Weak sectional category

Based on a Whitehead-type characterization of the sectional category we develop the notion of weak sectional category. This is a new lower bound of the sectional category, which is inspired by the notion of weak category in the sense of Berstein-Hilton. We establish several properties and inequalities, including the fact that the weak sectional category is a better lower bound for the sectional category than the classical one given by the nilpotency of the kernel of the induced map in cohomology. Finally, we apply our results in the study of the topological complexity in the sense of Farber.