Timeline for What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
Current License: CC BY-SA 3.0
21 events
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| Sep 9 at 6:35 | answer | added | Guillaume Laplante-Anfossi | timeline score: 7 | |
| Nov 29, 2017 at 13:18 | history | edited | Simon Henry |
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| Nov 29, 2017 at 13:18 | comment | added | Simon Henry | @HarryGindi: I have no Ideas. I believe I wanted to put "algebraic topology" and just clicked on the wrong one and never noticed. I'll edit that. | |
| Nov 29, 2017 at 13:16 | comment | added | Harry Gindi | @SimonHenry Great question, but I'm wondering if the Algebraic Geometry tag is correct. | |
| Nov 29, 2017 at 13:15 | history | edited | Simon Henry | CC BY-SA 3.0 |
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| Nov 29, 2017 at 13:01 | vote | accept | Simon Henry | ||
| Nov 29, 2017 at 13:00 | answer | added | Simon Henry | timeline score: 47 | |
| Mar 31, 2016 at 19:27 | answer | added | Edward Dunne | timeline score: 67 | |
| Mar 29, 2016 at 21:53 | answer | added | Yonatan Harpaz | timeline score: 60 | |
| Mar 29, 2016 at 20:11 | comment | added | Daniel Gerigk | @Charles Rezk Perhaps you're thinking of "Weak identity arrows in higher categories" by Joachim Kock? | |
| Mar 25, 2016 at 16:49 | comment | added | Simon Henry | Sorry, I was not clear: That's not what I meant . The Eckman-Hilton argument, is basically the argument in Simpson's paper. What I meant is what goes wrong when one try to strictify using "generalized Moore path" indexed by pasting diagrams as suggested by Kapranov and Voevodsky. Do we fail to get a strict groupoid ? do we fail to get a groupoid at all ? or do we get a groupoid with the wrong homotopy type and why ? | |
| Mar 25, 2016 at 16:48 | comment | added | Charles Rezk | ... I seem to remember some kind of work about some form of higher category with a weakened unit condition (maybe by Eugenia Cheng?) but I can't find it now. | |
| Mar 25, 2016 at 16:47 | comment | added | Charles Rezk | ... Commutativity is bad for the homotopy hypothesis, since so few homotopy types support a strict commutative monoid structure. So to get a homotopy hypothesis, we should weaken something about the composition operations in our infinity groupoid, so as to banish Eckmann-Hilton. The common approach is to not have them be actually operations anymore; or to not require that they commute on the nose. Simpson seems to be suggesting that removing units, but keeping compatible composition laws, is sufficient ... | |
| Mar 25, 2016 at 16:44 | comment | added | Charles Rezk | I have no idea what goes wrong in the KV paper, but here's why units might be bad. It is because of our arch-enemy, the Eckmann-Hilton argument, which says given a set with two binary unital operations which commute with each other, they are actually both the same commutative monaid law. This shows, for instance, that n-endomorphisms of an object in a strict infinity category always form a commutative monoid, for $n\geq2$. ... | |
| Mar 25, 2016 at 16:06 | comment | added | Simon Henry | Indeed, but I was thinking that if it was indeed the case then maybe now that 18 years have passed Simpson's conjecture would have become a theorem ^^ More seriously, even if Simpson is right, it is not clear to me how units makes Kapranov-Voevodsky construction wrong (except because we have a counterexample ! ). | |
| Mar 25, 2016 at 15:22 | comment | added | Dylan Wilson | The way Simpson actually states his conjecture is to say that he thinks that's what the correct parts in Kapranov-Voevodsky actually prove. So Simpson's conjecture is a guess of an answer to your question of what can be salvaged. | |
| Mar 25, 2016 at 12:29 | comment | added | Asaf Karagila♦ | Well. Not being a constructivist I am free to say "Yes, there is such person. But its existence might not be constructive!" ;-) | |
| Mar 25, 2016 at 12:21 | history | edited | David White | CC BY-SA 3.0 |
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| Mar 25, 2016 at 12:13 | comment | added | Simon Henry | Well, that is true, but we are only sure for one person in this case : Carlos Simpson only gave a counterexample to the main result and was not sure (at least at the time he wrote his 1998 paper) what was Kapranov and Voevodsky mistakes and what could be salvaged from their paper. | |
| Mar 25, 2016 at 12:08 | comment | added | Asaf Karagila♦ | In a typical mathematician fashion, yes. Someone can explain what is wrong in the paper. The fact someone has claimed to find a mistake, and that mistake has been eventually acknowledged by one of the authors means that at least two people can explain what's wrong. :-) | |
| Mar 25, 2016 at 11:47 | history | asked | Simon Henry | CC BY-SA 3.0 |