Timeline for Why are we interested in the Fundamental Groupoid of a Space?
Current License: CC BY-SA 3.0
7 events
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| Jan 9, 2019 at 12:19 | comment | added | Ronnie Brown | @HJRW On further thought, covering space work is mainly about pullbacks and van Kampen is about pushouts (in both cases, of spaces or groupoids). One aspect of the latter is modelling identification of points: one uses the bifibration of categories $Ob: Groupoids \to Set$: I keep on being surprised from questions on these web sites that algebraic topology students often are not told told how to model topology by fibrations, covering morphisms, universal morphisms,, orbit morphisms, ..., of groupoids. See my books. | |
| Jan 8, 2019 at 17:59 | comment | added | Ronnie Brown | @HJRW I got into groupoids through writing a text on topology in the 1960s and getting dissatisfied with a van Kampen theorem that did not compute the fundamental group of the circle. It turned out that one needed 2 base points for this! Going down to 1 base point does not work for this example! After a 1967 meeting with George Mackey, I decided to write an account of covering spaces via groupoids, as in the first edition. Later, after work of Higgins and Taylor, I added orbit spaces. Your guess seems wrong to me! And higher dim groupoids can generalise deep work of JHCW! | |
| Jan 2, 2019 at 1:05 | comment | added | HJRW | @RonnieBrown : I would argue that (carefully) throwing away information is exactly what algebraic topology is all about. After all, that’s what an invariant is. Your point about pathological spaces is well taken. But, for non-pathological spaces, I still guess that the two proofs are essentially the same. | |
| Jan 1, 2019 at 22:40 | comment | added | Ronnie Brown | @HJRW The van Kampen theorem for a set of base points calculates examples of unions where the subspaces may be bad locally and so have no reasonable covering spaces. Further, as Grothendieck writes in 1984 in "Esquisses ..." Section 2 : "...people still obstinately persist,when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation,.." Why throw away information? | |
| Oct 30, 2018 at 13:23 | comment | added | HJRW | Of course, one of the things that's going on here is that covering spaces are very closely related to groupoids: the fundamental group lifts to a groupoid of paths between the preimages of the base point. Probably (though I haven't checked) the groupoid and covering space versions are more or less the same proof from different points of view. | |
| Jul 15, 2014 at 19:06 | comment | added | Jorge António | I really appreciate your answer. Thanks! | |
| Jul 15, 2014 at 3:53 | history | answered | Omar Antolín-Camarena | CC BY-SA 3.0 |