Timeline for What can be an appropriate notion of principal bundle over a category (with an appropriate notion of local trivialisation)?
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26 events
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| May 19, 2020 at 12:37 | vote | accept | Adittya Chaudhuri | ||
| May 17, 2020 at 16:04 | comment | added | Adittya Chaudhuri | @HarryGindi Thank you very much for the explanation . | |
| May 17, 2020 at 16:02 | comment | added | Harry Gindi | @AdittyaChaudhuri If you read more of the paper, they say that local triviality in their sense becomes automatic, which is what I said. 'As before, this means that the local triviality clause appearing in the traditional definition of principal bundles is not so much a characteristic of principality as such, as rather a condition that ensures that a given quotient taken in a category of geometric spaces coincides with the “refined” quotient obtained when regarding the situation in the ambient ∞-topos.' This will be my last message on the subject. | |
| May 17, 2020 at 15:54 | comment | added | Adittya Chaudhuri | @HarryGindi Please note I assumed that "they have assumed a notion of local triviality" when they talked about principal bundle over a groupoid or a stack in the last paragraph of the page 18 of the paper mentioned in the link(in the previous comment). | |
| May 17, 2020 at 15:49 | comment | added | Adittya Chaudhuri | @HarryGindi In the page 18 , in section 3.1 in arxiv.org/pdf/1207.0248.pdf they mention that local triviality can be understood in the context of higher geometry... Also in the last paragraph of the page they mention that it's possible to have a notion of Principal bundle over a base groupoid or stack. Then according to the previous comments you made the notion of local triviality they are referring must be different from the usual one I have. Since I have very little background in infinity category it would be very helpful if you clarify "their" notion of "local triviality". | |
| May 17, 2020 at 15:35 | comment | added | Adittya Chaudhuri | @HarryGindi Thank you very much for your valuable comments. | |
| May 17, 2020 at 15:34 | comment | added | Harry Gindi | @AdittyaChaudhuri more or less. | |
| May 17, 2020 at 15:31 | comment | added | Adittya Chaudhuri | @HarryGindi Ok. That means you are saying we can expect a notion of local triviality in the case of categories internal to $Diff_{infty}$ because they carry a topology coming from its object set and morphism set. But for a general category there is no reason to expect such notion of local triviality... Am I right? | |
| May 17, 2020 at 15:28 | comment | added | Harry Gindi | @AdittyaChaudhuri Those are internal categories in $\mathbf{Diff}_\infty$, so they do carry a topology. These are much more complicated structures than what you are contemplating. | |
| May 17, 2020 at 15:25 | comment | added | Adittya Chaudhuri | @HarryGindi Please note these definitions are present in the section 2.2 in the page "contents" of the paper mentioned in the link(in the previous comment). Thank you. | |
| May 17, 2020 at 15:20 | comment | added | Adittya Chaudhuri | @HarryGindi Ok. Thank you. I am trying to understand. But if you see the definition 2.12 ,2.13 ,2.14 in arxiv.org/pdf/hep-th/0412325.pdf then is it not natural to expect a notion of locally trivial principal bundle over a general category? (I agree the fact that in this paper they have worked only with categories internal to some generalised smooth spaces(like diffeological space or Chen space))...But can we expect to have an appropriate meaning of locally trivial principal bundle if we extend the notion from category int to smooth space to a general category? | |
| May 17, 2020 at 15:08 | comment | added | Harry Gindi | @AdittyaChaudhuri The property is just the property of being a Kan fibration. | |
| May 17, 2020 at 15:06 | comment | added | Adittya Chaudhuri | @HarryGindi Ok. But I guess the infinity groupoid (taking homotopy between homotopies and so on.....) contains a sufficient information for the reconstruction of the space upto weak equivalence (provided the space is a CW complex) .. Then why can't we have an appropriate notion of local triviality for a principal bundle over an infinity groupoid? | |
| May 17, 2020 at 15:00 | comment | added | Harry Gindi | @AdittyaChaudhuri The fundamental groupoid can only see the locally constant sheaves on your space. It's a special property of locally constant sheaves of sets that they can be reconstructed up to isomorphism from their homotopical data assuming that your original space is nice enough. | |
| May 17, 2020 at 14:54 | comment | added | Adittya Chaudhuri | @HarryGindi In response to your comment " Any two points that are connected by a path are homotopically indistinguishable, so it destroys the topology."... but in that case we can talk about fundamental groupoid of the space..(which captures the information of 1st homotopy group and the path components of the space)... Then if we have an appropriate notion of a locally trivial principal bundle over a path groupoid then I guess we can expect that it will also capture a portion of topological information present in a principal bundle over the original space. | |
| May 17, 2020 at 14:47 | comment | added | Praphulla Koushik | @HarryGindi The statement "Any two points that are connected by a path are homotopically indistinguishable, so it destroys the topology" is acceptable, but, I do not see how that answers my question :O Can you elaborate how it answers my question? | |
| May 17, 2020 at 14:26 | comment | added | Harry Gindi | @PraphullaKoushik Any two points that are connected by a path are homotopically indistinguishable, so it destroys the topology. | |
| May 17, 2020 at 14:14 | comment | added | Praphulla Koushik | What does it mean to say "Local triviality doesn't really make sense homotopically/ categorically. It's not a fully homotopy-invariant property"? Does it mean one can find maps $f,g:X\rightarrow Y$ that are homotopy equivalent such that $f$ is locally trivial in some sense, say locally trivial fibre bundle, but $g$ is not a locally trivial fibre bundle? | |
| May 17, 2020 at 9:59 | comment | added | Adittya Chaudhuri | I can understand . But if I take a Grothendieck Pretopology on Cat then we can choose a cover for the category $C$ with respect to the pretopology. Then if there is a notion of Cech Groupoid $\tilde{Ch}(U)$ corresponding to this cover on $C$ and if we consider a functor/2-functor $\tilde{Ch}(U) \rightarrow B^2G$ then what we are expecting to get? (Note here I am just using basic category theory with no notion of infinity category) | |
| May 17, 2020 at 9:51 | comment | added | Harry Gindi | Local triviality doesn't really make sense homotopically/categorically. It's not a fully homotopy-invariant property. The fact that this works for topological spaces is because you are working with nice spaces that have a close relationship between their topological and homotopical structure, somehow. | |
| May 17, 2020 at 9:48 | comment | added | Adittya Chaudhuri | Please note I mentioned about local triviality in (2) in the question. | |
| May 17, 2020 at 9:44 | comment | added | Adittya Chaudhuri | Thank you for the answer. Although I know very basics of infinity categories but I am curious to know whether the notion of local triviality in "Principal bundle over a category" is automatically followed from the notion you gave in the answer? | |
| May 17, 2020 at 9:41 | comment | added | Harry Gindi | In complete generality, being a principal G-space is an additional structure that has to be specified and checked to be compatible with the data in the fibration, but this is the rough gist of it. | |
| May 17, 2020 at 9:34 | history | edited | Harry Gindi | CC BY-SA 4.0 |
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| May 17, 2020 at 9:28 | history | edited | Harry Gindi | CC BY-SA 4.0 |
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| May 17, 2020 at 9:20 | history | answered | Harry Gindi | CC BY-SA 4.0 |