Timeline for A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?
Current License: CC BY-SA 4.0
21 events
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| May 7, 2021 at 21:54 | comment | added | Charles Rezk | There's one dangling thread here: what is the "universal $\infty$-connected object". The "universal object" is the inclusion functor $i\colon \mathcal{S}^{\mathrm{fin}}\to \mathcal{S}$ as an object in $\mathrm{Fun}(\mathcal{S}^{\mathrm{fin}},\mathcal{S})$. According to what is here, the "universal $\infty$-connected object" is the sheafification of $i$ with respect to the atomic topology on $(\mathcal{S}^{\mathrm{fin}})^{op}$. What does this look like? | |
| S May 21, 2018 at 20:06 | history | bounty ended | Tim Campion | ||
| S May 21, 2018 at 20:06 | history | notice removed | Tim Campion | ||
| May 20, 2018 at 21:43 | answer | added | Denis Nardin | timeline score: 14 | |
| May 20, 2018 at 20:52 | answer | added | Tim Campion | timeline score: 9 | |
| May 20, 2018 at 20:32 | comment | added | Tim Campion | @SimonHenry Please feel free to revert my addendum to the title / tag edits if you don't like them. I thought it might be appropriate to highlight the main connection that seemed to emerge in the comments. | |
| May 20, 2018 at 20:31 | history | edited | Tim Campion | CC BY-SA 4.0 |
edited title
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| May 20, 2018 at 20:08 | history | edited | Tim Campion |
edited tags
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| S May 20, 2018 at 20:06 | history | bounty started | Tim Campion | ||
| S May 20, 2018 at 20:06 | history | notice added | Tim Campion | Improve details | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 30, 2015 at 15:08 | comment | added | Jacob Lurie | The localization can be described more concretely: it's sheaves with respect to the Grothendieck topology where every morphism generates a covering. Equivalently, it's functors F: {finite spaces} -> {spaces} (or you can do a pointed version if you like) with the property that for any X, F(X) is the totalization of the cosimplicial space given by applying F to the "Cech nerve" of the map X->* (in the opposite of finite spaces). This is much larger than the class of 1-excisive functors: it contains all n-excisive functors for any n, and more (such as products of n-excisive functors as n varies). | |
| Aug 28, 2015 at 17:17 | comment | added | Marc Hoyois | As you say, any $1$-excisive functor $\mathcal{S}^\mathrm{fin}_* \to \mathcal{S}$ belongs to this localization, since $(*,S^0)$ is ∞-connected, but I don't know if there's an unpointed analog of this (what's $\mathrm{Exc}^1(\mathcal{S}^\mathrm{fin},\mathcal{S})$ anyway? torsors under bundles of spectra?). In any case, I don't see why the reverse inclusion should hold, but I'm no expert in calculus either... | |
| Aug 28, 2015 at 17:16 | comment | added | Marc Hoyois | Right, the classifying ∞-topos for (pointed) $n$-connected objects is the left exact localization of $\mathrm{Fun}(\mathcal{S}^\mathrm{fin}_{(*)}, \mathcal{S})$ generated by the map $\tau_{\leq n} \to *$, and for $n=\infty$ it's their intersection. | |
| Aug 28, 2015 at 14:32 | comment | added | Simon Henry | If I'm correct (but I'm not an expert on Goodwillie calculus so I might be wrong), the "universal object" (corresponding to the sheaf "represented" by the terminal object) is the system of spectra $(X,S)$ with $X=\{*\}$ and $S$ the sphere spectrum. So it is in particular $\infty$-connected, so the $\infty$-topos of system of spectra should at least be a subtopos of the classyfing topos for $\infty$-connected object. (so in particular a subtopos of the topos of $(-1)$-connected object... But looking at thos classyfing topos for $n$-connected object for finite $n$ seem to be a very good idea ! | |
| Aug 28, 2015 at 13:20 | comment | added | Marc Hoyois | So the ∞-topos that classifies $(-1)$-connected objects is the subtopos of $\mathrm{Fun}(\mathcal{S}^\mathrm{fin},\mathcal{S})$ consisting of functors that are RKE of their restriction to nonempty spaces. Does a $1$-excisive functor have this property? | |
| Aug 28, 2015 at 13:08 | comment | added | Simon Henry | Well I wouldn't mind if one get a description of the classifying topos for pointed $\infty$-connected spaces instead. but even like that I don't see reason for point parametrized spectra to be the solution...... | |
| Aug 28, 2015 at 12:59 | comment | added | Marc Hoyois | Being a localization of presheaves on $\mathcal{S}^\mathrm{fin,op}_*$, the ∞-topos of parametrized spectra should classify pointed objects with some property. So it can't be exactly the classifying topos you're looking for. | |
| Aug 27, 2015 at 11:25 | history | edited | Simon Henry | CC BY-SA 3.0 |
edited title
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| Aug 27, 2015 at 10:05 | comment | added | Zhen Lin | Tangentially, what does the $\infty$-topos of local systems of spectra classify? | |
| Aug 27, 2015 at 9:44 | history | asked | Simon Henry | CC BY-SA 3.0 |