Timeline for When is the dual of a limit the same as the colimit of the duals?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| S Jun 26, 2020 at 13:03 | history | bounty ended | CommunityBot | ||
| S Jun 26, 2020 at 13:03 | history | notice removed | CommunityBot | ||
| Jun 23, 2020 at 21:46 | comment | added | Daniel Robert-Nicoud | @paulgarrett Thanks, that's an interesting example :) | |
| Jun 18, 2020 at 17:56 | comment | added | paul garrett | Not an answer in the context you want, but, in the category of locally convex topological (complex) vector spaces, I think it is not always true that the dual of a limit is the colimit of the duals (even just as sets or as vector spaces without topologies). Namely, the (only) proof that I know seems to require that the limitands in the limit be Banach spaces. I do not have a counter-example to prove the necessity of this condition, because I've not needed a more general statement (yet?) Anyway, it does not seem true for "general" reasons. | |
| Jun 18, 2020 at 16:14 | comment | added | Daniel Robert-Nicoud | @SamGunningham Of course, limits in $C_{ft}$ are the same as limits in $C$ (when they exist). But what happens in more general cases? Is there a clean explanation of this phenomenon? (Am I doing something wrong? Am I missing something obvious?) | |
| Jun 18, 2020 at 16:04 | history | edited | Daniel Robert-Nicoud | CC BY-SA 4.0 |
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| Jun 18, 2020 at 16:03 | comment | added | Daniel Robert-Nicoud | @SamGunningham The point is that I am not taking the limits and colimits in the category of finite type chain complexes but in the category of all chain complexes instead and asking for $\lim D$ to have finite type. I'll try to make this clearer in the OP. Also a question would be what are possible good definitions of "duality" and "objects of finite type" in other categories. | |
| Jun 18, 2020 at 12:29 | comment | added | Sam Gunningham | Perhaps I am missing something, but is this phenomenon not just happening because the dual functor gives an equivalence of categories $C_{ft} \simeq C_{ft}^{op}$? So it takes (co)limits in $C_{ft}$ to (co)limits in $C_{ft}^{op}$. | |
| S Jun 18, 2020 at 11:41 | history | bounty started | Daniel Robert-Nicoud | ||
| S Jun 18, 2020 at 11:41 | history | notice added | Daniel Robert-Nicoud | Authoritative reference needed | |
| Jun 17, 2020 at 16:11 | history | edited | Daniel Robert-Nicoud | CC BY-SA 4.0 |
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| Jun 15, 2020 at 15:30 | history | asked | Daniel Robert-Nicoud | CC BY-SA 4.0 |