Rational relative sectional category

Abstract

We develop an algebraic model for the relative sectional category of a continuous map in rational homotopy theory using commutative differential graded algebras (CDGAs). Our main result establishes that for formal maps, the rational relative sectional category can be computed purely from cohomology, using ideal nilpotency. We also show that this equality may fail in general topological settings. Applying this framework, we obtain purely algebraic characterizations for the rational Lusternik-Schnirelmann category and the rational higher topological complexity of a map. Finally, we provide an algebraic description of the rational homotopic distance between formal maps.