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Nov 29, 2019 at 16:45 comment added SashaP @AchimKrause Yes, you are absolutely right. Thanks for the correction!
Nov 29, 2019 at 16:45 history edited SashaP CC BY-SA 4.0
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Nov 29, 2019 at 15:39 comment added Ronnie Brown The use of groupoids leads to "higher homotopy groupoids" which are usually "nonabelian" but to get that needs, in my view, more structure on a space than just a base point, e.g. a filtered space, or n-cube of pointed spaces.
Nov 29, 2019 at 15:31 comment added Ronnie Brown It's an interesting fact that the fundamental and homotopy groups are dependant on having a base point, See also this discussion:mathoverflow.net/questions/40945
Nov 29, 2019 at 6:45 comment added Achim Krause They always differ for nonabelian $\pi_1$, but for an easy example just think of a perfect group.
Nov 29, 2019 at 6:45 comment added Achim Krause Unpointed homotopy classes $[S^1,Y]$ are given by the set of conjugacy classes in $\pi_1(Y)$, not $H_1(Y)$. (The former is the orbits of the conjugacy action of $\pi_1$ on itself in the category of sets, the latter in the category of groups (i.e. the abelianisation.)
Nov 28, 2019 at 22:17 vote accept Zhaoting Wei
Nov 28, 2019 at 21:51 history answered SashaP CC BY-SA 4.0