The sectional category of a map

We study a generalization of the Svarc genus of a fiber map. For an arbitrary collection E of spaces and a map f:X-->Y, we define a numerical invariant, the E-sectional category of f, in terms of open covers of Y. We obtain several basic features of E-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple properties which characterize the E-sectional category. In the final section we obtain inequalities for the E-sectional category of a composition and inequalities relating the E-sectional category to the Fadell-Husseini category of a map and the Clapp-Puppe category of a map.