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\cup_Z Y) = \Pi_1(X) *_{\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.

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 groups $\pi_1(X,x)$ you have to chose one base point for each connected component. In terms of fundamental groupoïds though, it's elementary: $\Pi_1(X\cup_Z Y) = \Pi_1(X) *_{\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.

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.

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\cup_Z Y) = \Pi_1(X) *_{\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.