Functors on the category of quasi-fibrations (Q930744)

It is known that a continuous endofunctor \(\Phi\) of the category of topological spaces gives rise to a fibrewise continuous endofunctor on the category of continuous maps if \(X\subset \Phi(X)\). The authors prove that any continuous endofunctor of the category of topological spaces, without any additional condition, induces a fibrewise continuous endofunctor on the category of cell-wise trivial fibre spaces over polyhedra preserving the class of quasi-fibrations.