Timeline for Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2021 at 20:10 | comment | added | Emily | @Achim and Eric It is a weird name! I did have in mind another related statement as Eric said (that $\mathbb{Z}$ corepresents the group of units functor $(-)^\times\colon\mathsf{CMon}\to\mathsf{Sets}$ from commutative monoids to sets). | |
| Sep 21, 2021 at 20:10 | history | edited | Emily | CC BY-SA 4.0 |
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| Sep 21, 2021 at 17:50 | comment | added | Eric Peterson | @Achim I was also puzzled, but it is a good name for functor on rings corepresented by the group ring on Z. Given some of the other adjunctions floating around, perhaps that’s closer to what was intended? | |
| Sep 21, 2021 at 5:42 | comment | added | Achim Krause | $(-)^\times$ is a pretty weird name for what seems to be the forgetful functor from abelian groups to sets... | |
| Sep 21, 2021 at 3:38 | vote | accept | Emily | ||
| Sep 21, 2021 at 3:31 | answer | added | Tyler Lawson | timeline score: 8 | |
| Sep 21, 2021 at 2:36 | history | edited | Emily | CC BY-SA 4.0 |
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| Sep 21, 2021 at 2:31 | comment | added | Emily | @skd Sorry for the confusing notation! I didn't mean the mod 2 Moore spectrum | |
| Sep 21, 2021 at 2:31 | history | edited | Emily | CC BY-SA 4.0 |
added 211 characters in body; edited title
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| Sep 21, 2021 at 2:31 | comment | added | Emily | @TimCampion I was writing "$\mathbb{S}/2$" to mean the connective spectrum associated to the $\mathbb{E}_\infty$-group $\Omega Q\mathbb{RP}^\infty$ because it satisfies an analogous universal property to the one for $\mathbb{Z}/2$: while morphisms of monoids from $\mathbb{Z}$ and $\mathbb{Z}/2$ to a monoid $A$ are the same as invertible/involutory elements of $A$, symmetric monoidal functors from $QS^0$ and $\Omega Q\mathbb{RP}^\infty$ to $\mathcal{C}$ are the same as invertible/involutory objects of $\mathcal{C}$. I've edited the question to use a less confusing notation. Thanks, Tim! | |
| Sep 21, 2021 at 2:04 | comment | added | Tim Campion | "$\mathbb S / 2$" would standardly denote the mod 2 Moore spectrum, which is the same as $\Sigma^{\infty-1} \mathbb R \mathbb P^2$... but you've defined "$\mathbb S/2$" to be $\Omega Q (\mathbb R \mathbb P^\infty)$... first of all $Q(X)$ usually means $\Omega^\infty \Sigma^\infty X$, so what you've written is a space and not a spectrum, thought the (unstable) homotopy groups of $\Omega Q \mathbb R \mathbb P^\infty$ are the same as the (stable) homotopy groups of $\Sigma^{\infty-1} \mathbb R \mathbb P^\infty$. Still, that's a different spectrum from the mod 2 Moore spectrum. Could you clarify? | |
| Sep 21, 2021 at 1:27 | comment | added | skd | S/2 doesn't have a unital multiplication. In general, S/p does have an A_{p-1}-structure, but not an A_p-structure. | |
| Sep 21, 2021 at 1:16 | history | asked | Emily | CC BY-SA 4.0 |