@Fernando: @Todd: I'd just like to add to Todd's remark on the classification of groupoids up to isomorphism. It was early realised that any groupoids is the disjoint union of its connected components; and that given any $cx \in Ob(G)$ for a connected groupoid $G$ ithen $G$ s isomorphic to $G(x) * T$ where $G(x)$ is the vertex, or object group, at $x$ and $T$ is a "tree groupoid", i.e. $T(y,z)$ is a singleton for all $y,z \in Ob(G)$. However this determination depends on first choosing the object $x$ and then for each $ y \ne x$ in $Ob(G)$, choosing an element in $G(x,y)$. So there are lots of choices. As Fernando remarks, a single connected groupoid is up to homotopy "the same as" a group.

However the relation of groupoids to other areas of mathematics is interesting.

(source)

Now what the objects of a groupoid add to a group is a kind of "spatial" character. This allows between different groupoids all sorts of new possible interactions, quite unlike those of groups. This is especially relevant to van Kampen type situations. Further, the choices involved in the above determination imply that the classification of **diagrams** of groupoids does not reduce to the classification of diagrams of groups.

Further, morphisms of groupoids have much more variety than do those for groups: for groupoids we have equivalences, fibrations, covering morphisms (related to actions on sets), quotient morphisms (factor by a normal subgroupoid), universal morphisms (identify objects in some way), orbit morphisms, .... So it is often in the relations **between** groupoids rather than the classification of single groupoids that we should see the benefit of their use. This reflects the categorical viewpoint.

March 27, 2015: I discussed this matter in the 1980s with Alex Heller and he remarked:"We have long passed the days when the classification of objects up to isomorphism was the sole object of mathematics. Thus the classification of vector spaces is trivial; the classification of vector spaces with one morphism is interesting, and is the rational canonical form; the classification of vector spaces with two endomorphisms is difficult; and with three endomorphisms is unknown."

Groupoids internal to a given category are of much interest, partly because groupoids generalise equivalence relations, and so the idea of quotienting.

A discussion on mathoverflow on many base points seems relevant.

2 May, 2020 I had in the 1980's a discussion on the subject of "groupoids reduce to groups" with Alex Heller. He argued that "classification up to isomorphism" is not always the only worthwhile objective. After all classification of finite dimensional complex vector spaces is well known; but linear algebra remains in the syllabus! Further, classification of such vector spaces with an endomorphism is interesting (normal forms!), classification with two endomorphisms is hard, and with three is unknown. In the case of groupoids, maybe the problem is to formulate the interesting questions! One also needs to look at examples, perhaps such as Conway groupoids (cf Wikipedia, for example), and to study the history. My own interest was partly in the fact that a natural formulation of "higher dimensional group" yielded just "abelian groups" (Eckmann-Hilton, cf homotopy groups), while even "2-dimensional groupoids" are intriguingly complicated. What should one make of that?

pointedconnected groupoids. See the appendix in Baez and Shulman's article: arxiv.org/abs/math/0608420, p. 50 and following. $\endgroup$ – Todd Trimble♦ Feb 27 '13 at 18:19