Return to Answer

Writing the 1968 edition of my book now called Topology and Groupoids (T&G) (available on amazon.com and e-version from my web site) convinced me that all of 1-dimensional homotopy theory was better expressed in terms of groupoids rather than groups, in that one obtained more powerful theorems with simpler proofs. Later results on the fundamental groupoid of orbit spaces (Chapter 11 of T&G) are more awkward to express in terms of groups; this elaborates on the point by Dustin Clausen. See further details below.

The point is that a group acting on a space $X$ acts also on the fundamental groupoid $\pi_1 X$. If $X$ is Hausdorff, the action is properly discontinuous, and $X$ has a universal cover, then the fundamental groupoiud of the orbit space $X/G$ is the orbit groupoid of $\pi_1 X$. This is the groupoid expression of Armstrong's results. See Chapter 11 of Topology and Groupoids.

Writing the 1968 edition of my book now called Topology and Groupoids (T&G) (available on amazon.com and e-version from my web site) convinced me that all of 1-dimensional homotopy theory was better expressed in terms of groupoids rather than groups, in that one obtained more powerful theorems with simpler proofs. Later results on the fundamental groupoid of orbit spaces (Chapter 11 of T&G) are more awkward to express in terms of groups; this elaborates on the point by Dustin Clausen. See further details below.

The point is that a group acting on a space $X$ acts also on the fundamental groupoid $\pi_1 X$. If $X$ is Hausdorff, the action is properly discontinuous, and $X$ has a universal cover, then the fundamental groupoiud of the orbit space $X/G$ is the orbit groupoid of $\pi_1 X$. This is the groupoid expression of Armstrong's results. See Chapter 11 of Topology and Groupoids.

I hope the following file Grothendieck of a Beamer presentation of a talk for a zoom conference on Grothendieck organised by John Alexander Cruz Morales and Colin McLarty for Aug 27-28 2020 will be helpful: it contains an extensive quote of Grothendieck's comments from my From groups to groupoids survey article and also suggestions of relations with Conway groupoids, and of uses of say thousands of base points.

I hope the following file Grothendieck of a Beamer presentation of a talk for a zoom conference on Grothendieck organised by John Alexander Cruz Morales and Colin McLarty for Aug 27-28 2020 will be helpful: it contains an extensive quote of Grothendieck's comments from my From groups to groupoids survey article and also suggestions of relations with Conway groupoids, and of uses of say thousands of base points.

However Aleksandrov and Hopf were surely right! Now we know that "groupoids in the category of groupoids" can be more complicated than groupoids, and so on in higher dimensions. The fascination with the study of homotopy groups, which are defined only for spaces with base point, seems to have been a factor in ignoring the idea of a set of base points. The possible definition of strict higher homotopy groupoids seems to need more structure on a space, and so much work has taken place on the study of non strict higher homotopy groupoids. For a part of the story of the strict case, see my 2018 Indagationes paper referred to above.

However Aleksandrov and Hopf were surely right! Now we know that "groupoids in the category of groupoids" can be more complicated than groupoids, and so on in higher dimensions. The fascination with the study of homotopy groups, which are defined only for spaces with base point, seems to have been a factor in ignoring the idea of a set of base points. The possible definition of strict higher homotopy groupoids seems to need more structure on a space, and so much work has taken place on the study of non strict higher homotopy groupoids. For a part of the story of the strict case, see my 2018 Indagationes paper referred to above.