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Groupoïds don't add much in theory but they do make some statements much simpler and they are sometimes necessary to get some intuition.

A typical example is the Van Kampen theorem. To state it in terms of fundamental groupes $\pi_1(X,x)$ you have to chose one base point for each connected component. In terms of fundamental groupoïds though, it's elementary: $\Pi_1(X\times \Pi_1(X\cup_Z Y) = \Pi_1(X) \times *_{\Pi_1(Z)} \Pi_1(Y)$.

The reason the usual statement of the theorem is equivalent to this one is that any groupoïd is equivalent to a disjoint sum of groups. But equivalent does not mean equal; isomorphic does not mean canonicaly isomorphic. A statement about groupoïds translate into a statement about groups up to conjugacy and this kind of subtlety can get very tricky (and/or interesting) in practice.

Deligne's theory of the motivic unipotent fundamental group "Le groupe fondamental de la droite projective moins trois points" gives a great illustration of this fact. It gives a thoery of groupoïds and their represenations in fibered categories. It also shows that even in the elementary case of $P^1- { 0,1,\infty }$ you have to work with fundamental groupoïds to really understand the arithmetic aspects of the fundamental groups.

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Groupoïds don't add much in theory but they do make some statements much simpler and they are sometimes necessary to get some intuition.

A typical example is the Van Kampen theorem. To state it in terms of fundamental groupes $\pi_1(X,x)$ you have to chose one base point for each connected component. In terms of fundamental groupoïds though, it's elementary: $\Pi_1(X\times Y) = \Pi_1(X) \times \Pi_1(Y)$.

The reason the usual statement of the theorem is equivalent to this one is that any groupoïd is equivalent to a disjoint sum of groups. But equivalent does not mean equal; isomorphic does not mean canonicaly isomorphic. A statement about groupoïds translate into a statement about groups up to conjugacy and this kind of subtlety can get very tricky (and/or interesting) in practice.

Deligne's theory of the motivic unipotent fundamental group "Le groupe fondamental de la droite projective moins trois points" gives a great illustration of this fact. It gives a thoery of groupoïds and their represenations in fibered categories. It also shows that even in the elementary case of $P^1- { 0,1,\infty }$ you have to work with fundamental groupoïds to really understand the arithmetic aspects of the fundamental groups.