The example you refer to (April 22) uses the idea of a placement of a body $B$ in an environment $E$, and notes that a path in the space of placements corresponds uniquely to a placement of $B$ in the space of paths, because both correspond to a lower-order map from $I\times B$ to $E$ itself. These correspondences are invertible, as well as smooth, recursive, etc. (i.e. they preserve whatever structure characterizes the ambient category of spaces). But the transformations are far from banal because it is the smoothness of functionals on the map spaces that the sought structure controls, and composing paths or placements with possible functionals yield new quantities of very diverse needed kinds.
This is described for example in Article V, Session 30, and Session 31 of Conceptual Mathematics (Lawvere & Schanuel). My paper about Volterra's functionals (published in the RCM Palermo in 2000 and downloadable from my Buffalo homepage) discusses the unsuitability for functional analysis (as well as for homotopy theory) of the attempt to characterize continuity or cohesion using open sets or other contravariant structure.
That unsuitability was pointed out by Hurewicz who formulated the necessary exponential laws and inspired Fox (BAMS 1945) to make use of the compact open construction that had been defined by Lefschetz in 1942. Hurewicz himself introduced, in Princeton lectures in the late 1940's, the k-spaces; these k-spaces seemed to be an adequate repair of the difficulty and so led to later studies with various refinements by Kelley, Steenrod, and (of course) yourself.
However, an equally serious flaw in the traditional definition of continuity had already been revealed by Peano's construction of space-filling curves, suggesting that a return to Frechet's covariant conception based on paths or on Eilenberg's singular figures should be considered. This is almost trivially the basis for the success of exponentials in simplicial sets and similar categories, including those studied in synthetic differential geometry; but the analysis given by Johnstone as a prelude to his important topological topos (which of course still contains the Peano pathologies) reveals that there are still significant foundational issues involved in the appropriate definition of a monoid of continuous reparameterizations of paths; these issues were emphasized by Grothendieck in his Tame Topology and have now been partially addressed by the o-minimal theorists.
(Hurewicz used the k to abbreviate 'kompakt erzeugte')
