Timeline for Why the third stage of Cech nerve a Lie 2-groupoid?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2020 at 15:47 | vote | accept | Adittya Chaudhuri | ||
| Aug 22, 2020 at 15:47 | |||||
| Mar 5, 2020 at 17:06 | comment | added | Adittya Chaudhuri | @DmitriPavlov Thank you Sir. | |
| Mar 5, 2020 at 14:21 | comment | added | Dmitri Pavlov | A high-level discussion can be found in ncatlab.org/nlab/show/…. The article by Rezk math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf contains details in Section 7. | |
| Mar 5, 2020 at 6:30 | comment | added | Adittya Chaudhuri | @DmitriPavlov Thank you Sir. If possible can you please fill little detail about how "all simplicial levels of the source above level 3 get thrown away, whereas the 3rd level gets collapsed onto the 2nd"?.. Also since I am a beginner in this area it would be of immense help if you can suggest some literature which deals with this kind of question(setup). | |
| Mar 5, 2020 at 0:12 | comment | added | Dmitri Pavlov | All simplicial levels of the Čech nerve are nontrivial. However, if you map into a 2-truncated object, then all simplicial levels of the source above level 3 get thrown away, whereas the 3rd level gets collapsed onto the 2nd (i.e., two 2-simplices get identified if they are connected by a homotopy presented by a 3-simplex). | |
| Mar 4, 2020 at 18:52 | comment | added | Adittya Chaudhuri | @DmitriPavlov Thanks. (As I already mentioned that I am not well versed with the language of higher category yet.). So I am trying to understand your comment. I can understand that $B^{2}U(1)$ is 2-truncated. But my question is "when you dont consider the whole Cech nerve" then are you assuming that all the higher morphisms(greater than 2) in Cech nerve are trivial?? (I apologize priorly if I sound stupid. I started learning higher category very recently. ) | |
| Mar 4, 2020 at 18:33 | comment | added | Dmitri Pavlov | Not simplicial sets, but rather simplicial presheaves on the site of smooth manifolds. Both Č(U) and B^2 U(1) are such simplicial presheaves. There is no need to consider the whole Čech nerve because the target B^2 U(1) is 2-truncated, so only simplicial levels 0, 1, 2, and 3 of Č(U) matter. | |
| Mar 4, 2020 at 18:11 | comment | added | Adittya Chaudhuri | @DmitriPavlov Thanks. So that means in the construction of Principal 2-bundle as mentioned in ncatlab.org/nlab/show/… the morphism $g:C(U)\rightarrow B^{2}U(1)$ is actually a morphism of simplicial sets?(where C(U) is the Cech Nerve and $B^{2}U(1)$ is the one object 2 groupoid)...Then I guess they actually treated $B^{2}U(1)$ as the nerve of Lie 2-groupoid $B^{2}U(1)$?? But then it means they actually consider the whole Cech nerve ( not only upto its 3rd stage) as the former is a simplicial set and the later is not. | |
| Mar 4, 2020 at 17:48 | comment | added | Dmitri Pavlov | Nerves of bicategories were characterized by Duskin in Theorem 8.6 of his paper Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. However, there is no reason to prioritize the bicategorical model over the simplicial model; in fact, the latter is what typically is used in the realm of higher Lie theory and stacks. | |
| Mar 4, 2020 at 17:24 | history | edited | Adittya Chaudhuri | CC BY-SA 4.0 |
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| Mar 4, 2020 at 12:08 | history | edited | Adittya Chaudhuri | CC BY-SA 4.0 |
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| Mar 4, 2020 at 11:17 | history | answered | Adittya Chaudhuri | CC BY-SA 4.0 |