Timeline for What's a groupoid? What's a good example of a groupoid?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 16, 2016 at 13:08 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| May 22, 2015 at 10:02 | comment | added | Ronnie Brown | This discussion should be linked to mathoverflow.net/questions/40945/… | |
| Oct 18, 2013 at 1:09 | comment | added | David Roberts♦ | Ok, I mean the bicategory of proper etale Lie groupoids with anafunctors as 1-arrows (or equivalently, right principal bibundles or ...) | |
| Oct 17, 2013 at 6:49 | comment | added | Maik Köster | Constructing the 2-category David Roberts mentions is an intermediate step in the construction of the 1-categories which are actually isomorphic (not just equivalent). One should remark that in this 1-category of groupoids, the objects are not the groupoids but rather equivalence classes of groupoids. These groupoids are even more extended to be a pair of a groupoid and a topological space to be able to distinguish orbifolds which are diffeomorphic but not identical. | |
| Oct 17, 2013 at 6:45 | comment | added | Maik Köster | However, if you use groupoids as objects on the one side and orbifolds as objects on the other side, the categories are just equivalent not isomorphic. Switching between the categories forgets the information on diffeomorphisms. This is not sufficient if you want to investigate the diffeomorphism group of an orbifold. | |
| Oct 17, 2013 at 6:45 | comment | added | Maik Köster | This is fine if you are mainly interested in properties of groupoids. However, when you are interested in orbifolds as geometric objects, you need your objects to be proper orbifolds (as you have manifolds as objects in the category of manifolds and not topological spaces and a choice of covering charts). | |
| Oct 17, 2013 at 6:39 | comment | added | Maik Köster | @David: It depends what you want to do. In the 2-category and bi-category approaches, the objects on the groupoid side are actual groupoids while the objects on the orbifold side are pairs consisting of the topological space underlying the orbifold and a choice of an orbifold atlas representing the orbifold structure. In this case, each orbifold gives rise to infinitely many objects. | |
| Oct 17, 2013 at 0:25 | comment | added | David Roberts♦ | A note for readers: it's better to use the 2-category of groupoids, suitably localised, than the 1-category :-) | |
| Oct 16, 2013 at 18:45 | history | answered | Maik Köster | CC BY-SA 3.0 |