Timeline for Why Grothendieck's Homotopy Hypothesis is so difficult?
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| May 25, 2017 at 0:31 | comment | added | user40276 | Ok, thanks for the response. I'm aware that braided monoidal categories can't be strictified, but what exactly is the extent in "To some extent yes,..."? In other words, is there a precise theorem stating that the rectification can always be turned into a $E_n$ kind of obstruction? | |
| May 8, 2017 at 18:55 | comment | added | Yonatan Harpaz | To some extent yes, it is the higher commutativity that is the problem. For example, every monoidal groupoid is equivalent to a strict monoidal groupoid (i.e., a monoidal groupoid in which the associativity constraints hold "on the nose"), but not every braided monoidal groupoid is equivalent to one in which the commutativity constraints hold on the nose. Indeed, if the braiding $T_x: x \otimes x \to x \otimes x$ of an object $x \in X$ is not the identity then $X$ cannot be strictified. That's another way of thinking of what happens with the $S^2$ example. | |
| May 7, 2017 at 22:06 | comment | added | user40276 | Now about your example of $S^2$, so this group of automorphism is strictly associative? In other words, only the commutativity holds up to homotopy, right? But is this a more general phenomenon? In other words, in the examples where I can't strictly stuff, are there some $E_n$ constraints implicitly making all the problem? You see, since I can rectify $A_{\infty}$-algebras and, more generally, I can rectify $A_{\infty}$-categories to DG-categories, it seems that there are some constraints not related to associativity implicitly somewhere. | |
| May 7, 2017 at 22:02 | comment | added | user40276 | Thanks for your answer. As you mentioned, I was somehow convinced that the simplicial counterpart was rectifiable while the globular side wasn't. And this made me think that this was the main difficult in stablishing the Homotopy Hypothesis. So we can rectify the 0 and 1 dimensional levels. It seems weird that one cannot go any step further. What's so special about these levels? I think that this is the same problem of why 3-categories can be rectified while 2-categories can't. | |
| May 7, 2017 at 15:45 | history | edited | Yonatan Harpaz | CC BY-SA 3.0 |
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| May 7, 2017 at 13:15 | history | answered | Yonatan Harpaz | CC BY-SA 3.0 |