Timeline for Why Grothendieck's Homotopy Hypothesis is so difficult?
Current License: CC BY-SA 3.0
10 events
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| May 7, 2017 at 22:48 | history | edited | user40276 | CC BY-SA 3.0 |
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| May 7, 2017 at 22:45 | comment | added | user40276 | @QiaochuYuan Yes, I noticed it after reading Yonatan answer. However (as I commented in Yonatan's answer) it really bugs me that only the 0 and 1 level can be rectified. What's so special about 2-categories? | |
| May 7, 2017 at 22:33 | comment | added | Qiaochu Yuan | @user40276: okay, in that case there's no contradiction with the observation that $\Pi_3(S^2)$ can't be rectified to a strict $3$-groupoid, because the $3$-truncation of a simplicial category isn't a strict $3$-category. | |
| May 7, 2017 at 21:44 | comment | added | user40276 | @QiaochuYuan Well, I was thinking about Lurie's comment in HTT page 6 and 7, where he says that a category enriched over $\infty$-groupoids with a coherent associative composition can be rectified into an ordinary category enriched over simplicial objects. I think that he means something like Cordier nerve construction (and its adjoint) or the rectification of categories enriched over $A_{\infty}$-spaces. But I'm not sure if the latter is really true. | |
| May 7, 2017 at 13:45 | history | edited | Qfwfq | CC BY-SA 3.0 |
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| May 7, 2017 at 13:15 | answer | added | Yonatan Harpaz | timeline score: 15 | |
| May 7, 2017 at 12:53 | comment | added | Dmitri Pavlov | @AntonFetisov: You can rectify any ∞-category to a bifibrant simplicial category, and the space of functors between such categories is equivalent to the space of functors between the original ∞-categories. | |
| May 7, 2017 at 11:50 | comment | added | Anton Fetisov | The problem is not the $(\infty,1)$-categories themselves but functors between them. Even if you rectify the categories you must still consider the functors in a weak sense, with all possible homotopy coherent data. I think this is explained well in the introduction of Lurie's Higher Topos Theory. By the way, the homotopy hypothesis per se isn't about globular sets, it is about an existence of some notion of $(\infty,1)$-category where the subcategory of groupoids would be equivalent to homotopy types. Globular groupoids is just one possible model. | |
| May 7, 2017 at 7:49 | comment | added | Qiaochu Yuan | What is a strict $(\infty, 1)$-category? If you mean something like a category strictly enriched over strict $\infty$-groupoids then there's no way your rectification result holds. If not, there's no contradiction. | |
| May 7, 2017 at 1:08 | history | asked | user40276 | CC BY-SA 3.0 |