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At the end (at 42:11) of his Voevodsky Memorial talk Pierre Deligne poses a question (for the HoTT community?).
Given α:πn+k(Sn), or an element in [Sn+k,Sn] does it induce [α]:a=na→a=n+ka, and do so universally?
a=na for a:A stands for the iterated loop space of A, Ωn(A).
So is he asking whether α induces a map [[Sn,A],[Sn+k,A]] (which it surely does), and so a map from Ωn(A) to Ωn+k(A)?
What’s the challenge?
The maps of fundamnetal groups are truncated to the set level. I think he is asking if the truncation can be rid of. Can this be extended to a map of untruncated loop spaces.
There was some confusion at the meeting about this too. It’s already an interesting question at the level Ali interpreted it; but it seems that Deligne intended his “πn+k(Sn)” to refer to the homotopy groups of spheres as defined in ZFC. In other words, if we know classically that we have some element of a homotopy group of spheres, can we use that to define in type theory a corresponding map on loop spaces.
And what can we say to the question as posed in #2 and #3? Is there something one can ’reasonably’ ask for, that can’t be delivered?
If it’s “reasonable” to ask for a map πn+k(Sn)→ΩnX→Ωn+kX, then yes; I don’t think we can deliver that. But even in ZFC such a map doesn’t really exist canonically. You can choose a representative of any α∈πn+k(Sn), which in turn induces a map ΩnX→Ωn+kX, and equivalent representatives of α induce homotopic maps, but not uniquely homotopic ones. So the axiom of choice can give you a particular map, but it’s not canonically determined.
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