Timeline for What are the reflective subcategories of the category of presentable categories?
Current License: CC BY-SA 4.0
17 events
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| Jul 23, 2022 at 20:28 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Aug 5, 2018 at 5:02 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Aug 4, 2018 at 22:23 | history | edited | Tim Campion | CC BY-SA 4.0 |
Expanded a bit. Also added back credit to Ivan, as his answer is undeleted now.
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| Jul 31, 2018 at 15:50 | comment | added | Ivan Di Liberti | Alexander Cambell observed in a comment that Lex does not have an initial object. I think that the problem is that the candidate initial object is just pseudo-initial, when looking at Lex as a two category. I am not a 2-category theory expert thus I cancelled the answer. I will make it reappear but plz, readd his comment. | |
| Jul 31, 2018 at 10:20 | comment | added | KotelKanim | Wait, what? Do you mean it is only equivalent to a presentable category or does "technically" means something more serious than that? Btw, I don't know why you cancelled your answer. It was a nice answer and I think it can definitely be useful for someone who stumbles across this thread. | |
| Jul 31, 2018 at 8:22 | comment | added | Ivan Di Liberti | Technically, $Pr^L_{\mu}$ is not presentable. This is the reason for which cancelled my answer. | |
| Jul 30, 2018 at 16:14 | vote | accept | KotelKanim | ||
| Jul 30, 2018 at 16:02 | comment | added | KotelKanim | I pretty much convinced myself in the fact I needed convincing. The thing is that filtered colimits of categories aren't that bad. The space of objects is the filtered colimit of the spaces of objects and the mapping spaces are also easy to describe, so this indeed (as you implied) basically boils down to the fact that every presentable category is $\kappa$-compactly generated for some $\kappa$ and every cocontinuous functor between presentable categories preserves $\kappa$-compact objects for some $\kappa$ (though, it need not be the same $\kappa$). | |
| Jul 30, 2018 at 15:36 | comment | added | KotelKanim | You are right... sorry. The sizes are really confusing. $\Pr^L_{\kappa}$ is of course large (but not huge, which is what was important). | |
| Jul 30, 2018 at 15:33 | comment | added | Tim Campion | Well, $Pr^L_\kappa$ is large (but locally small), though that doesn't affect your argument. I suppose the argument I gave above is kind of going through the argument that $Pr^L$ is indeed the colimit of this chain, which makes it much messier! | |
| Jul 30, 2018 at 15:30 | comment | added | KotelKanim | Btw, if $Pr^L$ can indeed be defined as the colimit of a large chain of large (even small) categories $Pr^L_{\kappa}$ as you say, i.e. one that takes place in the (huge) category of large categories, then it is automatic that $Pr^L$ is large (not huge). But I still need to convince myself of this fact. | |
| Jul 30, 2018 at 11:54 | comment | added | KotelKanim | Those are very good points. And, it might be even enough for what I need, but I'll have to think about it. | |
| S Jul 30, 2018 at 10:05 | history | suggested | Ivan Di Liberti | CC BY-SA 4.0 |
There was a mistake in my answe. Lex is not locally presentable. I think that it only has pseudo-limits. My name was not needed.
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| Jul 30, 2018 at 8:09 | review | Suggested edits | |||
| S Jul 30, 2018 at 10:05 | |||||
| Jul 29, 2018 at 22:01 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jul 29, 2018 at 21:52 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jul 29, 2018 at 21:40 | history | answered | Tim Campion | CC BY-SA 4.0 |