Whitehead and Ganea constructions for fibrewise sectional category

We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibrewise sectional category is the analogue of the ordinary sectional category in the fibrewise setting and also the natural generalization of the fibrewise unpointed LS category in the sense of Iwase-Sakai. On the other hand the fibrewise pointed version is the generalization of the fibrewise pointed LS category in the sense of James-Morris. After giving their main properties we also establish some comparisons between such two versions.