On page 390 of Topology and Groupoids he discusses the equivalence of the category of Covering Groupoids and the category of Covering Spaces.
Could someone give me some examples on how it would be useful to translate topological problems into problems about covering groupoids? I do not immediately see the use.
There is a lovely space called the configuration space $C_n(\mathbb{R}^2)$ of $n$ unordered distinct points in $\mathbb{R}^2$ with fundamental group the braid group $B_n$. It has a lovely covering space, namely the ordered configuration space $OC_n(\mathbb{R}^2)$ of $n$ ordered distinct points in $\mathbb{R}^2$; its fundamental group is the pure braid group $P_n$, and these groups fit into a short exact sequence
$$1 \to P_n \to B_n \to S_n \to 1.$$
Now, here is a curious observation: the symmetric group $S_n$ clearly acts freely on $OC_n(\mathbb{R}^2)$ by permuting the points, and the quotient by this action is the covering map to the unordered configuration space. This covering map is in turn classified by $P_n$ as a subgroup of $B_n$, using the standard classification of covering spaces in terms of subgroups. However, the standard classification of covering spaces in terms of subgroups is not an equivalence of categories, which means it cannot address the following: the action of $S_n$ on $OC_n$ does not give rise to an action on its fundamental group, because the action has no fixed points.
However, it does act on the fundamental groupoid of $OC_n$, and so working with covering groupoids allows you to talk about this cover in a way that allows you to continue talking about the $S_n$ action on it as well.
In general, the basic reason to care about groupoids as opposed to thinking only in terms of collections of groups is that groupoids with extra structure (in this case, a group action) are generally much more interesting than collections of groups with extra structure.
This question somewhat overlaps with this stackexchnge question.
Just to tell some personal history on this, in 1967 I gave a talk to a British Math. Colloquium on the groupoid van Kampen theorem, and afterwards a person came up to me at tea and said: "That was very interesting. I have been using groupoids for years. My name is Mackey."
He told me of his work on ergodic groupoids which involved actions of groups. I gradually realised that if two people could come across this idea from two quite different directions, then there was more in this groupoid idea than had previously met my eye. Later, it could be seen that Mackey's work and that of his students was influential on the development of the area of noncommutative geometry.
I decided that it was an error to miss out on actions of groups, groupoids and covering spaces for my topology book, and in the summer of 1967 wrote the chapter giving a groupoid approach to covering spaces. I was happy with the result; later editions have developed it somewhat, but this approach has been only partly adopted elsewhere, and seen as idiosyncratic.
Writing this chapter also led to the notion of fibration of groupoids, which turned out to be useful, and appeared in later editions of the book. See also arXiv:1207.6404 for recent uses of groupoids.
For the question under discussion, my own argument is that an algebraic model of a covering map is better seen as a covering morphism rather than an action of a group, a description which also involves a choice of base point. The notion of lifting maps is then conveniently algebraically modelled by lifting morphisms.
But I insist that it is up to readers to make their own comparisons and judgements!
Nov 11, 2017 A further point is that the first main use of groupoids in the first edition was the van Kampen theorem for the fundamental groupoid on a set of base points, so yielding easily the fundamental group of the circle, and much more. It was then of interest to see how the use of groupoids can be helpful in other areas of algebraic topology, such as covering spaces, and orbit spaces. In algebraic topology we investigate algebraic models of topological situations. For further discussion, see this recent paper.
