Timeline for Coslices of $\mathbb D$-presentable categories
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2021 at 0:01 | comment | added | Tim Campion | It may be delicate -- if you wanted to go further and ask about $\mathbb D$-accessibility, then classically taking a slice or coslice can change the index of accessibility, so you don't stay in the same doctrine. So special features of the locally $\mathbb D$-presentable case must be used. | |
| Jan 27, 2021 at 23:48 | history | edited | varkor | CC BY-SA 4.0 |
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| Jan 27, 2021 at 23:47 | comment | added | varkor | @TimCampion: thanks, I had completely overlooked that observation in ABLR. I've edited the question to make my intent clearer. | |
| Jan 27, 2021 at 23:43 | history | edited | varkor | CC BY-SA 4.0 |
Fixed some typos
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| Jan 27, 2021 at 23:39 | comment | added | Tim Campion | Okay right -- I learned that uncountable-siftedness is the same as uncountable-filteredness from ABLR. The place it's proved is apparently Adamek,Koubek, and Velebil. It's interesting that Centazzo's counterexample is the empty doctrine. I wonder if it might suffice to assume the doctrine is sound and nonempty? (EDIT: Er -- it would have to be a bit more subtle than that, e.g. the doctrine of idempotents will also be a counterexample) | |
| Jan 27, 2021 at 23:36 | comment | added | varkor | @TimCampion: thanks for pointing that out. I suppose I was looking for a more conceptual categorical reason. Perhaps it would be worth instead asking for a way to see that both locally $\kappa$-presentable and locally strongly presentable categories have this coslice property. | |
| Jan 27, 2021 at 23:33 | history | edited | varkor | CC BY-SA 4.0 |
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| Jan 27, 2021 at 23:31 | comment | added | Tim Campion | Assuming that "$\omega$-sifted" just means "sifted", the answer in that case should be "yes", just thinking about it in terms of universal algebra -- add some generators and relations to your variety coming from the map out of $X$. But it would be nice to have a conceptual reason which applies to a clear class of sound doctrines. | |
| Jan 27, 2021 at 23:28 | comment | added | Tim Campion | What does "$\kappa$-sifted" mean? If $I$-colimits commute with $\kappa$-small products in $Set$ and $\kappa > \omega$, then $I$ is $\kappa$-filtered, if I recall correctly. If that's what "$\kappa$-sifted" means, then "$\kappa$-sifted" is only special when $\kappa = \omega$. | |
| Jan 27, 2021 at 23:24 | history | asked | varkor | CC BY-SA 4.0 |