Timeline for I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2014 at 13:40 | comment | added | Mike Shulman | @ZhenLin That doesn't mean that Rezk completion is a sort of skeletality, though. | |
| Apr 30, 2014 at 1:05 | vote | accept | Theo Johnson-Freyd | ||
| Apr 28, 2014 at 15:25 | answer | added | Chris Schommer-Pries | timeline score: 11 | |
| Apr 23, 2014 at 6:04 | comment | added | Rune Haugseng | I suppose the article arxiv.org/abs/1312.3178 contains one approach to your (1)-(5), though that model also involves completion. | |
| Apr 23, 2014 at 3:13 | comment | added | Theo Johnson-Freyd | ... to present to you the space of objects, and for this I need already to know (or at least have some meaningful way to describe) the isomorphisms. So, for example, a complicated "generators and relations" presentation of my category won't do, as it might be hard to know whether something is invertible. A more important example: if the only definition of "category" that I knew was CSS, then I would be forced to decide what "Morita equivalent" meant without knowing that tensor product of bimodules was associative. (Then again, I think this is what happened in practice!) | |
| Apr 23, 2014 at 1:23 | comment | added | Theo Johnson-Freyd | Karol, Mike, and Zhen: Perhaps I should have qualified my comments about completeness. One dimension along which skeletality and completeness differ is that the latter is "non-evil". But I think this is the main difference, and I do think completeness is a "non-evil" version of skeletality. From the point of view of setting up a smoothly operating theory, I entirely understand why completeness is a natural condition. But it is (for me) a hard condition to satisfy as a user. If I want to present a CSS to you that is supposed to be the category of some naturally-occuring objects, I need ... | |
| Apr 23, 2014 at 1:18 | comment | added | Theo Johnson-Freyd | @DavidRoberts I have, of course, read Leinster's foundational paper, but not recently — I'll take another look at it. It was written at a time when the theory of $(\infty,1)$-categories had not yet been settled, and long before comparison results like Barwich–Schommer-Pries were available. Eventually, I would like to apply some known results from the theory of $(\infty,n)$-categories (established via, say, $n$-fold complete Segal spaces) to my gadgetry. So when I ask for "axioms to check", I secretly mean "a model satisfying B–SP's axioms", so that I can transfer the known results. | |
| Apr 22, 2014 at 21:34 | comment | added | Mike Shulman | There is a certain apparent similarity between completeness and skeletality, but it's misleading to call them similar because of what Karol said. Completeness is actually a very natural condition. Fortunately, there is a universal way to "complete" an incomplete category (which also distinguishes it from skeletality). | |
| Apr 22, 2014 at 20:32 | comment | added | Zhen Lin | @KarolSzumiło That's only if you know in advance to take the "classifying diagram" of the category, however! I am inclined to agree with Theo Johnson-Freyd that "completeness" is a kind of skeletality – of course, here I am using equality in the sense of homotopy type theory... | |
| Apr 22, 2014 at 11:03 | comment | added | Karol Szumiło | Just a remark: completeness is not a homotopy theoretic version of skelatality. When reduced to a plain $1$-category completeness says that every isomorphism is isomorphic in the category of arrows to an identity, which is trivially satisfied by every $1$-category. This condition is not unnatural, it merely asks for homotopical data not to "contradict" the categorical data in a Segal space. | |
| Apr 22, 2014 at 10:36 | comment | added | Adam Gal | There is also Kapranov and Dyckerhoff's n-Segal spaces (not the same as n-fold segal spaces) | |
| Apr 22, 2014 at 7:13 | comment | added | Zhen Lin | Did you have a look at Simpson's [Homotopy theory of higher categories]? He defines higher Segal categories by iterated weak enrichment, so it might be a good fit for your problem. | |
| Apr 22, 2014 at 4:32 | comment | added | David Roberts♦ | Have you tried turning it off and on... I mean, have you checked Tom Leinster's arxiv.org/abs/math.CT/0107188? | |
| Apr 22, 2014 at 0:36 | history | asked | Theo Johnson-Freyd | CC BY-SA 3.0 |