Timeline for Why is the definition of the higher homotopy groups the "right one"?

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Oct 22, 2017 at 8:34	comment	added	David Roberts♦		@Qiaochu my thesis paved the road that would lead to the land where the 2-connected cover is a(n infinite-dimensional) Lie groupoid, equivalently a kind of differentiable stack. Higher analogues of this are doable with truncated internal Kan complexes.
Oct 21, 2017 at 23:45	history	made wiki			Post Made Community Wiki by Todd Trimble
Oct 21, 2017 at 22:30	comment	added	Mike Shulman		Of course, if you don't know why to care about higher homotopy groups, then you probably don't know why to care about classifying things up to weak homotopy equivalence...
Oct 21, 2017 at 18:51	comment	added	Qiaochu Yuan		So, one major difference between this story and the story of the fundamental group is that it is much more explicitly a story about homotopy types, whereas the fundamental group is also important at the level of spaces; e.g. the covering spaces of a manifold are still manifolds and so forth. I don't know any analogue of this sort of connection to geometric structures for the higher homotopy groups; typically the $n$-connected covers of a manifold, $n \ge 2$, can't be represented by manifolds, for example.
Oct 21, 2017 at 18:46	history	edited	Qiaochu Yuan	CC BY-SA 3.0	edited body
Oct 21, 2017 at 18:31	history	edited	Qiaochu Yuan	CC BY-SA 3.0	added 276 characters in body
Oct 21, 2017 at 18:26	history	answered	Qiaochu Yuan	CC BY-SA 3.0