Fibrewise smash product and fibrewise mapping spaces (Q1295250)

An object \(X\) of a category \({\mathcal C}\) with finite products is called exponentiable provided that the endofunctor \((-)\times X:{\mathcal C}\to{\mathcal C}\) is coadjoint (i.e., has a right adjoint). For the category \({\mathcal T}op/B\) of fibrewise topological spaces (over a base space \(B)\) the exponentiable objects have been characterized by \textit{S. B. Niefield} [J. Pure Appl. Algebra 23, 147-167 (1982; Zbl 0475.18011)]. The authors characterize the smash-exponentiable objects in the category \({\mathcal T}op/B_*\) of fibrewise pointed topological spaces \((X,p,s)\) (over \(B\) with a fixed section) by showing that \((X,p,s)\) is smash-exponentiable (i.e., the smash product functor \((-)\wedge_B X:{\mathcal T}op/B_*\to {\mathcal T}op/B_*\) is coadjoint) iff \((X,p)\) is exponentiable. In particular the smash-exponentiability of \((X,p,s)\) does not depend on the section \(s\).