , A ) (\mathcal {X},\mathcal {A}) where X = ( X , p , B ) \mathcal {X} = (X,p,B) and A = ( A , p | A , B ) \mathcal {A} = (A,p|A,B) are Hurewicz fibrations mapping onto B and A ⊂ X A \subset X . It is proved that ( X , A ) (\mathcal {X},\mathcal {A}) is a cofibration if and only if ( X ∪ f Y , Y ) (\mathcal {X}{ \cup _f}\mathcal {Y},\mathcal {Y}) is a strongly-paired fibration for each fibration Y = ( Y , q , B ) \mathcal {Y} = (Y,q,B) and fiber map f : A → Y f:\mathcal {A} \to \mathcal {Y} . It follows as a corollary that the notions of fiber homotopy equivalence and strong fiber homotopy equivalence [5] coincide for all Hurewicz fibrations. That ( X , A ) (\mathcal {X},\mathcal {A}) be “strongly-paired” requires more than that each lifting function for A \mathcal {A} be extendable to X \mathcal {X} . This and other notions of pairing are studied.
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