Timeline for Examples of bilimits that aren't 2-limits, and some related questions
Current License: CC BY-SA 4.0
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| Oct 6, 2023 at 23:28 | comment | added | Emily | @MartinBrandenburg It's hard not to accept the answer when it's so good =) As Kevin noted though, there seems to be some problems with how Q3 is currently formulated, and since I haven't thought about bilimits and the topics surrounding then in a long long time, I think I'll try reading a bit about them and see if I can rephrase Q3 in a way that makes sense. If that goes well I'll then ask it as a separate question :) Thank you so much again, Martin! | |
| Oct 6, 2023 at 12:36 | comment | added | Alec Rhea | @MartinBrandenburg Convention within a discipline can be confusing to people not already familiar with it; in a setting like this with potentially thousands of readers in the years to come (not all of whom will have the same ‘standard conventions’ as you), I believe it’s best to be as explicit as possible. More to the point, perhaps there is hidden circularity in your characterization in light of the fact that the colimit size condition makes reference to being an object of the category of sets. Still cool! | |
| Oct 6, 2023 at 8:37 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
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| Oct 6, 2023 at 8:18 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
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| Oct 6, 2023 at 8:12 | comment | added | Martin Brandenburg | Yes my (quite standard) conventions are: All categories are locally small unless otherwise stated. Cocomplete means the existence of small colimits. | |
| Oct 6, 2023 at 3:45 | comment | added | Alec Rhea | You’re obviously considering a $2$-category of locally small categories if ${\bf Set}$ is a member of your $2$-category; ignoring the foundational issues because I’m sure you know how to remedy them, you’re also sweeping the word ‘small’ under the rug when you say cocomplete, no? | |
| Oct 5, 2023 at 23:07 | comment | added | Martin Brandenburg | I hope that someone else can answer Q3. I saw that you accepted my answer, even though I didn't answer Q3. Maybe ask a separate question? | |
| Oct 5, 2023 at 20:08 | comment | added | Emily | @MartinBrandenburg Wow, I was expecting there might be only artificial examples for Q2, but the examples you mentioned are *so* nice. Thank you for the answer, Martin! :) | |
| Oct 5, 2023 at 19:59 | vote | accept | Emily | ||
| Oct 5, 2023 at 16:10 | comment | added | Alec Rhea | Very interesting; am I mistaken, or is your example a non-circular $2$-categorical characterization of the category of sets with Cartesian products? I say non-circular because I’ve seen it argued that characterizations like ‘the terminal Grothendieck topos’ are circular because the definition of a Grothendieck topos involves the category of sets, but that doesn’t seem to be the case here. (maybe in the cocompleteness condition?) | |
| Oct 5, 2023 at 12:55 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
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| Oct 5, 2023 at 12:49 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
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| Oct 5, 2023 at 11:29 | comment | added | Peter LeFanu Lumsdaine | Nice clear counterexamples — and perhaps worth adding that your example for Q2 is a very typical kind of example: in 2-categories of “categories with structure”, whose maps preserve that structure not-necessarily-on-the-nose, the colimits will generally be only bicategorical not strictly 2-categorical. | |
| Oct 5, 2023 at 10:02 | comment | added | Martin Brandenburg | PS: I prefer the term "bicategorical limit" instead of "bilimit", because it is more explicit and cannot be confused with other notions where "bi" stands for "twosided". | |
| Oct 5, 2023 at 10:00 | history | answered | Martin Brandenburg | CC BY-SA 4.0 |