Timeline for Is continuity of a functor stable under pullback?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 16, 2014 at 4:33 | comment | added | Mike Shulman | I forget who it was who said that the only thing worse than bad terminology is continually changing terminology. Cartesian square also means pullback. | |
| Apr 15, 2014 at 16:32 | comment | added | Adam Gal | It just doesn't work as a verb - "pulling back x along y" is much clearer than "taking the comma object and considering the resulting morphism" | |
| Apr 15, 2014 at 16:09 | comment | added | Zhen Lin | "Comma object" is at worst as bad as "comma category". Do you object to that too? | |
| Apr 15, 2014 at 12:12 | comment | added | Adam Gal | I agree, but I really dislike the term "comma object" :-/ Does "Cartesian square" mean final square though? or is this also synonymous with pullback in your meaning? | |
| Apr 14, 2014 at 13:37 | comment | added | Mike Shulman | Of course it's a matter of terminology. (-: But there are good reasons to use terminology in one way rather than another, such as consistency with the literature and fitting into a general framework. | |
| Apr 13, 2014 at 11:03 | comment | added | Adam Gal | AFAIK the only context where the diagonal map is used is when you deal with biproducts, which is not really the general case. | |
| Apr 13, 2014 at 11:01 | comment | added | Adam Gal | @MikeShulman I guess it is a matter of terminology. I always thought of pullbacks in a regular category as final squares, so that's the generalization I made. | |
| Apr 9, 2014 at 19:58 | comment | added | Mike Shulman | And of course a comma object is the same as a final square. My point is that neither comma objects or final squares are the same as "lax pullbacks". A lax pullback is the lax limit of a cospan. | |
| Apr 9, 2014 at 19:57 | comment | added | Mike Shulman | @AdamGal, the definition I linked to is for 2-categories (i.e. (2,2)-categories) where there is really only one model. If you wanted an analogous definition for $(\infty,2)$-categories, you'd have to formulate it in an appropriate way for your chosen model. | |
| Apr 8, 2014 at 19:04 | comment | added | Adam Gal | @MikeShulman I think the definition of lax pullback in nlab is not very model agnostic. If you just define the lax pullback as a final square (with the appropriate assumptions) then I think the comma category is exactly a lax pullback. | |
| Apr 8, 2014 at 16:42 | comment | added | Dimitri Chikhladze | Yes, this did occur to me. Comma objects used to be called lax pullbacks too. Although terminology might be standardizing now. So my remark is really about comma objects in Cat. | |
| Apr 8, 2014 at 15:54 | comment | added | Mike Shulman | It depends on what you mean by "lax pullback". The use of "lax pullback" to mean "comma object" is better avoided, since comma objects are not actually any sort of lax limit. See ncatlab.org/nlab/show/2-limit#lax . | |
| Apr 8, 2014 at 9:59 | comment | added | Dimitri Chikhladze | "2.16 Limits in comma categories" in "Francis Borceux, Handbook of Categorical Algebra 1, Cambridge University Press". In Cat lax pullbacks are comma categories. | |
| Apr 7, 2014 at 23:03 | comment | added | Adam Gal | I suspected as much :-) can you give a reference for it? Maybe there is a proof which extends easily. | |
| Apr 7, 2014 at 16:19 | comment | added | Dimitri Chikhladze | If everything is a 1-category, then what is true is that if C and D have limits, and i and p preserve them, then the lax pullback has limits and both projections preserve them. | |
| Apr 6, 2014 at 21:55 | history | asked | Adam Gal | CC BY-SA 3.0 |