A note on paired fibrations

Abstract:

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