Gottlieb's theorem for \({\mathcal F}\)-fibrations (Q5936532)

For a given fixed fibre \(F\), the compact-open space of (base point preserving) self homotopy equivalences of \(F\) is \(aut F\) (respectively \(aut_*F\)). Gottlieb constructed a fibration NEWLINE\[NEWLINEF\hookrightarrow B aut_* F\to B aut FNEWLINE\]NEWLINE which classifies fibre-homotopy classes of fibrations with fibre \(F\). Beginning with a category \(\mathcal F\) which has as objects compactly-generated spaces with perhaps additional structure and as morphisms homotopy equivalences, May developed a general theory of \(\mathcal F\)-fibrations. With a multi-layered definition of \(\mathcal F\)-fibrations, May ensured that fibres turn out to be objects in the category \(\mathcal F\). The author generalizes Gottlieb's theorem to \(\mathcal F\)-fibrations.