Timeline for Lie groupoids being homotopy equivalent
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 20, 2020 at 17:54 | vote | accept | Praphulla Koushik | ||
| Mar 1, 2020 at 16:05 | comment | added | Praphulla Koushik | @SebastianGoette Please let me know if you have any favorite reference for this set up? | |
| Feb 28, 2020 at 9:51 | comment | added | Sebastian Goette | @PraphullaKoushik I am sorry, I am just learning groupoids myself. But here is my naive comment anyway: If you want to view a Lie groupoid with a single object as a Lie group $G$, then the version in the answer fits nicely. If you want to view the very same groupoid as a model for the classifying space $BG$ (as a differentiable stack), then you should use some version that is compatible with principal bibundles. It seems to me that both versions make sense, depending on the context. | |
| Feb 27, 2020 at 4:47 | comment | added | Praphulla Koushik | @SebastianGoette So, can you please tell me what you feel could be another choice for the definition of Homotopy equivalence? | |
| Feb 26, 2020 at 20:32 | comment | added | Sebastian Goette | @PraphullaKoushik you said in a comment on the question that you want to keep the space of objects. If you consider principal bibundles as morphisms instead of the morphisms in the answer, you only keep information about orbits of points and about their stabilisers. | |
| Feb 24, 2020 at 6:44 | comment | added | Praphulla Koushik | I believe that is the standard way to talk about Homotopy equivalences. But, I was thinking if we should consider the category of Lie groupoids whose morphisms are morphisms of Lie groupoids or that of bibundles (I have the set up of differentiable stacks in mind).. So, in that case should we involve the notion of bibundle (generalized morphism) when defining the notion of Homotopy equivalence? | |
| Feb 23, 2020 at 19:28 | comment | added | John Greenwood | Yeah, this is the standard notion of homotopy for groupoids. In fact this construction using the "cylinder object" $X\times[0,1]$ is the standard way to define "homotopy" in any model category see pg 233-234 of Quillen's "Rational Homotopy theory" | |
| Feb 23, 2020 at 2:59 | comment | added | Praphulla Koushik | Yes, that one I am aware of associating a topological groupoid for a topological space, Lie groupoid for a manifold.. Has this formal notion of Homotopy equivalence used any where else in literature, in similar way a notion of homotopy equivalence of topological spaces used? | |
| Feb 23, 2020 at 2:22 | comment | added | John Greenwood | it might be confusing because it's formally identical to "homotopy equivalence" of topological spaces. But in the above all the objects are topological groupoids and all maps between them are maps of groupoids. Note that a topological space gives rise to a pretty canonical topological groupoid by declaring that there are no non-identity morphisms, i.e. the space $X$ becomes the groupoid $X\rightrightarrows X$. | |
| Feb 23, 2020 at 1:36 | comment | added | Praphulla Koushik | I think I am missing something here...You are only saying about Homotopy equivalence of topological spaces... There is a +1 to your answer, so, there must be something interesting in your answer, it is just that I am misunderstanding | |
| Feb 22, 2020 at 19:07 | history | answered | John Greenwood | CC BY-SA 4.0 |