Extension of functors to fibrewise pointed spaces

Abstract

We describe a new general method for the fibrewise extension of a given endofunctor on the category of pointed topological spaces to the category of fibrewise pointed spaces. We derive some properties of the construction and show how it can be profitably used to build the Whitehead-Ganea framework for the fibrewise Lusternik-Schnirelmann category and the topological complexity.