Timeline for The homotopy category of the category of enriched categories
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2019 at 18:04 | comment | added | Kevin Carlson | @MikeShulman Sure, done. | |
| Nov 21, 2019 at 16:23 | comment | added | Mike Shulman | @KevinCarlson Yes, that's true: groupoids are explicit enough that we can just look explicitly at the free $A_3$ groupoid on one object and see clearly that it's not $A_4$. Actually I think it would be worth adding that as an additional answer to this question. | |
| Nov 21, 2019 at 16:00 | comment | added | Kevin Carlson | @MikeShulman Yes, that would be nice. It seems to me that the category whose objects are rooted binary trees and whose morphisms are sequences of moves generated by the associator is an $A_3$ groupoid which manifestly doesn’t satisfy the pentagon identity, so at least an example of an $A_3$ structure not extending to an $A_4$. | |
| Nov 20, 2019 at 6:27 | comment | added | Mike Shulman | @KevinCarlson Okay. In particular it would be nice to know whether there is an example when $\mathcal{C} = \rm Gpd$ rather than $\rm SSet$, which would necessarily be $A_3$-but-not-$A_4$ since an $A_4$ groupoid is already $A_\infty$. | |
| Nov 20, 2019 at 0:28 | comment | added | Kevin Carlson | @MikeShulman I believe these are all the examples in the paper, so that one gets $A_4$-but-no-$A_5$ and $A_2$-but-no-$A_3$, but misses $A_3$-but-no-$A_4$. It would certainly be nice to have examples at every level of coherence. | |
| Nov 19, 2019 at 21:22 | comment | added | Mike Shulman | @KevinCarlson Assuming that $p$ must be a prime, that gives an example of a space that admits a (say) $A_4$-structure (hence also an $A_3$-structure) but no $A_5$-structure (hence no $A_\infty$-structure). Is there an example of a space admitting an $A_3$-structure but no $A_4$-structure? | |
| Nov 18, 2019 at 18:58 | history | edited | Harry Gindi | CC BY-SA 4.0 |
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| Nov 18, 2019 at 17:50 | comment | added | Kevin Carlson | It may not be easy to come up with examples, but Stasheff did it (citing some help from Adams) in the paper introducing $A_n$ spaces. According to Theorem 17 of Homotopy Associativity of H-spaces, I, any odd-dimensional Moore space of $\mathbb Z$ localized away from $p$ admits an $A_{p-1}$ structure but no $A_p$ structure. I have never heard of any other examples, though I hope more are known to somebody! | |
| Nov 18, 2019 at 17:04 | history | edited | Harry Gindi | CC BY-SA 4.0 |
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| Nov 18, 2019 at 14:27 | history | undeleted | Harry Gindi | ||
| Nov 18, 2019 at 14:27 | history | edited | Harry Gindi | CC BY-SA 4.0 |
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| Nov 18, 2019 at 13:59 | history | deleted | Harry Gindi | via Vote | |
| Nov 18, 2019 at 13:26 | history | edited | Harry Gindi | CC BY-SA 4.0 |
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| Nov 18, 2019 at 13:16 | history | answered | Harry Gindi | CC BY-SA 4.0 |