Timeline for Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?
Current License: CC BY-SA 3.0
22 events
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| Oct 22, 2012 at 7:45 | history | edited | Jonathan Chiche | CC BY-SA 3.0 |
Corrected a silly error in the title.
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| Oct 22, 2012 at 7:30 | comment | added | Jonathan Chiche | Thanks to those who expressed their opinion by upvoting Andrej Bauer's comment. Special thanks to Mike, who made me aware that I was not the only one not to get the LaTeX parsed in the main questions list. I have now edited the title accordingly. | |
| Oct 22, 2012 at 7:28 | history | edited | Jonathan Chiche | CC BY-SA 3.0 |
Title shortened and LaTeX removed from it, added final remark.
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| Oct 1, 2012 at 0:06 | comment | added | Mike Shulman | I agree that your title would be improved by a little concision, even if not as extremely so as Andrej's. I also think it looks ugly to have dollar signs in a title. TeX is parsed when displaying the title of the question on this page, but not in the main questions list. | |
| Sep 30, 2012 at 13:32 | comment | added | Michal R. Przybylek | Jonathan, I do think that Andrej's title is far better than yours:-) | |
| Sep 30, 2012 at 13:30 | comment | added | Michal R. Przybylek | It seems that the difference between your characterisation of $z$ and the above characterisation is that you lack the stability condition. Notice also (as I have written in my first comment), that your property for $\mathbb{W}$ is weaker than saying "there exists an object with has an internal terminal point" --- clearly, there may be a lot of inequivalent reasons that $\mathbb{W}$ satisfies your property. | |
| Sep 30, 2012 at 13:30 | comment | added | Michal R. Przybylek | Jonathan, I tried to explain Eric's comment in (perhaps) a more familiar setting. Because, the 2-category $\mathbf{Cat}^{\mathbb{W}^{op}}$ is always (bi)cartesian closed, you may define the concept of internal propositional connectives in any 2-category $\mathbb{W}$. In our case we may say that $w$ has a terminal point (or internal truth point) if the map $\hom(-, z) \rightarrow 1$ has a right adjoint, where $1$ is a terminal object in $\mathbf{Cat}^{\mathbb{W}^{op}}$. (cont) | |
| Sep 30, 2012 at 7:52 | comment | added | Jonathan Chiche | Thnaks for your explanation. I do not want to discuss further this particular case, but I want to apologize if you have found the tone of my comments harsh. I just would like to point out that I recalled one of the moderators (I forgot which one) once explained that MO users are given plenty of space to write the title of the questions and that he thought users should take advantage of that to make title precise. Therefore, it was unclear for me whose opinion the change you proposed reflected. (I would like to apologize for the many grammar mistakes in my previous comments too.) | |
| Sep 30, 2012 at 3:34 | comment | added | Andrej Bauer | Here is something for people to upvote or not: your title is too long, a shorter one would be better. | |
| Sep 30, 2012 at 3:32 | comment | added | Andrej Bauer | Your title is quite long and it contains the entire question. It looked ugly to me so I changed it to a shorter one. You did not like the change, so you reverted it. I don't care enough about this to discuss it. I aplogize if you found my change inappropriate. | |
| Sep 30, 2012 at 1:26 | answer | added | Michael Barr | timeline score: 4 | |
| Sep 29, 2012 at 23:43 | history | edited | Jonathan Chiche | CC BY-SA 3.0 |
Going back to the original title. See my comments.
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| Sep 29, 2012 at 23:41 | comment | added | Jonathan Chiche | If people want to object then I will be glad to hear their objections that they could wite in comments. Other users can upvote comments and I will certainly listen to objections if the community back them up. | |
| Sep 29, 2012 at 23:40 | comment | added | Jonathan Chiche | (continued) deleting question marks has the effect that the title is not a question anymore, and I want the title of my questions to be real questions. (The button on the right of my screen is called "Ask Question".) In addition, I want them to be precise. One of the reasons for this choice is that I would not like people to waste their time reading my questions if they could know by the title that they are not interested by them. The modification makes the title lose its precision as well as its character of being a question. Therefore, I am going back to the original one. (Continued.) | |
| Sep 29, 2012 at 23:29 | comment | added | Jonathan Chiche | I appreciate that some people try to make others' questions better by fixing typos and LaTeX, correcting language mistakes, suggesting changes and so on, but, in this case, I would have liked to know why Andrej Bauer thought the title of my question would be better the way he has put it. In the past, I have edited the title of a question of mine after someone suggested I do so in a comment. If such a choice have been made here (suggesting instead of modifying arbitrarily), I could have replied (before the title was changed) that, for one thing, (continued) | |
| Sep 29, 2012 at 23:19 | comment | added | Jonathan Chiche | Thanks for your comments. Perhaps I'm missing something (it's quite late here), but don't they both assume that the 2-category has a "terminal object" (for Eric Wofsey) or an object called "1" (Michal R. Przybylek) the definition of which is not clear for me right now? I'll try to understand tomorrow if I'm missing something indeed. But, yes, I agree that such an object need not be unique. | |
| Sep 29, 2012 at 18:59 | comment | added | Michal R. Przybylek | Extending Eric's comment --- one may say, that an object $z$ has an internal terminal point if the unique arrow $z \rightarrow 1$ has a right adjoint $\top \colon 1 \rightarrow z$. By 2-Yoneda this means that the transformation $\hom(-, z) \rightarrow \hom(-, 1)$ has a right adjoint, which, modulo a kind of Beck-Chevalley condition (the stability condition), is equivalent to saying that for each $x$ the category $\hom(x, z)$ has a terminal object. | |
| Sep 29, 2012 at 18:19 | comment | added | Eric Wofsey | I don't know whether this is useful in your context, but an alternate definition of "category with a terminal object" you might consider is an object $z$ such that there exists an adjunction between $z$ and the terminal object of your 2-category. | |
| Sep 29, 2012 at 15:13 | comment | added | Michal R. Przybylek | Jonathan, I guess the problem with your property is that "an object $z$ such that $\hom(x, z)$ has a terminal object" need not be unique --- even up to equivalence. | |
| Sep 29, 2012 at 13:41 | history | edited | Andrej Bauer | CC BY-SA 3.0 |
edited title; edited title
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| Sep 29, 2012 at 12:33 | history | edited | Jonathan Chiche | CC BY-SA 3.0 |
deleted 60 characters in body; edited title
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| Sep 29, 2012 at 9:10 | history | asked | Jonathan Chiche | CC BY-SA 3.0 |