Timeline for Is every conservative, left exact left adjoint comonadic, $\infty$-categorically?
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| Dec 3, 2021 at 20:17 | answer | added | Tim Campion | timeline score: 4 | |
| Oct 3, 2021 at 16:16 | comment | added | Tim Campion | Regarding question (2), here is one possibility. Say that $F : A \to B$ detects constancy if the natural map $A \to Pro(A) \times_{Pro(B)} B$ is essentially surjective. Then if $F$ is a conservative, left exact left adjoint between left exact categories, and if $F$ detects constancy, we may deduce that $F$ is comonadic. (In the pro-categories, it suffices to take just pro-objects coming from $\omega^{op}$-towers.) I don't know how checkable this condition is in practice, nor how often it is satisfied... | |
| Oct 3, 2021 at 15:38 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Oct 3, 2021 at 15:30 | comment | added | Ivan Di Liberti | I asked myself the same question when working on my thesis, indeed this lemma would be key to show that sober infinity-ionads correspond to infinity-topoi with enogh points (Indeed I think it is the only missing thing). See 3.2.6 and 4.0.3 in the ArXiv version of my paper "Towards Higher Topology". While I was thinking about it, I remember I had a negative feeling about it. | |
| Oct 3, 2021 at 15:24 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Oct 3, 2021 at 15:16 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Oct 3, 2021 at 14:41 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Oct 3, 2021 at 14:35 | history | asked | Tim Campion | CC BY-SA 4.0 |