Timeline for Limits in an $(\infty,1)$-category
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 3, 2011 at 13:10 | vote | accept | Guillaume Brunerie | ||
| Nov 1, 2011 at 4:19 | answer | added | Mike Shulman | timeline score: 3 | |
| Oct 31, 2011 at 2:58 | answer | added | Akhil Mathew | timeline score: 3 | |
| Oct 31, 2011 at 2:50 | answer | added | Moosbrugger | timeline score: 2 | |
| Oct 31, 2011 at 1:18 | answer | added | Hiro Lee Tanaka | timeline score: 22 | |
| Oct 30, 2011 at 23:30 | comment | added | Moosbrugger | is equal to the limit of $K\to\mathcal{C}$. | |
| Oct 30, 2011 at 23:30 | comment | added | Moosbrugger | I only know the quasi-categories model, but as long as you have a notion of limit between functors of $\infty$-categories you're okay. If you have $K\to \mathcal{C}$ a map from a simplicial set $K$ to a quasi-category, take a categorical equivalence $K\simeq J$ where $J$ is a quasi-category with a functor $J\to\mathcal{C}$ such that the diagram commutes. Then, using whatever equivalence between quasi-categories and you're other model, transfer $J\to\mathcal{C}$ to some $J'\to\mathcal{C}'$ in your other world and take the limit. The point is that the limit of $J\to\mathcal{C}$... | |
| Oct 30, 2011 at 21:43 | comment | added | David Roberts♦ | ...I couldn't say what you'll get, but that's a start... | |
| Oct 30, 2011 at 21:42 | comment | added | David Roberts♦ | Well, an $(\infty,1)$-category, in the quasi-category version, is a simplicial set, so in that case restricting to a quasi-category would be restricting your options. But I don't have a good reason why you do need that greater generality. It's possibly due to using the model category structure on simplicial sets such that the fibrant objects are quasicategories. As far as using different models goes, if one has a Quillen equivalence to another model category which is also a presentation for $(\infty,1)$-categories, then one can simply transport the diagram along that Quillen equivalence... | |
| Oct 30, 2011 at 21:22 | comment | added | Yosemite Sam | *and by 'why' above I meant 'where'! | |
| Oct 30, 2011 at 21:21 | comment | added | Yosemite Sam | I know next to nothing about oo-stuff, but I've heard from experts that passing from one kind of oo-categories to another can morally be done, but technically you would have to reprove all theorems (and apparently this is why Voevodsky's univalent maths should come in). It's a vague comment I know, but I wanted to throw it in there as I only recently learned of these issues! | |
| Oct 30, 2011 at 20:07 | comment | added | Guillaume Brunerie | For example if I have a Batanin-Leinster $(\infty,1)$-category $C$, what is a limit in $C$? I can't use a simplicial set $K$ and a map $K\to C$ anymore, because $C$ is globular and I can't easily describe the maps between a simplicial set and something globular. | |
| Oct 30, 2011 at 19:50 | comment | added | Moosbrugger | I'm not quite sure what it would mean for a definition to be independent of the model. But Lurie's book proves that limits and colimits are invariant under categorical equivalences in the "$K$-variable," which I think answers your question. However, sometimes in the context of quasi-categories, it's more convenient to allow an arbitrary simplicial set, just like sometimes when doing homotopy theory with simplicial sets it's convenient to allow non-Kan sets. E.g., the quantity of generators and relations is much smaller. | |
| Oct 30, 2011 at 19:37 | history | asked | Guillaume Brunerie | CC BY-SA 3.0 |