Timeline for Compare three 2-categories of (Lie) groupoids
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Dec 8, 2011 at 18:32 | comment | added | David Carchedi | Sure thing David. I still need to set that up. As soon as I do, I'll update it on here as well. | |
| Dec 6, 2011 at 0:16 | comment | added | David Roberts♦ | David - could you update your MO user page with your new MPIM home page when it is ready? Thanks. | |
| Nov 22, 2011 at 10:44 | comment | added | David Carchedi | @Ma Ming: It appears they keep changing the link location. Just google "Categorical Properties of Topological and Differentiable Stacks". This week, I will also make a new home page here at the MPIM and put a copy up on there. | |
| Nov 22, 2011 at 9:33 | comment | added | Ma Ming | @David Carchedi Sorry, your link is broken or denial of access for outsiders. | |
| Nov 22, 2011 at 9:04 | comment | added | Ma Ming | @David Carchedi Thank you! O.K. It seems that I ignored there is an equivalent relationship of 2-cells in (1), I will be back after I found it. | |
| Nov 21, 2011 at 20:21 | comment | added | David Carchedi | ... one shows that G and H represent the same $2$-cell, hence there is a unique $2$-cell which is the endomorphism of the identity span on the terminal groupoid, showing that it is still terminal in the bicategory of spans. | |
| Nov 21, 2011 at 20:19 | comment | added | David Carchedi | @Ma Ming: After looking again at your proposed counter-example, I think you have misunderstood how the $2$-cells work. See "Etendues and stacks as a bicategory of fractions" by Pronk. If $*$ is the terminal groupoid, and we consider the identity span (as you suggest), then any (single, not pair) Lie groupoid SUCH THAT its unique map to $*$ is a Morita equivalence (and hence a strong equivalence), "gives" an endo-2-cell of the identity span. But, there is an equivalence relation we must mod-out by, and if G and H are two such groupoids, using strong equivalences $* \to G$ (and to H$)... | |
| Nov 21, 2011 at 19:56 | history | edited | David Carchedi | CC BY-SA 3.0 |
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| Nov 21, 2011 at 18:40 | comment | added | David Carchedi | @Ben: Clearly smooth functors was supposed to be smooth functions, woops. | |
| Nov 21, 2011 at 18:37 | comment | added | David Carchedi | @Ma Ming: TO convince you more: Make a 1'') where each Morita equivalence $K \to H$ used comes equipped with a choice of local sections of $t \circ pr_1:H_1 \times_{H_0} K_0 \to H_0$. So let this be part off a span H<-K->G. Then, if the cover is U, you can construct a span H<-H_U->G which is isomorphic (in the morphism groupoid in spans) to your original. But this is the same data as a principal bundle for G over H + a choice of local sections, so we get a 2-functor from (1'') to (2)'. It's "inverse" to the one (implied) in my answer. | |
| Nov 21, 2011 at 18:24 | comment | added | David Carchedi | @Ma Ming: I'm not sure I understand what you are saying (it's hard to squish this into a comment). Feel free to email me instead. Anyhow, one thing that I didn't mention is that there is also the "data" of the natural isomorphism between the "two different induced maps" $H_{U \cap U'} \to H$ but these two maps actually agree on the nose. | |
| Nov 21, 2011 at 18:21 | comment | added | David Carchedi | @Konrad: Dorette Pronk's thesis is a good reference for comparing the bicategory of fractions approach to the one in terms of stacks. She only proves it for etale stacks, but it works without this assumption if you remove her use of topoi and rephrase things in terms of stacks of torsors. If I may be so bold, I would suggest my thesis for the proof that (2) is equivalent to differentiable stacks, as I work this out in some detail: igitur-archive.library.uu.nl/dissertations/2011-0830-200501/… | |
| Nov 21, 2011 at 18:14 | comment | added | David Carchedi | @Benjamin: Not quite. First, you need these etendues to be "smooth", i.e. be locally isomorphic to $R^n$ with its ring of smooth functors (just like arbitrary locally ringed spaces need nor be manifolds). Secondly, you have to restrict to Lie groupoids which are (Morita equivalent to) etale Lie groupoids. This breaks down when stuff isn't etale. | |
| Nov 21, 2011 at 18:11 | comment | added | Benjamin Steinberg | I believe that these are also the same in 2-category and as ringed etendues. | |
| Nov 21, 2011 at 17:05 | comment | added | Ma Ming | If my understand of biequivalent of bicategories are correct: locally equivalent and surjecttive-up-to-equivalence on objects, then I have a simple counterexample illustrated that 2-cells in (1) is much more than (2). Let G and H be the most trivial groupoid , morphism and objects spaces are both one point manifold. Let M and N also be *. Then the induced spans of Lie groupoids are *<<---->* and <<---->*. However any pair Lie groupoid will give a 2-cell between such spans. | |
| Nov 21, 2011 at 16:53 | comment | added | Konrad Waldorf | What is a good reference for the equivalence of these four bicategories? | |
| Nov 21, 2011 at 16:41 | history | answered | David Carchedi | CC BY-SA 3.0 |