Timeline for 2-completeness of stacks
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2023 at 11:40 | comment | added | Luca Mesiti | Remember that we are still missing a reference for the fact that a bireflective bicategory inside a bicategorically (co)complete bicategory is bicategorically (co)complete. Regarding your question in the Edit part, there is something in Section 7 of "Flexible limits for 2-categories" of Bird, Kelly, Power, Street. But it seems to me it requires a paper "Flexibility for 2-monads" of Kelly and Power that was in preparation at the time. I found this at.yorku.ca/c/a/j/f/41.htm but it is just an abstract for a talk by Kelly. | |
| Jun 4, 2023 at 11:09 | comment | added | Nico | I have edited my question to include everything I learned from your answer! | |
| Jun 2, 2023 at 16:30 | comment | added | Luca Mesiti | I have just had an idea for commas in stacks. Start from a diagram for a comma in stacks. If you embed the diagram in Ps[C^op,Cat], you can find the comma there. Assume that it's true that stacks have all bilimits calculated in Ps[C^op,Cat]. Then since the comma in Ps[C^op,Cat] that we have found also satisfies the universal property of a bicomma, it needs to be a stack. So it is a comma in stacks for the original diagram. Can this be extended to general flexible limits? Maybe saying that products are also biproducts, inserters are also biinserters and so on? | |
| Jun 2, 2023 at 12:53 | comment | added | Nico | @LucaMesiti i think it is irrelevant if the limit classifies up to equivalence or isomophism, the only important part is if the universal property itself is evil. i am very sure that stacks have comma objects, i will write down my proof later when i am at home | |
| Jun 2, 2023 at 11:19 | comment | added | Ivan Di Liberti | I agree with everything Luca is saying. Moderately relevant literature for this conversation: - Street, Joyal: Pullbacks equivalent to pseudopullbacks. - Lack, Bourke: On 2-categorical infity-cosmoi, Appendix A. | |
| Jun 2, 2023 at 9:53 | comment | added | Luca Mesiti | The universal property of the comma is with identities in the triangles giving an isomorphism of categories, while the universal property of the bicomma has isomorphisms in the triangles giving an equivalence of categories. Bicommas are bilimits so I believe stacks have them, even if we need some more references. Commas are instead strict 2-limits and for this might be problematic. But they are PIE, so there's hope. Canonaco wrote in "Lectures on algebraic stacks" that stacks have isocommas (called there "fibred products"). But what about commas? | |
| Jun 2, 2023 at 3:45 | comment | added | Nico | @KevinArlin i know about the last part. You can use the left adjoint splitting to change $Ps(J,Cat)(W,K(A,D-))$ into $[J,Cat](LW,K(A,D-))$, that is what I was thinking about. Not all of the PIE limits are of those form though, right? I was just rly confused by the statement that comma objects might be problematic | |
| Jun 1, 2023 at 16:56 | comment | added | Kevin Carlson | Flexible limits are homotopically meaningful strict 2-limits. I think the only difference between a bicomma and a comma is that a bicomma might only induce an equivalence of categories between maps in and cones, whereas a comma induces an isomorphism of categories. It's almost a category error to ask whether a PIE limit is pseudo, because PIE limits come up in the context of strict weighted limits in a category-enriched category. It just so happens that pseudo limits, with an isomorphism of categories in the defining universal property, can be defined as certain (strict!) weighted limits. | |
| May 31, 2023 at 17:29 | comment | added | Nico | After inspecting my proof a little I think I would have gotten in trouble with 2-universal properties which involve $=$-2-cells. Some of the PIE-limits are not pseudo, right? It might be a good idea to check them by hand. | |
| May 31, 2023 at 17:27 | comment | added | Nico | I just checked comma objects, and they seem to be okay. $f/g$ is a stack (respectively prestack) when domains and codomain of $f$ and $g$ are (pre)stacks. But I think that it works because comma objects are homotopical okay, there are no equalities appearing in the universal property. What is a bicomma object? | |
| May 31, 2023 at 17:05 | comment | added | Nico | Oh lol, I was always under the impression that comma objects are homotopical meaningful limits. | |
| S May 30, 2023 at 14:00 | review | First answers | |||
| May 30, 2023 at 14:58 | |||||
| S May 30, 2023 at 14:00 | history | answered | Luca Mesiti | CC BY-SA 4.0 |