I was browsing through survey on Lusternik-Schnirelman category and I became curious is it possible to give the definition of the category using a more invariant approach than the classical definition?
In particular, it would be nice if we could define the category as a functor on certain stable homotopy category of spaces. Of course, we could not expect the usual notion of stability to work here since the suspension destroys any meaningful information about the category. Perhaps this could be fixed by applying some functor other than suspension?
Or maybe a notion of the category is of purely unstable nature in all reasonable senses and this is a meaningless question. (after all it comes from differential geometry)