  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.
