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, and therefore of the topological complexity of Farber. On the other hand the fibrewise pointed version is the generalization of the fibrewise pointed LS category in the sense of James–Morris, and therefore of monoidal topological complexity of Iwase–Sakai. After giving the main properties for the pointed and unpointed fibrewise sectional category we also establish a comparison between such two versions. We remark a theorem that gives sufficient conditions so that the unpointed and pointed versions agree. As a corollary we obtain the known corresponding result for topological complexity and the monoidal topological complexity given by Dranishnikov.