Timeline for What is the group completion of the groupoid of even finite sets and even permutations?
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6 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2021 at 13:16 | comment | added | user171227 | The $\mathbb{E}_\infty$ structure on the group completion corresponds to delooping the space in Nardin's comment to a spectrum. There is a unique homotopy class of spectrum maps $H\mathbb{Z} \to \tau_{\leq 1} S$ inducing multiplication by 2 on $\pi_0$, I would think the relevant spectrum fits in a pullback of the form $H\mathbb{Z} \to \tau_{\leq 1} S \leftarrow S$. | |
| Oct 12, 2021 at 12:17 | comment | added | archipelago | Similarly, $BE_\infty^+$ is the cover of $\Omega^2_0S^2$ that corresponds to the subgroup $2\cdot \mathbb{Z}\le \mathbb{Z}=\pi_1(\Omega_0^2S^2)$. | |
| Oct 12, 2021 at 12:08 | comment | added | archipelago | @NeilStrickland After taking plus-constructions, yes. | |
| Oct 12, 2021 at 8:11 | comment | added | Neil Strickland | Do we not just get $BA_\infty=\Omega^\infty S[2,\infty)$? | |
| Oct 12, 2021 at 6:06 | comment | added | Denis Nardin | The group completion theorem tells us that the underlying space of the group completion is $2\mathbb{Z}×BA_\infty^+$. I'm trying to think if we can say something about the corresponding spectrum | |
| Oct 12, 2021 at 5:31 | history | asked | Emily | CC BY-SA 4.0 |