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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 28th 2018
   • (edited Sep 28th 2018)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAt the end (at 42:11) of his Voevodsky Memorial [talk](https://www.youtube.com/watch?v=WfDcrN5_1wA) Pierre Deligne poses a question (for the HoTT community?). Given $\alpha: \pi_{n+k}(S^n)$, or an element in $[S^{n+k}, S^n]$ does it induce $[\alpha]: a =_n a \to a =_{n+k} a$, and do so universally? $a =_n a$ for $a:A$ stands for the iterated loop space of $A$, $\Omega^n(A)$. So is he asking whether $\alpha$ induces a map $[[S^n, A], [S^{n+k}, A]]$ (which it surely does), and so a map from $\Omega^n(A)$ to $\Omega^{n+k}(A)$? What's the challenge?

   At the end (at 42:11) of his Voevodsky Memorial talk Pierre Deligne poses a question (for the HoTT community?).

   Given , or an element in does it induce , and do so universally?

   for stands for the iterated loop space of , .

   So is he asking whether induces a map (which it surely does), and so a map from to ?

   What’s the challenge?

2. • CommentRowNumber2.
   • CommentAuthorAli Caglayan
   • CommentTimeSep 28th 2018
   • (edited Sep 28th 2018)
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   Author: Ali Caglayan
   Format: MarkdownItexThe 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeSep 28th 2018
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   Author: Mike Shulman
   Format: MarkdownItexThere 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 "$\pi_{n+k}(S^n)$" 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.

   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 “” 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.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 30th 2018
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   Author: David_Corfield
   Format: MarkdownItexAnd 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?

   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?

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeSep 30th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIf it's "reasonable" to ask for a map $\pi_{n+k}(S^n) \to \Omega^n X \to \Omega^{n+k} X$, 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 $\alpha \in \pi_{n+k}(S^n)$, which in turn induces a map $\Omega^n X \to \Omega^{n+k} X$, and equivalent representatives of $\alpha$ 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.

   If it’s “reasonable” to ask for a map , 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 , which in turn induces a map , and equivalent representatives of 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.