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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)

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    $\begingroup$ What do you mean by a more 'invariant' approach? Why exactly do you want it to be some functor on some 'stable' homotopy category? $\endgroup$
    – Tyrone
    Dec 6, 2019 at 12:55
  • $\begingroup$ @Tyrone Just seems to me like a direction in which "good" concepts in homotopy theory work. It looks like usually stable invariants are more accessible to computation. Perhaps I'm wrong and you can educate on the subject. $\endgroup$
    – Grisha Taroyan
    Dec 7, 2019 at 20:45
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    $\begingroup$ Stable invariants, by definition, are invariants of stable phenomena. As you point out, LS category may not really be a stable phenomenon. This was the impetus for my question: it seems more natural to me to ask what information needs to be filtered out of the homotopy category to improve calculability, rather than to assume that this information must be of stable nature. In fact, when you get to chapter 6 and study Hopf invariants, you will come to understand that LS category is very much an unstable phenomenon. $\endgroup$
    – Tyrone
    Dec 8, 2019 at 14:13
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    $\begingroup$ Being honest, this is one reason why I personally don't find stable homotopy to be very interesting, since it tends to destroy a lot of the actual interseting mathematics going on. $\endgroup$
    – Tyrone
    Dec 8, 2019 at 14:13
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    $\begingroup$ My short answer: no one knows what information to filter away to make LS category more calculable and still retain all its information. This is why the book defines about 10000 other related invariants and spends time comparing them. $\endgroup$
    – Tyrone
    Dec 8, 2019 at 14:16

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