Timeline for Are there any interesting classes of limits containing finite limits?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 28 at 10:48 | vote | accept | Morgan Rogers | ||
| Mar 27 at 3:44 | answer | added | Tim Campion | timeline score: 4 | |
| Mar 26 at 18:02 | answer | added | Denis T | timeline score: 5 | |
| Mar 26 at 16:43 | comment | added | Morgan Rogers | @TimCampion your first comment (specifically the 1-categorical part, although you can include the infinity-categorical part if you like) is exactly what I was looking for. Do you have an example/construction of a category with $\aleph_1$-cofiltered limits and finite limits? If so, please post it as an answer :) | |
| Mar 26 at 16:39 | history | edited | Morgan Rogers | CC BY-SA 4.0 |
clarified second bullet point.
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| Mar 23 at 7:03 | comment | added | Tim Campion | I don’t understand the second bullet — as written the answer is obviously yes, because of the sort of examples Tom points out. I’d like to know the answer to the first bullet, but I’m reminded of the number of times I’ve driven myself nuts thinking about accessible categories which don’t have sequential colimits. | |
| Mar 23 at 6:49 | comment | added | Tim Campion | If you allow weighted limits, then I suppose a category can have all finite limits as well as all powers without having all countable limits. Or conically, a category can have all finite limits as well as all $\aleph_1$-cofiltered limits without having all countable limits. For another “more exotic” example, $\infty$-categorically it’s interesting to talk about $\pi$-finite limits. It’s also interesting $\infty$-categorically that idempotent splitting is not a finite limit, but an $\infty$-category can have finite limits and split idempotents without having all countable limits. | |
| Feb 23 at 9:30 | comment | added | Morgan Rogers | Edited, indeed I was aware of the commutation classes which include filtered colimits, but I'm asking about the ones which are subclasses of filtered colimits (so commute with $\Phi$-diagrams for a limit class $\Phi$ as in the remainder of the question). | |
| Feb 23 at 9:28 | history | edited | Morgan Rogers | CC BY-SA 4.0 |
clarified second bullet point in response to comment
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| Feb 23 at 9:25 | comment | added | varkor | My interpretation was that the first bullet point is intended to be read as "other limit-colimit-commutation classes containing the finite limits", and the second bullet point should include "where $\Phi$ contains the finite limit diagrams" (but this should be clarified in the question). | |
| Feb 22 at 23:33 | comment | added | Tom Leinster | Right; as I said, my comment's not an answer to the question in the title. | |
| Feb 22 at 22:30 | comment | added | Kevin Carlson | While it's true that there are infinitely many interesting classes contained in finite limits, it's interesting that neither of the linked papers mentions any that contain finite limits, other than the $\lambda$-small ones. I can't think of any myself--it feels as if basically anything you might add (some discrete infinite categories, some cofiltered ones...) gets you to something that's interchangeable with some $\Phi_\kappa$ for most purposes. | |
| Feb 22 at 22:08 | comment | added | Tom Leinster | In answer to your first bullet point (but not the question in the title), yes, there are tons of other limit-colimit commutation classes. See Bjerrum, Johnstone, Leinster & Sawin, Notes on commutation of limits and colimits and Adámek, Borceux, Lack and Rosicky, A classification of accessible categories. But maybe you know all this and I've misunderstood what you're asking | |
| Feb 22 at 21:41 | history | asked | Morgan Rogers | CC BY-SA 4.0 |