Timeline for What are the reflective subcategories of the category of presentable categories?
Current License: CC BY-SA 4.0
21 events
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| Nov 22, 2018 at 8:10 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Jul 31, 2018 at 19:00 | comment | added | Ivan Di Liberti | Now I feel much better. | |
| Jul 31, 2018 at 18:57 | comment | added | KotelKanim | Well, I am mainly interested in the $\infty$-categorical situation anyway, and I assume that arguments which are "formal" enough will apply equally to both the $\infty$-categorical and the 1-categorical situation. But the meaning of this is that the 1-categorical $Pres$ (or $Pr^L$) should be considered as a $(2,1)$-category, in which case it has an initial object in the precise sense that the mapping space from it (actually, groupoid in this case) to any other is contractible. Moreover, the Gabriel-Ulmer duality is an equivalence of 2-categories. So, it is not a problem from my point of view. | |
| Jul 31, 2018 at 15:50 | history | undeleted | Ivan Di Liberti | ||
| Jul 30, 2018 at 3:25 | history | deleted | Ivan Di Liberti | via Vote | |
| Jul 30, 2018 at 2:21 | comment | added | Ivan Di Liberti | This is a very big problem in my answer. I might withdraw tomorrow morning. | |
| Jul 30, 2018 at 2:15 | comment | added | Alexander Campbell | @IvanDiLiberti, Lex does not have an initial object as a 1-category, since every category with finite limits admits at least two finite limit preserving functors to the free-living isomorphism (namely the two constant functors). Hence Lex is not locally presentable as a 1-category. | |
| Jul 29, 2018 at 16:43 | comment | added | KotelKanim | This is vaguely familiar, but I need to digest this. It seems like a good place to start! | |
| Jul 29, 2018 at 16:26 | comment | added | Harry Gindi | Ah, what if you set the cardinal to be a strongly inaccessible cardinal representing a Grothendieck Universe? | |
| Jul 29, 2018 at 16:23 | comment | added | Tim Campion | Yes. See Simon Henry's comment. | |
| Jul 29, 2018 at 16:22 | comment | added | Harry Gindi | Can this be generalized to arbitrary regular cardinals in some way? | |
| Jul 29, 2018 at 16:15 | comment | added | Ivan Di Liberti | (But I wrote it very explicitly in the answer). | |
| Jul 29, 2018 at 16:13 | comment | added | Tim Campion | I agree this seems useful. I suppose I object to the notation -- calling it "$Pres$" strongly suggests to me that at least its objects are exactly the presentable categories, which is not the case -- they are only the locally finitely presentable categories. | |
| Jul 29, 2018 at 16:10 | comment | added | Ivan Di Liberti | I completely agree Pres is not $Pr^L$. I thought that still, this might be useful. | |
| Jul 29, 2018 at 15:57 | comment | added | Tim Campion | Careful! $Lex^{op}$ is equivalent to the category $Pr^L_{\aleph_0}$of locally _finitely_ presentable categories (and left adjoint functors preserving finitely presentable objects). This is what Simon Henry pointed out in his comment. But maybe it's reasonable to consider only localizations which preserve $Pres^L_\kappa$ for sufficiently large $\kappa$, in which case this is a good starting point... The Vopenka's Principle point is a good one, I think. | |
| Jul 29, 2018 at 15:04 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Jul 29, 2018 at 14:51 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Jul 29, 2018 at 14:46 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Jul 29, 2018 at 14:36 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Jul 29, 2018 at 14:28 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Jul 29, 2018 at 14:21 | history | answered | Ivan Di Liberti | CC BY-SA 4.0 |