Timeline for Do locally convex topological vector spaces embed into diffeological spaces?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 17, 2015 at 5:10 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Added reference to complementary answer
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| Jun 16, 2015 at 2:29 | comment | added | David Roberts♦ | Thanks, Pedro. I may ask Helge and others in his circle of colleagues direction if they know of any result in that direction. | |
| Jun 16, 2015 at 2:28 | vote | accept | David Roberts♦ | ||
| Jun 16, 2015 at 1:25 | comment | added | Pedro Lauridsen Ribeiro | I have edited my answer to better suit your original question, based on your comments. I've also added a remark at the end concerning your remaining question, to which unfortunately I don't know an answer. Papers who cite Glöckner's work quoted above, according to MathSciNet, apparently don't even raise that question, let alone answer it. | |
| Jun 16, 2015 at 1:13 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Improved explanation in view of OP's comments, superfluous discussion on continuous linear maps removed
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| Jun 16, 2015 at 0:41 | comment | added | David Roberts♦ | I was thinking of the category of lctvs and MB-smooth maps, and the category of diffeological spaces. My understanding is that diffeological maps between lctvs are the same thing as convenient smooth maps. So, all up, I think you are correct in your summary of my residual question. I can ask a new question, if this is too much, and accept this answer if you like, since it did cover my original question. | |
| Jun 15, 2015 at 23:04 | comment | added | Pedro Lauridsen Ribeiro | So, you were considering smooth maps (in either sense) as the arrows of the category of lctvs in your original question, and your residual question is whether the answer changes if you restrict the arrows to be smooth diffeomorphisms. Is this corrrect? | |
| Jun 15, 2015 at 22:59 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Improved explanation in view of comments by the OP
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| Jun 15, 2015 at 22:39 | comment | added | David Roberts♦ | I wasn't thinking of linear maps, and I was trying to compare to the statement for Fréchet spaces, which by embedding means a full and faithful functor (and taking all smooth maps between Fréchet spaces). My residual query is whether a convenient diffeomorphism is necessarily MB-smooth, which is intermediate between the inclusion being fully faithful and being just faithful. | |
| Jun 15, 2015 at 14:43 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
small rewording
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| Jun 15, 2015 at 14:43 | comment | added | Pedro Lauridsen Ribeiro | @DavidRoberts so, what is the definition of "embedding of categories" you have in mind in your question? Do you put any restriction on the arrows in the category of lctvs besides being continuous linear maps (e.g. do you require them to be topological isomorphisms, or something)? | |
| Jun 15, 2015 at 6:17 | comment | added | David Roberts♦ | "there are maps between E and F which are smooth in the diffeological sense and need not even be continuous, let alone Michal-Bastiani smooth." <-- ah, this is what I wanted to know! Although it wasn't clear from how I asked it, since 'embedding between categories' doesn't have a universal definition. So I guess the only thing to worry about is if there diffeological isomorphisms (not necc. linear) that are not MB-smooth. This seems like a difficult question. | |
| Jun 15, 2015 at 5:53 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Conclusion rectified in order to match previously incompletely understood hypothesis, reasoning unchanged
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| Jun 15, 2015 at 5:36 | history | answered | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |