Timeline for real and complex vector spaces as topological categories
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 24, 2016 at 8:08 | vote | accept | KotelKanim | ||
| Feb 23, 2016 at 19:32 | comment | added | Denis Nardin | @SebastianGoette One reason for desiring such a thing is that it would produce a notion of "vector bundle over a stratified space", where the gluing map between each strata can be any linear map, since stratified spaces can be represented by $\infty$-categories. Of course we've seen that the desired $\infty$-category does not exists, so the point is moot. | |
| Feb 23, 2016 at 16:34 | comment | added | Sebastian Goette | I thought that people usually pass to $\infty$-categories if strict categories do not work for a certain problem. Here, I have the impression that all the OP asks for holds in strict categories, enriched over $Top$. Hence, I asked for additional motivation in my first comment. | |
| Feb 23, 2016 at 15:49 | comment | added | Yonatan Harpaz | If you take the $\infty$-category associated to the usual enrichment then the adjunction works but all the groups of self-equivalences are trivial (in fact, the entire $\infty$-category is equivalent to a point). If you first take the maximal sub-groupoids of $Vect_{\mathbb{C}}$ and $Vect_{\mathbb{R}}$ (with their usual enrichment), and then take the associated $\infty$-categories then the groups of self-equivalences are what you want, but you don't have an adjunction any more. The question is if there exists an $\infty$-category which gives both. | |
| Feb 23, 2016 at 15:37 | comment | added | Denis Nardin | @SebastianGoette I don't understand. Of course $\infty$-adjuctions are the same things as ordinary adjunctions if the categories are strict. Sure, the classical enrichments do the job if the categories in question are strict, but we are trying to make the categories not to be strict (so to forget all additional structure like monomorphisms and epimorphisms that allow you to reconstruct $O_n$ from the ordinary adjunction). I feel I'm missing something of what you're saying | |
| Feb 23, 2016 at 15:20 | comment | added | Sebastian Goette | @DenisNardin Excuse my naïvity. I thought that the "classical" adjunction of extension/restriction is compatible with the topologies, and that "$\infty$-adjunction" reduces to "adjunction" if the categories happen to be strict? The rank is a strange notion. It means that morphisms don't factor over some "small" objects, and I don't see how adjunctions would see this. | |
| Feb 23, 2016 at 15:11 | comment | added | Denis Nardin | The usual approach is to utilize the topological category given by the isometric embeddings. This works well enough but it does not satisfy the condition on the adjunction. | |
| Feb 23, 2016 at 15:09 | comment | added | Denis Nardin | @SebastianGoette In $\infty$-categories there's no notion of monomorphism or epimorphism. The question is asking whether there's a topological enrichment of $Vect$ such that the interior of the resulting $\infty$-category is $\coprod_n BO_n$ and such that the natural adjunction between real and complex vector spaces can be upgraded to an adjunction of $\infty$-categories. It seems a very reasonable question to me. | |
| Feb 23, 2016 at 13:40 | answer | added | Yonatan Harpaz | timeline score: 5 | |
| Feb 23, 2016 at 13:20 | comment | added | Sebastian Goette | I don't understand what the rank condition has to do with your actual question (the "real" rank seems to depend on the real structure one chooses on the complex vector space that is the source of a morphism. It gives additional structure). I also have the impression that the "classical" enrichments do the job even in the strict categorical sense. Finally, although the mapping spaces are contractible, they carry more structure. For example, the subspaces of mono- or epimorphisms are typically not contractible. Maybe some additional motivation would help me to see your point. | |
| Feb 23, 2016 at 8:33 | history | edited | KotelKanim | CC BY-SA 3.0 |
deleted 4 characters in body
|
| Feb 23, 2016 at 7:17 | history | asked | KotelKanim | CC BY-SA 3.0 |