Timeline for Corepresentability of involutory objects in monoidal $\infty$-categories
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 21, 2021 at 6:41 | comment | added | Maxime Ramzi | If $Q= \Omega^\infty \Signa^\infty$ then yeah, the same proof works | |
| Sep 21, 2021 at 0:00 | comment | added | Emily | @MaximeRamzi Thanks! Regarding the shift, will it be $\Omega Q\mathbf{B}\mathbb{Z}/2\cong\Omega Q\mathbb{RP}^\infty$ for the $E_\infty$-case? | |
| Sep 20, 2021 at 7:23 | comment | added | Maxime Ramzi | @skd : it's almost that, although I think there's a shit: it should be $\Omega^k\Sigma^{k-1}B\mathbb Z/2$, because $map_{E_k}(\Omega^k\Sigma^{k-1} BX, Y) = map_{E_{k-1}}(\Omega^{k-1}\Sigma^{k-1} BX, BY) = map_{S_*}(BX, BY) = map_{E_1}(X,Y)$. That being said, I agree with you that I don't see why there would be a more explicit description, except maybe for replacing $B\mathbb Z/2$ with $\mathbb RP^\infty$ | |
| Sep 20, 2021 at 7:20 | comment | added | Maxime Ramzi | Indeed, $map_{E_k}(\mathrm{Ind}(G), X) = map_{E_1}(G,X) = map_{E_1}(G,X^{inv}) = map_{E_k}(\mathrm{Ind}(G), X^{inv})$ where $X^{inv}$ is the full sub-$E_k$-groupoid on invertible elements | |
| Sep 20, 2021 at 7:19 | comment | added | Maxime Ramzi | Emily : 1) it's because any monoidal morphism between strong monoidal functors out of a grouplike monoidal category is invertible. But in fact, even if that weren't so, because you want it to land in $S$ you would have to take $map$ rather than $Fun$ (and the $C^\simeq$ is because $\mathbb Z/2$ is a groupoid). 2) I'm viewing it as some form of extension of scalars, so it's a similar notation as for extension of scalars along a ring map $A\to B$: $\mathrm{Ind}_A^B$. 3) Yes: for any $E_1$-group $G$ this will be the case | |
| Sep 19, 2021 at 21:19 | comment | added | skd | Unless I'm misunderstanding something, isn't $\mathrm{Ind}^{E_k}_{E_1}(\mathbf{Z}/2)$ supposed to be $\Omega^k \Sigma^k B\mathbf{Z}/2$? If so, I doubt one can get a more explicit description of this object | |
| Sep 19, 2021 at 21:02 | comment | added | Emily | (Also, I updated the question to correct the E_k vs E_k-1 issue; thanks!) | |
| Sep 19, 2021 at 21:02 | comment | added | Emily | Thanks, Maxime! Would it be okay to ask a few questions? 1) Why do we have $Fun^\otimes(\mathbb{Z}/2,C)\cong map_{E_1}(\mathbb{Z}/2,C^\simeq)$? 2) Why did you choose the notation $\mathrm{Ind}^{E_k}_{E_1}$? 3) Is there a reason to expect $\mathrm{Ind}^{E_k}_{E_1}(\mathbb{Z}/2)$ to be an $E_k$-group? | |
| Sep 19, 2021 at 21:02 | vote | accept | Emily | ||
| Sep 19, 2021 at 11:13 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |