Timeline for Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2020 at 22:12 | comment | added | Simon Henry | ... for stricter version of $3$-groupoids, like Gray-groupoids. Finally, I don't know if I should mention it (because it is still a bit speculative) but last summer Christopher Dean announced in a talk new results about constructing an "$\infty$-category of $\infty$-category" that might (if I understood his talk correctly) be enough to finish the proof of the homotopy hypothesis following my previous work, but these are not out yet, so I don't know yet. So... there is still some hope for a proof in the near future. | |
| Aug 21, 2020 at 22:03 | comment | added | Simon Henry | No still not. except the result I mentioned in this post, the only new things is the construction by Lanari of the model structure for Grothendieck 3-groupoids (arxiv.org/abs/1809.07923) and a joint paper of Lanari and I where we generalize the result mentioned above to a proof that the existence of the "folk" model struture on n-groupoids implies the version of homotopy hypothesis for n-groupoids (arxiv.org/abs/1905.05625). Together, they proove the homotopy hypothesis for 3-groupoids, which is nice, but not so striking as this was already know... | |
| Aug 21, 2020 at 2:09 | comment | added | David Roberts♦ | Simon, do we know the homotopy hypothesis yet for Grothendieck's definition of $\infty$-groupoid? I vaguely recall progress, but not enough to remember where and what. | |
| Nov 26, 2018 at 19:31 | comment | added | Simon Henry | @MikeShulman : to me kan complex are endowed with chosen lift (because either you assume and you don't care, or you don't assume choice and you will need this assumption) and what you are refering to is only a distinction between considering $\infty$-groupoids with strict functors between them, or with pseudo-functors between them. But I definitely agree this is debatable.... And anyway the issue is solved by Nikolaus' paper. | |
| Nov 26, 2018 at 14:10 | comment | added | Mike Shulman | I don't think it's really correct to call Kan complexes a "purely algebraic notion". Algebraic notions are given by operations, whereas the definition of Kan complex includes axioms "for every horn there exists a (non-specified) filler". Using AC one may choose fillers, but they will not be preserved by morphisms of Kan complexes, while morphisms of "algebraic" structures preserve all the operations. So I think the closest one can come to a "purely algebraic notion" from this direction is "algebraic Kan complexes", whose homotopy hypothesis is also true: arxiv.org/abs/1003.1342 | |
| Apr 11, 2018 at 20:51 | vote | accept | Qfwfq | ||
| Apr 11, 2017 at 17:31 | history | edited | David White | CC BY-SA 3.0 |
Fixed many typos
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| Apr 9, 2017 at 13:01 | history | edited | Simon Henry | CC BY-SA 3.0 |
added 20 characters in body
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| Apr 9, 2017 at 12:28 | history | edited | Simon Henry | CC BY-SA 3.0 |
added 14 characters in body
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| Apr 9, 2017 at 12:20 | history | answered | Simon Henry | CC BY-SA 3.0 |