Timeline for Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?
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| May 22, 2022 at 22:18 | answer | added | Peter May | timeline score: 2 | |
| May 22, 2022 at 20:08 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 22, 2022 at 20:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 24, 2021 at 20:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 25, 2021 at 19:42 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 25, 2021 at 19:38 | answer | added | Emily | timeline score: 1 | |
| Aug 25, 2021 at 19:38 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 23, 2021 at 21:38 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 23, 2021 at 21:31 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 21, 2021 at 2:01 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 21, 2021 at 1:56 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 11, 2021 at 22:24 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 10, 2021 at 17:03 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 10, 2021 at 17:00 | comment | added | Emily | @DenisNardin Thanks! | |
| Aug 10, 2021 at 15:23 | comment | added | Denis Nardin | @Emily Yes, the functor $Ω^∞$ is lax symmetric monoidal (since it is the right adjoint to the symmetric monoidal functor $Σ^∞$), and so it sends $\mathcal{O}$-algebras in $\operatorname{Sp}$ to $\mathcal{O}$-algebras in $\mathcal{S}_\ast$ for every $\infty$-operad $\mathcal{O}$. Note that this gives a negative answer to your question about the characterization of $E_\infty$-monoids in $\mathcal{S}_\ast$, since $\Omega^\infty E$ usually is not of the form $X_+$ for any $X$ (as all connected components are equivalent as spaces) | |
| Aug 10, 2021 at 15:20 | history | edited | Emily | CC BY-SA 4.0 |
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| Aug 10, 2021 at 15:19 | comment | added | Emily | Finally, about the example you mentioned, is it correct to say that we can make the assignment $E\mapsto\Omega^\infty E$ into a functor $\mathsf{Alg}_{\mathbb{E}_{k}}(\mathsf{Sp})\to\mathsf{Alg}_{\mathbb{E}_{k}}(\mathcal{S}_*)$? (Sorry if I'm being silly asking this, I'm still learning most of this material) | |
| Aug 10, 2021 at 15:18 | comment | added | Emily | I'm unsure if this is still true in the $\infty$-categorical case, though. (I asked this as a separate question here). If it doesn't, then we can also just consider $\mathbb{E}_{\infty}$-Hopf monoids in $\mathcal{S}_*$, rather than group objects there (though this would make things less interesting, i think :/) | |
| Aug 10, 2021 at 15:15 | comment | added | Emily | So, any monoid $A$ in $\mathsf{Sets}_*$ can be made into a bimonoid in it in a unique way, and since antipodes are also unique if they exist, the statement that $A$ has a Hopf monoid structure becomes a property, rather than extra structure. So in this sense we may say that $A$ is a group object in $\mathsf{Sets}_*$ iff it is a Hopf monoid in $\mathsf{Sets}_*$. | |
| Aug 10, 2021 at 15:14 | comment | added | Emily | @MaximeRamzi About $\mathbb{E}_{\infty}$-groups, I've been thinking about this, though I'm not yet sure: in the $1$-categorical case, it makes sense to speak of "group objects in $\mathsf{Sets}_*$" (as a property of monoids in $\mathsf{Sets}_*$), even though it is non-Cartesian: by a result of Péroux and Shipley (Lemma 2.4 of arXiv:1708.02592), every comonoid in $(\mathsf{Sets}_*,\wedge,S^0)$ comes uniquely from a comonoid in $(\mathsf{Sets},\times,\mathrm{pt})$, freely adjoined with a basepoint. | |
| Aug 10, 2021 at 12:03 | comment | added | Maxime Ramzi | How do you make sense of $E_\infty$-group in a non-cartesian monoidal ($\infty$-,but it is irrelevant for my question)category ? For $E_\infty$-monoids, this is a good question but I'm not sure there's a more satisfying answer than "they are $E_\infty$-monoids in that category"... Unlike for sets, the basepoint here need not be "added". Examples of such monoids are of course given by the multiplicative structure on $\Omega^\infty E$, for $E$ a commutative ring spectrum | |
| Aug 9, 2021 at 22:24 | history | asked | Emily | CC BY-SA 4.0 |